diff --git "a/random/test.json" "b/random/test.json"
new file mode 100644--- /dev/null
+++ "b/random/test.json"
@@ -0,0 +1 @@
+[{"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Monoid.lean", "full_name": "exists_open_nhds_one_split", "start": [475, 1], "end": [479, 61], "traced_tactics": [{"tactic": "have : (fun a : M \u00d7 M => a.1 * a.2) \u207b\u00b9' s \u2208 \ud835\udcdd ((1, 1) : M \u00d7 M) :=\n  tendsto_mul (by simpa only [one_mul] using hs)", "annotated_tactic": ["have : (fun a : M \u00d7 M => a.1 * a.2) \u207b\u00b9' s \u2208 \ud835\udcdd ((1, 1) : M \u00d7 M) :=\n    <a>tendsto_mul</a> (by simpa only [<a>one_mul</a>] using hs)", [{"full_name": "tendsto_mul", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [113, 9], "def_end_pos": [113, 20]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\ns : Set M\nhs : s \u2208 \ud835\udcdd 1\n\u22a2 \u2203 V, IsOpen V \u2227 1 \u2208 V \u2227 \u2200 (v : M), v \u2208 V \u2192 \u2200 (w : M), w \u2208 V \u2192 v * w \u2208 s", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\ns : Set M\nhs : s \u2208 \ud835\udcdd 1\nthis : (fun a => a.1 * a.2) \u207b\u00b9' s \u2208 \ud835\udcdd (1, 1)\n\u22a2 \u2203 V, IsOpen V \u2227 1 \u2208 V \u2227 \u2200 (v : M), v \u2208 V \u2192 \u2200 (w : M), w \u2208 V \u2192 v * w \u2208 s"}, {"tactic": "simpa only [prod_subset_iff] using exists_nhds_square this", "annotated_tactic": ["simpa only [<a>prod_subset_iff</a>] using <a>exists_nhds_square</a> this", [{"full_name": "Set.prod_subset_iff", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [100, 9], "def_end_pos": [100, 24]}, {"full_name": "exists_nhds_square", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [674, 9], "def_end_pos": [674, 27]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\ns : Set M\nhs : s \u2208 \ud835\udcdd 1\nthis : (fun a => a.1 * a.2) \u207b\u00b9' s \u2208 \ud835\udcdd (1, 1)\n\u22a2 \u2203 V, IsOpen V \u2227 1 \u2208 V \u2227 \u2200 (v : M), v \u2208 V \u2192 \u2200 (w : M), w \u2208 V \u2192 v * w \u2208 s", "state_after": "no goals"}, {"tactic": "simpa only [one_mul] using hs", "annotated_tactic": ["simpa only [<a>one_mul</a>] using hs", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Monoid M\ninst\u271d : ContinuousMul M\ns : Set M\nhs : s \u2208 \ud835\udcdd 1\n\u22a2 s \u2208 \ud835\udcdd (1 * 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Manifold/ContMDiff.lean", "full_name": "ContMDiffAt.comp_of_eq", "start": [1039, 1], "end": [1041, 31], "traced_tactics": [{"tactic": "subst hx", "annotated_tactic": ["subst hx", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3\u2076 : NormedAddCommGroup E\ninst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b3\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3\u00b3 : TopologicalSpace M\ninst\u271d\u00b3\u00b2 : ChartedSpace H M\ninst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b3\u2070 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b2\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2\u2077 : TopologicalSpace M'\ninst\u271d\u00b2\u2076 : ChartedSpace H' M'\ninst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2\u00b9 : TopologicalSpace M''\ninst\u271d\u00b2\u2070 : ChartedSpace H'' M''\nF : Type u_11\ninst\u271d\u00b9\u2079 : NormedAddCommGroup F\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_12\ninst\u271d\u00b9\u2077 : TopologicalSpace G\nJ : ModelWithCorners \ud835\udd5c F G\nN : Type u_13\ninst\u271d\u00b9\u2076 : TopologicalSpace N\ninst\u271d\u00b9\u2075 : ChartedSpace G N\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N\nF' : Type u_14\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\nG' : Type u_15\ninst\u271d\u00b9\u00b9 : TopologicalSpace G'\nJ' : ModelWithCorners \ud835\udd5c F' G'\nN' : Type u_16\ninst\u271d\u00b9\u2070 : TopologicalSpace N'\ninst\u271d\u2079 : ChartedSpace G' N'\ninst\u271d\u2078 : SmoothManifoldWithCorners J' N'\nF\u2081 : Type u_17\ninst\u271d\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081\nF\u2082 : Type u_18\ninst\u271d\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082\nF\u2083 : Type u_19\ninst\u271d\u00b3 : NormedAddCommGroup F\u2083\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083\nF\u2084 : Type u_20\ninst\u271d\u00b9 : NormedAddCommGroup F\u2084\ninst\u271d : NormedSpace \ud835\udd5c F\u2084\ne : LocalHomeomorph M H\ne' : LocalHomeomorph M' H'\nf f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx\u271d : M\nm n : \u2115\u221e\ng : M' \u2192 M''\nx : M\ny : M'\nhg : ContMDiffAt I' I'' n g y\nhf : ContMDiffAt I I' n f x\nhx : f x = y\n\u22a2 ContMDiffAt I I'' n (g \u2218 f) x", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3\u2076 : NormedAddCommGroup E\ninst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b3\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3\u00b3 : TopologicalSpace M\ninst\u271d\u00b3\u00b2 : ChartedSpace H M\ninst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b3\u2070 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b2\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2\u2077 : TopologicalSpace M'\ninst\u271d\u00b2\u2076 : ChartedSpace H' M'\ninst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2\u00b9 : TopologicalSpace M''\ninst\u271d\u00b2\u2070 : ChartedSpace H'' M''\nF : Type u_11\ninst\u271d\u00b9\u2079 : NormedAddCommGroup F\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_12\ninst\u271d\u00b9\u2077 : TopologicalSpace G\nJ : ModelWithCorners \ud835\udd5c F G\nN : Type u_13\ninst\u271d\u00b9\u2076 : TopologicalSpace N\ninst\u271d\u00b9\u2075 : ChartedSpace G N\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N\nF' : Type u_14\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\nG' : Type u_15\ninst\u271d\u00b9\u00b9 : TopologicalSpace G'\nJ' : ModelWithCorners \ud835\udd5c F' G'\nN' : Type u_16\ninst\u271d\u00b9\u2070 : TopologicalSpace N'\ninst\u271d\u2079 : ChartedSpace G' N'\ninst\u271d\u2078 : SmoothManifoldWithCorners J' N'\nF\u2081 : Type u_17\ninst\u271d\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081\nF\u2082 : Type u_18\ninst\u271d\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082\nF\u2083 : Type u_19\ninst\u271d\u00b3 : NormedAddCommGroup F\u2083\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083\nF\u2084 : Type u_20\ninst\u271d\u00b9 : NormedAddCommGroup F\u2084\ninst\u271d : NormedSpace \ud835\udd5c F\u2084\ne : LocalHomeomorph M H\ne' : LocalHomeomorph M' H'\nf f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx\u271d : M\nm n : \u2115\u221e\ng : M' \u2192 M''\nx : M\nhf : ContMDiffAt I I' n f x\nhg : ContMDiffAt I' I'' n g (f x)\n\u22a2 ContMDiffAt I I'' n (g \u2218 f) x"}, {"tactic": "exact hg.comp x hf", "annotated_tactic": ["exact hg.comp x hf", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3\u2076 : NormedAddCommGroup E\ninst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b3\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3\u00b3 : TopologicalSpace M\ninst\u271d\u00b3\u00b2 : ChartedSpace H M\ninst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b3\u2070 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b2\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2\u2077 : TopologicalSpace M'\ninst\u271d\u00b2\u2076 : ChartedSpace H' M'\ninst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2\u00b9 : TopologicalSpace M''\ninst\u271d\u00b2\u2070 : ChartedSpace H'' M''\nF : Type u_11\ninst\u271d\u00b9\u2079 : NormedAddCommGroup F\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_12\ninst\u271d\u00b9\u2077 : TopologicalSpace G\nJ : ModelWithCorners \ud835\udd5c F G\nN : Type u_13\ninst\u271d\u00b9\u2076 : TopologicalSpace N\ninst\u271d\u00b9\u2075 : ChartedSpace G N\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N\nF' : Type u_14\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\nG' : Type u_15\ninst\u271d\u00b9\u00b9 : TopologicalSpace G'\nJ' : ModelWithCorners \ud835\udd5c F' G'\nN' : Type u_16\ninst\u271d\u00b9\u2070 : TopologicalSpace N'\ninst\u271d\u2079 : ChartedSpace G' N'\ninst\u271d\u2078 : SmoothManifoldWithCorners J' N'\nF\u2081 : Type u_17\ninst\u271d\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081\nF\u2082 : Type u_18\ninst\u271d\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082\nF\u2083 : Type u_19\ninst\u271d\u00b3 : NormedAddCommGroup F\u2083\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083\nF\u2084 : Type u_20\ninst\u271d\u00b9 : NormedAddCommGroup F\u2084\ninst\u271d : NormedSpace \ud835\udd5c F\u2084\ne : LocalHomeomorph M H\ne' : LocalHomeomorph M' H'\nf f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx\u271d : M\nm n : \u2115\u221e\ng : M' \u2192 M''\nx : M\nhf : ContMDiffAt I I' n f x\nhg : ContMDiffAt I' I'' n g (f x)\n\u22a2 ContMDiffAt I I'' n (g \u2218 f) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean", "full_name": "midpoint_vsub", "start": [133, 1], "end": [137, 44], "traced_tactics": [{"tactic": "rw [\u2190 vsub_sub_vsub_cancel_right p\u2081 p p\u2082, smul_sub, sub_eq_add_neg, \u2190 smul_neg,\n  neg_vsub_eq_vsub_rev, add_assoc, invOf_two_smul_add_invOf_two_smul, \u2190 vadd_vsub_assoc,\n  midpoint_comm, midpoint, lineMap_apply]", "annotated_tactic": ["rw [\u2190 <a>vsub_sub_vsub_cancel_right</a> p\u2081 p p\u2082, <a>smul_sub</a>, <a>sub_eq_add_neg</a>, \u2190 <a>smul_neg</a>,\n    <a>neg_vsub_eq_vsub_rev</a>, <a>add_assoc</a>, <a>invOf_two_smul_add_invOf_two_smul</a>, \u2190 <a>vadd_vsub_assoc</a>,\n    <a>midpoint_comm</a>, <a>midpoint</a>, <a>lineMap_apply</a>]", [{"full_name": "vsub_sub_vsub_cancel_right", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [174, 9], "def_end_pos": [174, 35]}, {"full_name": "smul_sub", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [988, 9], "def_end_pos": [988, 17]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "smul_neg", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [984, 9], "def_end_pos": [984, 17]}, {"full_name": "neg_vsub_eq_vsub_rev", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [156, 9], "def_end_pos": [156, 29]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "invOf_two_smul_add_invOf_two_smul", "def_path": "Mathlib/Algebra/Module/Basic.lean", "def_pos": [111, 9], "def_end_pos": [111, 42]}, {"full_name": "vadd_vsub_assoc", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [119, 9], "def_end_pos": [119, 24]}, {"full_name": "midpoint_comm", "def_path": "Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean", "def_pos": [73, 9], "def_end_pos": [73, 22]}, {"full_name": "midpoint", "def_path": "Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean", "def_pos": [43, 5], "def_end_pos": [43, 13]}, {"full_name": "AffineMap.lineMap_apply", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean", "def_pos": [514, 9], "def_end_pos": [514, 22]}]], "state_before": "R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx y z p\u2081 p\u2082 p : P\n\u22a2 midpoint R p\u2081 p\u2082 -\u1d65 p = \u215f2 \u2022 (p\u2081 -\u1d65 p) + \u215f2 \u2022 (p\u2082 -\u1d65 p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/ModelTheory/Satisfiability.lean", "full_name": "FirstOrder.Language.Theory.models_iff_finset_models", "start": [367, 1], "end": [381, 38], "traced_tactics": [{"tactic": "simp only [models_iff_not_satisfiable]", "annotated_tactic": ["simp only [<a>models_iff_not_satisfiable</a>]", [{"full_name": "FirstOrder.Language.Theory.models_iff_not_satisfiable", "def_path": "Mathlib/ModelTheory/Satisfiability.lean", "def_pos": [326, 9], "def_end_pos": [326, 35]}]], "state_before": "L : Language\nT : Theory L\n\u03b1 : Type w\nn : \u2115\n\u03c6 : Sentence L\n\u22a2 T \u22a8\u1d47 \u03c6 \u2194 \u2203 T0, \u2191T0 \u2286 T \u2227 \u2191T0 \u22a8\u1d47 \u03c6", "state_after": "L : Language\nT : Theory L\n\u03b1 : Type w\nn : \u2115\n\u03c6 : Sentence L\n\u22a2 \u00acIsSatisfiable (T \u222a {Formula.not \u03c6}) \u2194 \u2203 T0, \u2191T0 \u2286 T \u2227 \u00acIsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6})"}, {"tactic": "rw [\u2190 not_iff_not, not_not, isSatisfiable_iff_isFinitelySatisfiable, IsFinitelySatisfiable]", "annotated_tactic": ["rw [\u2190 <a>not_iff_not</a>, <a>not_not</a>, <a>isSatisfiable_iff_isFinitelySatisfiable</a>, <a>IsFinitelySatisfiable</a>]", [{"full_name": "not_iff_not", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [439, 9], "def_end_pos": [439, 20]}, {"full_name": "Classical.not_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [683, 24], "def_end_pos": [683, 31]}, {"full_name": "FirstOrder.Language.Theory.isSatisfiable_iff_isFinitelySatisfiable", "def_path": "Mathlib/ModelTheory/Satisfiability.lean", "def_pos": [107, 9], "def_end_pos": [107, 48]}, {"full_name": "FirstOrder.Language.Theory.IsFinitelySatisfiable", "def_path": "Mathlib/ModelTheory/Satisfiability.lean", "def_pos": [69, 5], "def_end_pos": [69, 26]}]], "state_before": "L : Language\nT : Theory L\n\u03b1 : Type w\nn : \u2115\n\u03c6 : Sentence L\n\u22a2 \u00acIsSatisfiable (T \u222a {Formula.not \u03c6}) \u2194 \u2203 T0, \u2191T0 \u2286 T \u2227 \u00acIsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6})", "state_after": "L : Language\nT : Theory L\n\u03b1 : Type w\nn : \u2115\n\u03c6 : Sentence L\n\u22a2 (\u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0) \u2194\n    \u00ac\u2203 T0, \u2191T0 \u2286 T \u2227 \u00acIsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6})"}, {"tactic": "push_neg", "annotated_tactic": ["push_neg", []], "state_before": "L : Language\nT : Theory L\n\u03b1 : Type w\nn : \u2115\n\u03c6 : Sentence L\n\u22a2 (\u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0) \u2194\n    \u00ac\u2203 T0, \u2191T0 \u2286 T \u2227 \u00acIsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6})", "state_after": "L : Language\nT : Theory L\n\u03b1 : Type w\nn : \u2115\n\u03c6 : Sentence L\n\u22a2 (\u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0) \u2194\n    \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u2192 IsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6})"}, {"tactic": "letI := Classical.decEq (Sentence L)", "annotated_tactic": ["letI := <a>Classical.decEq</a> (<a>Sentence</a> L)", [{"full_name": "Classical.decEq", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [983, 19], "def_end_pos": [983, 24]}, {"full_name": "FirstOrder.Language.Sentence", "def_path": "Mathlib/ModelTheory/Syntax.lean", "def_pos": [331, 5], "def_end_pos": [331, 13]}]], "state_before": "L : Language\nT : Theory L\n\u03b1 : Type w\nn : \u2115\n\u03c6 : Sentence L\n\u22a2 (\u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0) \u2194\n    \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u2192 IsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6})", "state_after": "L : Language\nT : Theory L\n\u03b1 : Type w\nn : \u2115\n\u03c6 : Sentence L\nthis : DecidableEq (Sentence L) := Classical.decEq (Sentence L)\n\u22a2 (\u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0) \u2194\n    \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u2192 IsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6})"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "L : Language\nT : Theory L\n\u03b1 : Type w\nn : \u2115\n\u03c6 : Sentence L\nthis : DecidableEq (Sentence L) := Classical.decEq (Sentence L)\n\u22a2 (\u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0) \u2194\n    \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u2192 IsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6})", "state_after": "case mp\nL : Language\nT : Theory L\n\u03b1 : Type w\nn : \u2115\n\u03c6 : Sentence L\nthis : DecidableEq (Sentence L) := Classical.decEq (Sentence L)\n\u22a2 (\u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0) \u2192\n    \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u2192 IsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6})\n\ncase mpr\nL : Language\nT : Theory L\n\u03b1 : Type w\nn : \u2115\n\u03c6 : Sentence L\nthis : DecidableEq (Sentence L) := Classical.decEq (Sentence L)\n\u22a2 (\u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u2192 IsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6})) \u2192\n    \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0"}, {"tactic": "intro h T0 hT0", "annotated_tactic": ["intro h T0 hT0", []], "state_before": "case mp\nL : Language\nT : Theory L\n\u03b1 : Type w\nn : \u2115\n\u03c6 : Sentence L\nthis : DecidableEq (Sentence L) := Classical.decEq (Sentence L)\n\u22a2 (\u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0) \u2192\n    \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u2192 IsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6})", "state_after": "case mp\nL : Language\nT : Theory L\n\u03b1 : Type w\nn : \u2115\n\u03c6 : Sentence L\nthis : DecidableEq (Sentence L) := Classical.decEq (Sentence L)\nh : \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0\nT0 : Finset (Sentence L)\nhT0 : \u2191T0 \u2286 T\n\u22a2 IsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6})"}, {"tactic": "simpa using h (T0 \u222a {Formula.not \u03c6})\n  (by\n    simp only [Finset.coe_union, Finset.coe_singleton]\n    exact Set.union_subset_union hT0 (Set.Subset.refl _))", "annotated_tactic": ["simpa using h (T0 \u222a {<a>Formula.not</a> \u03c6})\n      (by\n        simp only [<a>Finset.coe_union</a>, <a>Finset.coe_singleton</a>]\n        exact <a>Set.union_subset_union</a> hT0 (<a>Set.Subset.refl</a> _))", [{"full_name": "FirstOrder.Language.Formula.not", "def_path": "Mathlib/ModelTheory/Syntax.lean", "def_pos": [1031, 25], "def_end_pos": [1031, 28]}, {"full_name": "Finset.coe_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1399, 9], "def_end_pos": [1399, 18]}, {"full_name": "Finset.coe_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [725, 9], "def_end_pos": [725, 22]}, {"full_name": "Set.union_subset_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [846, 9], "def_end_pos": [846, 27]}, {"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [354, 9], "def_end_pos": [354, 20]}]], "state_before": "case mp\nL : Language\nT : Theory L\n\u03b1 : Type w\nn : \u2115\n\u03c6 : Sentence L\nthis : DecidableEq (Sentence L) := Classical.decEq (Sentence L)\nh : \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0\nT0 : Finset (Sentence L)\nhT0 : \u2191T0 \u2286 T\n\u22a2 IsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6})", "state_after": "no goals"}, {"tactic": "simp only [Finset.coe_union, Finset.coe_singleton]", "annotated_tactic": ["simp only [<a>Finset.coe_union</a>, <a>Finset.coe_singleton</a>]", [{"full_name": "Finset.coe_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1399, 9], "def_end_pos": [1399, 18]}, {"full_name": "Finset.coe_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [725, 9], "def_end_pos": [725, 22]}]], "state_before": "L : Language\nT : Theory L\n\u03b1 : Type w\nn : \u2115\n\u03c6 : Sentence L\nthis : DecidableEq (Sentence L) := Classical.decEq (Sentence L)\nh : \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0\nT0 : Finset (Sentence L)\nhT0 : \u2191T0 \u2286 T\n\u22a2 \u2191(T0 \u222a {Formula.not \u03c6}) \u2286 T \u222a {Formula.not \u03c6}", "state_after": "L : Language\nT : Theory L\n\u03b1 : Type w\nn : \u2115\n\u03c6 : Sentence L\nthis : DecidableEq (Sentence L) := Classical.decEq (Sentence L)\nh : \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0\nT0 : Finset (Sentence L)\nhT0 : \u2191T0 \u2286 T\n\u22a2 \u2191T0 \u222a {Formula.not \u03c6} \u2286 T \u222a {Formula.not \u03c6}"}, {"tactic": "exact Set.union_subset_union hT0 (Set.Subset.refl _)", "annotated_tactic": ["exact <a>Set.union_subset_union</a> hT0 (<a>Set.Subset.refl</a> _)", [{"full_name": "Set.union_subset_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [846, 9], "def_end_pos": [846, 27]}, {"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [354, 9], "def_end_pos": [354, 20]}]], "state_before": "L : Language\nT : Theory L\n\u03b1 : Type w\nn : \u2115\n\u03c6 : Sentence L\nthis : DecidableEq (Sentence L) := Classical.decEq (Sentence L)\nh : \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0\nT0 : Finset (Sentence L)\nhT0 : \u2191T0 \u2286 T\n\u22a2 \u2191T0 \u222a {Formula.not \u03c6} \u2286 T \u222a {Formula.not \u03c6}", "state_after": "no goals"}, {"tactic": "intro h T0 hT0", "annotated_tactic": ["intro h T0 hT0", []], "state_before": "case mpr\nL : Language\nT : Theory L\n\u03b1 : Type w\nn : \u2115\n\u03c6 : Sentence L\nthis : DecidableEq (Sentence L) := Classical.decEq (Sentence L)\n\u22a2 (\u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u2192 IsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6})) \u2192\n    \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u222a {Formula.not \u03c6} \u2192 IsSatisfiable \u2191T0", "state_after": "case mpr\nL : Language\nT : Theory L\n\u03b1 : Type w\nn : \u2115\n\u03c6 : Sentence L\nthis : DecidableEq (Sentence L) := Classical.decEq (Sentence L)\nh : \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u2192 IsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6})\nT0 : Finset (Sentence L)\nhT0 : \u2191T0 \u2286 T \u222a {Formula.not \u03c6}\n\u22a2 IsSatisfiable \u2191T0"}, {"tactic": "exact IsSatisfiable.mono (h (T0.erase (Formula.not \u03c6))\n  (by simpa using hT0)) (by simp)", "annotated_tactic": ["exact <a>IsSatisfiable.mono</a> (h (T0.erase (<a>Formula.not</a> \u03c6))\n      (by simpa using hT0)) (by simp)", [{"full_name": "FirstOrder.Language.Theory.IsSatisfiable.mono", "def_path": "Mathlib/ModelTheory/Satisfiability.lean", "def_pos": [80, 9], "def_end_pos": [80, 27]}, {"full_name": "FirstOrder.Language.Formula.not", "def_path": "Mathlib/ModelTheory/Syntax.lean", "def_pos": [1031, 25], "def_end_pos": [1031, 28]}]], "state_before": "case mpr\nL : Language\nT : Theory L\n\u03b1 : Type w\nn : \u2115\n\u03c6 : Sentence L\nthis : DecidableEq (Sentence L) := Classical.decEq (Sentence L)\nh : \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u2192 IsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6})\nT0 : Finset (Sentence L)\nhT0 : \u2191T0 \u2286 T \u222a {Formula.not \u03c6}\n\u22a2 IsSatisfiable \u2191T0", "state_after": "no goals"}, {"tactic": "simpa using hT0", "annotated_tactic": ["simpa using hT0", []], "state_before": "L : Language\nT : Theory L\n\u03b1 : Type w\nn : \u2115\n\u03c6 : Sentence L\nthis : DecidableEq (Sentence L) := Classical.decEq (Sentence L)\nh : \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u2192 IsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6})\nT0 : Finset (Sentence L)\nhT0 : \u2191T0 \u2286 T \u222a {Formula.not \u03c6}\n\u22a2 \u2191(Finset.erase T0 (Formula.not \u03c6)) \u2286 T", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "L : Language\nT : Theory L\n\u03b1 : Type w\nn : \u2115\n\u03c6 : Sentence L\nthis : DecidableEq (Sentence L) := Classical.decEq (Sentence L)\nh : \u2200 (T0 : Finset (Sentence L)), \u2191T0 \u2286 T \u2192 IsSatisfiable (\u2191T0 \u222a {Formula.not \u03c6})\nT0 : Finset (Sentence L)\nhT0 : \u2191T0 \u2286 T \u222a {Formula.not \u03c6}\n\u22a2 \u2191T0 \u2286 \u2191(Finset.erase T0 (Formula.not \u03c6)) \u222a {Formula.not \u03c6}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/Laurent.lean", "full_name": "LaurentPolynomial.isUnit_T", "start": [284, 1], "end": [285, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/PartENat.lean", "full_name": "PartENat.toWithTop_some", "start": [571, 1], "end": [572, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.coe_smul", "start": [1283, 1], "end": [1284, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Homotopy/HomotopyGroup.lean", "full_name": "GenLoop.continuous_fromLoop", "start": [240, 1], "end": [244, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Init/Data/Int/Basic.lean", "full_name": "Int.neg_eq_neg", "start": [77, 11], "end": [77, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Matrix/Block.lean", "full_name": "Matrix.blockDiagonal'_diagonal", "start": [718, 1], "end": [724, 15], "traced_tactics": [{"tactic": "ext \u27e8i, k\u27e9 \u27e8j, k'\u27e9", "annotated_tactic": ["ext \u27e8i, k\u27e9 \u27e8j, k'\u27e9", []], "state_before": "l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\np : Type u_5\nq : Type u_6\nm' : o \u2192 Type u_7\nn' : o \u2192 Type u_8\np' : o \u2192 Type u_9\nR : Type u_10\nS : Type u_11\n\u03b1 : Type u_12\n\u03b2 : Type u_13\ninst\u271d\u00b3 : DecidableEq o\ninst\u271d\u00b2 : Zero \u03b1\ninst\u271d\u00b9 : Zero \u03b2\ninst\u271d : (i : o) \u2192 DecidableEq (m' i)\nd : (i : o) \u2192 m' i \u2192 \u03b1\n\u22a2 (blockDiagonal' fun k => diagonal (d k)) = diagonal fun ik => d ik.fst ik.snd", "state_after": "case a.mk.h.mk\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\np : Type u_5\nq : Type u_6\nm' : o \u2192 Type u_7\nn' : o \u2192 Type u_8\np' : o \u2192 Type u_9\nR : Type u_10\nS : Type u_11\n\u03b1 : Type u_12\n\u03b2 : Type u_13\ninst\u271d\u00b3 : DecidableEq o\ninst\u271d\u00b2 : Zero \u03b1\ninst\u271d\u00b9 : Zero \u03b2\ninst\u271d : (i : o) \u2192 DecidableEq (m' i)\nd : (i : o) \u2192 m' i \u2192 \u03b1\ni : o\nk : m' i\nj : o\nk' : m' j\n\u22a2 blockDiagonal' (fun k => diagonal (d k)) { fst := i, snd := k } { fst := j, snd := k' } =\n    diagonal (fun ik => d ik.fst ik.snd) { fst := i, snd := k } { fst := j, snd := k' }"}, {"tactic": "simp only [blockDiagonal'_apply, diagonal]", "annotated_tactic": ["simp only [<a>blockDiagonal'_apply</a>, 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[790, 9], "def_end_pos": [790, 36]}]], "state_before": "n : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nj : Unit\n\u22a2 j = ()", "state_after": "no goals"}, {"tactic": "ext i j", "annotated_tactic": ["ext i j", []], "state_before": "case right\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\n\u22a2 toBlocks\u2082\u2081 (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M)) = 0", "state_after": "case right.a.h\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Unit\nj : Fin r\n\u22a2 toBlocks\u2082\u2081 (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M)) i j = OfNat.ofNat 0 i j"}, {"tactic": "have : i = unit := by simp only [eq_iff_true_of_subsingleton]", "annotated_tactic": ["have : i = <a>unit</a> := by simp only [<a>eq_iff_true_of_subsingleton</a>]", [{"full_name": "Unit.unit", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [135, 25], "def_end_pos": [135, 34]}, {"full_name": "eq_iff_true_of_subsingleton", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [790, 9], "def_end_pos": [790, 36]}]], "state_before": "case right.a.h\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Unit\nj : Fin r\n\u22a2 toBlocks\u2082\u2081 (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M)) i j = OfNat.ofNat 0 i j", "state_after": "case right.a.h\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Unit\nj : Fin r\nthis : i = ()\n\u22a2 toBlocks\u2082\u2081 (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M)) i j = OfNat.ofNat 0 i j"}, {"tactic": "simp [toBlocks\u2082\u2081, this, listTransvecCol_mul_mul_listTransvecRow_last_col M hM]", "annotated_tactic": ["simp [<a>toBlocks\u2082\u2081</a>, this, <a>listTransvecCol_mul_mul_listTransvecRow_last_col</a> M hM]", [{"full_name": "Matrix.toBlocks\u2082\u2081", "def_path": "Mathlib/Data/Matrix/Block.lean", "def_pos": [88, 5], "def_end_pos": [88, 15]}, {"full_name": "Matrix.Pivot.listTransvecCol_mul_mul_listTransvecRow_last_col", "def_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "def_pos": [497, 9], "def_end_pos": [497, 57]}]], "state_before": "case right.a.h\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Unit\nj : Fin r\nthis : i = ()\n\u22a2 toBlocks\u2082\u2081 (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M)) i j = OfNat.ofNat 0 i j", "state_after": "no goals"}, {"tactic": "simp only [eq_iff_true_of_subsingleton]", "annotated_tactic": ["simp only [<a>eq_iff_true_of_subsingleton</a>]", [{"full_name": "eq_iff_true_of_subsingleton", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [790, 9], "def_end_pos": [790, 36]}]], "state_before": "n : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Unit\nj : Fin r\n\u22a2 i = ()", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/Differentiation.lean", "full_name": "VitaliFamily.ae_tendsto_measure_inter_div", "start": [752, 1], "end": [770, 55], "traced_tactics": [{"tactic": "let t := toMeasurable \u03bc s", "annotated_tactic": ["let t := <a>toMeasurable</a> \u03bc s", [{"full_name": "MeasureTheory.toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [626, 17], "def_end_pos": [626, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1)"}, {"tactic": "have A :\n  \u2200\u1d50 x \u2202\u03bc.restrict s,\n    Tendsto (fun a => \u03bc (t \u2229 a) / \u03bc a) (v.filterAt x) (\ud835\udcdd (t.indicator 1 x)) := by\n  apply ae_mono restrict_le_self\n  apply ae_tendsto_measure_inter_div_of_measurableSet\n  exact measurableSet_toMeasurable _ _", "annotated_tactic": ["have A :\n    \u2200\u1d50 x \u2202\u03bc.restrict s,\n      <a>Tendsto</a> (fun a => \u03bc (t \u2229 a) / \u03bc a) (v.filterAt x) (\ud835\udcdd (t.indicator 1 x)) := by\n    apply <a>ae_mono</a> <a>restrict_le_self</a>\n    apply <a>ae_tendsto_measure_inter_div_of_measurableSet</a>\n    exact <a>measurableSet_toMeasurable</a> _ _", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "MeasureTheory.ae_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2456, 9], "def_end_pos": [2456, 16]}, {"full_name": "MeasureTheory.Measure.restrict_le_self", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 25]}, {"full_name": "VitaliFamily.ae_tendsto_measure_inter_div_of_measurableSet", "def_path": "Mathlib/MeasureTheory/Covering/Differentiation.lean", "def_pos": [739, 9], "def_end_pos": [739, 54]}, {"full_name": "MeasureTheory.measurableSet_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [645, 9], "def_end_pos": [645, 35]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\nA : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1)"}, {"tactic": "have B : \u2200\u1d50 x \u2202\u03bc.restrict s, t.indicator 1 x = (1 : \u211d\u22650\u221e) := by\n  refine' ae_restrict_of_ae_restrict_of_subset (subset_toMeasurable \u03bc s) _\n  filter_upwards [ae_restrict_mem (measurableSet_toMeasurable \u03bc s)] with _ hx\n  simp only [hx, Pi.one_apply, indicator_of_mem]", "annotated_tactic": ["have B : \u2200\u1d50 x \u2202\u03bc.restrict s, t.indicator 1 x = (1 : \u211d\u22650\u221e) := by\n    refine' <a>ae_restrict_of_ae_restrict_of_subset</a> (<a>subset_toMeasurable</a> \u03bc s) _\n    filter_upwards [<a>ae_restrict_mem</a> (<a>measurableSet_toMeasurable</a> \u03bc s)] with _ hx\n    simp only [hx, <a>Pi.one_apply</a>, <a>indicator_of_mem</a>]", [{"full_name": "MeasureTheory.ae_restrict_of_ae_restrict_of_subset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2606, 9], "def_end_pos": [2606, 45]}, {"full_name": "MeasureTheory.subset_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [633, 9], "def_end_pos": [633, 28]}, {"full_name": "MeasureTheory.ae_restrict_mem", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2586, 9], "def_end_pos": [2586, 24]}, {"full_name": "MeasureTheory.measurableSet_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [645, 9], "def_end_pos": [645, 35]}, {"full_name": "Pi.one_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [47, 9], "def_end_pos": [47, 18]}, {"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\nA : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\nA : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))\nB : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, indicator t 1 x = 1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1)"}, {"tactic": "filter_upwards [A, B] with x hx h'x", "annotated_tactic": ["filter_upwards [A, B] with x hx h'x", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\nA : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))\nB : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, indicator t 1 x = 1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\nA : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))\nB : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, indicator t 1 x = 1\nx : \u03b1\nhx : Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))\nh'x : indicator t 1 x = 1\n\u22a2 Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1)"}, {"tactic": "rw [h'x] at hx", "annotated_tactic": ["rw [h'x] at hx", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\nA : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))\nB : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, indicator t 1 x = 1\nx : \u03b1\nhx : Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))\nh'x : indicator t 1 x = 1\n\u22a2 Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1)", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\nA : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))\nB : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, indicator t 1 x = 1\nx : \u03b1\nhx : Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1)\nh'x : indicator t 1 x = 1\n\u22a2 Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1)"}, {"tactic": "apply hx.congr' _", "annotated_tactic": ["apply hx.congr' _", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\nA : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))\nB : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, indicator t 1 x = 1\nx : \u03b1\nhx : Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1)\nh'x : indicator t 1 x = 1\n\u22a2 Tendsto (fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\nA : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))\nB : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, indicator t 1 x = 1\nx : \u03b1\nhx : Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1)\nh'x : indicator t 1 x = 1\n\u22a2 (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) =\u1da0[filterAt v x] fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a"}, {"tactic": "filter_upwards [v.eventually_filterAt_measurableSet x] with _ ha", "annotated_tactic": ["filter_upwards [v.eventually_filterAt_measurableSet x] with _ ha", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\nA : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))\nB : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, indicator t 1 x = 1\nx : \u03b1\nhx : Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1)\nh'x : indicator t 1 x = 1\n\u22a2 (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) =\u1da0[filterAt v x] fun a => \u2191\u2191\u03bc (s \u2229 a) / \u2191\u2191\u03bc a", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\nA : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))\nB : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, indicator t 1 x = 1\nx : \u03b1\nhx : Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1)\nh'x : indicator t 1 x = 1\na\u271d : Set \u03b1\nha : MeasurableSet a\u271d\n\u22a2 \u2191\u2191\u03bc (t \u2229 a\u271d) / \u2191\u2191\u03bc a\u271d = \u2191\u2191\u03bc (s \u2229 a\u271d) / \u2191\u2191\u03bc a\u271d"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\nA : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))\nB : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, indicator t 1 x = 1\nx : \u03b1\nhx : Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1)\nh'x : indicator t 1 x = 1\na\u271d : Set \u03b1\nha : MeasurableSet a\u271d\n\u22a2 \u2191\u2191\u03bc (t \u2229 a\u271d) / \u2191\u2191\u03bc a\u271d = \u2191\u2191\u03bc (s \u2229 a\u271d) / \u2191\u2191\u03bc a\u271d", "state_after": "case h.e_a\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\nA : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))\nB : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, indicator t 1 x = 1\nx : \u03b1\nhx : Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1)\nh'x : indicator t 1 x = 1\na\u271d : Set \u03b1\nha : MeasurableSet a\u271d\n\u22a2 \u2191\u2191\u03bc (t \u2229 a\u271d) = \u2191\u2191\u03bc (s \u2229 a\u271d)"}, {"tactic": "exact measure_toMeasurable_inter_of_sigmaFinite ha _", "annotated_tactic": ["exact <a>measure_toMeasurable_inter_of_sigmaFinite</a> ha _", [{"full_name": "MeasureTheory.Measure.measure_toMeasurable_inter_of_sigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3608, 9], "def_end_pos": [3608, 50]}]], "state_before": "case h.e_a\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\nA : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))\nB : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, indicator t 1 x = 1\nx : \u03b1\nhx : Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd 1)\nh'x : indicator t 1 x = 1\na\u271d : Set \u03b1\nha : MeasurableSet a\u271d\n\u22a2 \u2191\u2191\u03bc (t \u2229 a\u271d) = \u2191\u2191\u03bc (s \u2229 a\u271d)", "state_after": "no goals"}, {"tactic": "apply ae_mono restrict_le_self", "annotated_tactic": ["apply <a>ae_mono</a> <a>restrict_le_self</a>", [{"full_name": "MeasureTheory.ae_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2456, 9], "def_end_pos": [2456, 16]}, {"full_name": "MeasureTheory.Measure.restrict_le_self", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))", "state_after": "case a\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\n\u22a2 {x | (fun x => Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))) x} \u2208 ae \u03bc"}, {"tactic": "apply ae_tendsto_measure_inter_div_of_measurableSet", "annotated_tactic": ["apply <a>ae_tendsto_measure_inter_div_of_measurableSet</a>", [{"full_name": "VitaliFamily.ae_tendsto_measure_inter_div_of_measurableSet", "def_path": "Mathlib/MeasureTheory/Covering/Differentiation.lean", "def_pos": [739, 9], "def_end_pos": [739, 54]}]], "state_before": "case a\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\n\u22a2 {x | (fun x => Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))) x} \u2208 ae \u03bc", "state_after": "case a.hs\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\n\u22a2 MeasurableSet t"}, {"tactic": "exact measurableSet_toMeasurable _ _", "annotated_tactic": ["exact <a>measurableSet_toMeasurable</a> _ _", [{"full_name": "MeasureTheory.measurableSet_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [645, 9], "def_end_pos": [645, 35]}]], "state_before": "case a.hs\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\n\u22a2 MeasurableSet t", "state_after": "no goals"}, {"tactic": "refine' ae_restrict_of_ae_restrict_of_subset (subset_toMeasurable \u03bc s) _", "annotated_tactic": ["refine' <a>ae_restrict_of_ae_restrict_of_subset</a> (<a>subset_toMeasurable</a> \u03bc s) _", [{"full_name": "MeasureTheory.ae_restrict_of_ae_restrict_of_subset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2606, 9], "def_end_pos": [2606, 45]}, {"full_name": "MeasureTheory.subset_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [633, 9], "def_end_pos": [633, 28]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\nA : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, indicator t 1 x = 1", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\nA : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (toMeasurable \u03bc s), indicator t 1 x = 1"}, {"tactic": "filter_upwards [ae_restrict_mem (measurableSet_toMeasurable \u03bc s)] with _ hx", "annotated_tactic": ["filter_upwards [<a>ae_restrict_mem</a> (<a>measurableSet_toMeasurable</a> \u03bc s)] with _ hx", [{"full_name": "MeasureTheory.ae_restrict_mem", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2586, 9], "def_end_pos": [2586, 24]}, {"full_name": "MeasureTheory.measurableSet_toMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [645, 9], "def_end_pos": [645, 35]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\nA : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc (toMeasurable \u03bc s), indicator t 1 x = 1", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\nA : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))\na\u271d : \u03b1\nhx : a\u271d \u2208 toMeasurable \u03bc s\n\u22a2 indicator t 1 a\u271d = 1"}, {"tactic": "simp only [hx, Pi.one_apply, indicator_of_mem]", "annotated_tactic": ["simp only [hx, <a>Pi.one_apply</a>, <a>indicator_of_mem</a>]", [{"full_name": "Pi.one_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [47, 9], "def_end_pos": [47, 18]}, {"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2075 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : SecondCountableTopology \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\n\u03c1 : Measure \u03b1\ninst\u271d : IsLocallyFiniteMeasure \u03c1\ns : Set \u03b1\nt : Set \u03b1 := toMeasurable \u03bc s\nA : \u2200\u1d50 (x : \u03b1) \u2202Measure.restrict \u03bc s, Tendsto (fun a => \u2191\u2191\u03bc (t \u2229 a) / \u2191\u2191\u03bc a) (filterAt v x) (\ud835\udcdd (indicator t 1 x))\na\u271d : \u03b1\nhx : a\u271d \u2208 toMeasurable \u03bc s\n\u22a2 indicator t 1 a\u271d = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Prod.lean", "full_name": "Filter.tendsto_fst", "start": [136, 1], "end": [137, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/NormedSpace/Extr.lean", "full_name": "IsMaxOn.norm_add_self", "start": [62, 1], "end": [63, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecificLimits/Normed.lean", "full_name": "NormedRing.summable_geometric_of_norm_lt_1", "start": [457, 1], "end": [462, 33], "traced_tactics": [{"tactic": "have h1 : Summable fun n : \u2115 \u21a6 \u2016x\u2016 ^ n := summable_geometric_of_lt_1 (norm_nonneg _) h", "annotated_tactic": ["have h1 : <a>Summable</a> fun n : \u2115 \u21a6 \u2016x\u2016 ^ n := <a>summable_geometric_of_lt_1</a> (<a>norm_nonneg</a> _) h", [{"full_name": "Summable", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [62, 5], "def_end_pos": [62, 13]}, {"full_name": "summable_geometric_of_lt_1", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [215, 9], "def_end_pos": [215, 35]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\nh : \u2016x\u2016 < 1\n\u22a2 Summable fun n => x ^ n", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\nh : \u2016x\u2016 < 1\nh1 : Summable fun n => \u2016x\u2016 ^ n\n\u22a2 Summable fun n => x ^ n"}, {"tactic": "refine' summable_of_norm_bounded_eventually _ h1 _", "annotated_tactic": ["refine' <a>summable_of_norm_bounded_eventually</a> _ h1 _", [{"full_name": "summable_of_norm_bounded_eventually", "def_path": "Mathlib/Analysis/Normed/Group/InfiniteSum.lean", "def_pos": [162, 9], "def_end_pos": [162, 44]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\nh : \u2016x\u2016 < 1\nh1 : Summable fun n => \u2016x\u2016 ^ n\n\u22a2 Summable fun n => x ^ n", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\nh : \u2016x\u2016 < 1\nh1 : Summable fun n => \u2016x\u2016 ^ n\n\u22a2 \u2200\u1da0 (i : \u2115) in cofinite, \u2016x ^ i\u2016 \u2264 \u2016x\u2016 ^ i"}, {"tactic": "rw [Nat.cofinite_eq_atTop]", "annotated_tactic": ["rw [<a>Nat.cofinite_eq_atTop</a>]", [{"full_name": "Nat.cofinite_eq_atTop", "def_path": "Mathlib/Order/Filter/Cofinite.lean", "def_pos": [156, 9], "def_end_pos": [156, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\nh : \u2016x\u2016 < 1\nh1 : Summable fun n => \u2016x\u2016 ^ n\n\u22a2 \u2200\u1da0 (i : \u2115) in cofinite, \u2016x ^ i\u2016 \u2264 \u2016x\u2016 ^ i", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\nh : \u2016x\u2016 < 1\nh1 : Summable fun n => \u2016x\u2016 ^ n\n\u22a2 \u2200\u1da0 (i : \u2115) in atTop, \u2016x ^ i\u2016 \u2264 \u2016x\u2016 ^ i"}, {"tactic": "exact eventually_norm_pow_le x", "annotated_tactic": ["exact <a>eventually_norm_pow_le</a> x", [{"full_name": "eventually_norm_pow_le", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [383, 9], "def_end_pos": [383, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\nh : \u2016x\u2016 < 1\nh1 : Summable fun n => \u2016x\u2016 ^ n\n\u22a2 \u2200\u1da0 (i : \u2115) in atTop, \u2016x ^ i\u2016 \u2264 \u2016x\u2016 ^ i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Dirac.lean", "full_name": "MeasureTheory.Measure.restrict_singleton", "start": [67, 1], "end": [73, 13], "traced_tactics": [{"tactic": "ext1 s hs", "annotated_tactic": ["ext1 s hs", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\n\u22a2 restrict \u03bc {a} = \u2191\u2191\u03bc {a} \u2022 dirac a", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s"}, {"tactic": "by_cases ha : a \u2208 s", "annotated_tactic": ["by_cases ha : a \u2208 s", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : a \u2208 s\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : \u00aca \u2208 s\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s"}, {"tactic": "have : s \u2229 {a} = {a} := by simpa", "annotated_tactic": ["have : s \u2229 {a} = {a} := by simpa", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : a \u2208 s\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : a \u2208 s\nthis : s \u2229 {a} = {a}\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s"}, {"tactic": "simp [*]", "annotated_tactic": ["simp [*]", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : a \u2208 s\nthis : s \u2229 {a} = {a}\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s", "state_after": "no goals"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : a \u2208 s\n\u22a2 s \u2229 {a} = {a}", "state_after": "no goals"}, {"tactic": "have : s \u2229 {a} = \u2205 := inter_singleton_eq_empty.2 ha", "annotated_tactic": ["have : s \u2229 {a} = \u2205 := <a>inter_singleton_eq_empty</a>.2 ha", [{"full_name": "Set.inter_singleton_eq_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1367, 9], "def_end_pos": [1367, 33]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : \u00aca \u2208 s\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : \u00aca \u2208 s\nthis : s \u2229 {a} = \u2205\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s"}, {"tactic": "simp [*]", "annotated_tactic": ["simp [*]", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type ?u.7488\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSpace \u03b2\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\na : \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nha : \u00aca \u2208 s\nthis : s \u2229 {a} = \u2205\n\u22a2 \u2191\u2191(restrict \u03bc {a}) s = \u2191\u2191(\u2191\u2191\u03bc {a} \u2022 dirac a) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Rat/Cast/CharZero.lean", "full_name": "Rat.cast_sub", "start": [59, 1], "end": [60, 100], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/Cyclotomic/Discriminant.lean", "full_name": "IsCyclotomicExtension.discr_prime_pow", "start": [144, 1], "end": [180, 46], "traced_tactics": [{"tactic": "cases' k with k k", "annotated_tactic": ["cases' k with k k", []], "state_before": "p : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhcycl : IsCyclotomicExtension {p ^ k} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ k)) K)\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (k - 1) * ((\u2191p - 1) * k - 1)))", "state_after": "case zero\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ Nat.zero} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero)\nhirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K)\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ Nat.zero) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1)))\n\ncase succ\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))"}, {"tactic": "simp only [coe_basis, _root_.pow_zero, powerBasis_gen _ h\u03b6, totient_one, mul_zero, mul_one,\n  show 1 / 2 = 0 by rfl, discr, traceMatrix]", "annotated_tactic": ["simp only [<a>coe_basis</a>, <a>_root_.pow_zero</a>, <a>powerBasis_gen</a> _ h\u03b6, <a>totient_one</a>, <a>mul_zero</a>, <a>mul_one</a>,\n      show 1 / 2 = 0 by rfl, <a>discr</a>, <a>traceMatrix</a>]", [{"full_name": "PowerBasis.coe_basis", "def_path": "Mathlib/RingTheory/PowerBasis.lean", "def_pos": [76, 9], "def_end_pos": [76, 18]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [639, 9], "def_end_pos": [639, 17]}, {"full_name": "IsPrimitiveRoot.powerBasis_gen", "def_path": "Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean", "def_pos": [122, 3], "def_end_pos": [122, 9]}, {"full_name": "Nat.totient_one", "def_path": "Mathlib/Data/Nat/Totient.lean", "def_pos": [45, 9], "def_end_pos": [45, 20]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "Algebra.discr", "def_path": "Mathlib/RingTheory/Discriminant.lean", "def_pos": [69, 19], "def_end_pos": [69, 24]}, {"full_name": "Algebra.traceMatrix", "def_path": "Mathlib/RingTheory/Trace.lean", "def_pos": [427, 19], "def_end_pos": [427, 30]}]], "state_before": "case zero\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ Nat.zero} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero)\nhirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K)\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ Nat.zero) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1)))", "state_after": "case zero\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ Nat.zero} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero)\nhirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K)\n\u22a2 Matrix.det (\u2191Matrix.of fun i j => BilinForm.bilin (traceForm K L) (\u03b6 ^ \u2191i) (\u03b6 ^ \u2191j)) =\n    \u2191((-1) ^ (\u03c6 \u21911 / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1)))"}, {"tactic": "have h\u03b6one : \u03b6 = 1 := by simpa using h\u03b6", "annotated_tactic": ["have h\u03b6one : \u03b6 = 1 := by simpa using h\u03b6", []], "state_before": "case zero\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ Nat.zero} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero)\nhirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K)\n\u22a2 Matrix.det (\u2191Matrix.of fun i j => BilinForm.bilin (traceForm K L) (\u03b6 ^ \u2191i) (\u03b6 ^ \u2191j)) =\n    \u2191((-1) ^ (\u03c6 \u21911 / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1)))", "state_after": "case zero\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ Nat.zero} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero)\nhirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K)\nh\u03b6one : \u03b6 = 1\n\u22a2 Matrix.det (\u2191Matrix.of fun i j => BilinForm.bilin (traceForm K L) (\u03b6 ^ \u2191i) (\u03b6 ^ \u2191j)) =\n    \u2191((-1) ^ (\u03c6 \u21911 / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1)))"}, {"tactic": "rw [h\u03b6.powerBasis_dim _, h\u03b6one, \u2190 (algebraMap K L).map_one,\n  minpoly.eq_X_sub_C_of_algebraMap_inj _ (algebraMap K L).injective, natDegree_X_sub_C]", "annotated_tactic": ["rw [h\u03b6.powerBasis_dim _, h\u03b6one, \u2190 (<a>algebraMap</a> K L).<a>map_one</a>,\n      <a>minpoly.eq_X_sub_C_of_algebraMap_inj</a> _ (<a>algebraMap</a> K L).<a>injective</a>, <a>natDegree_X_sub_C</a>]", [{"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "RingHom.map_one", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [559, 19], "def_end_pos": [559, 26]}, {"full_name": "minpoly.eq_X_sub_C_of_algebraMap_inj", "def_path": "Mathlib/FieldTheory/Minpoly/Basic.lean", "def_pos": [206, 9], "def_end_pos": [206, 37]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "RingHom.injective", "def_path": "Mathlib/Algebra/Field/Basic.lean", "def_pos": [255, 19], "def_end_pos": [255, 28]}, {"full_name": "Polynomial.natDegree_X_sub_C", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [1556, 9], "def_end_pos": [1556, 26]}]], "state_before": "case zero\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ Nat.zero} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero)\nhirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K)\nh\u03b6one : \u03b6 = 1\n\u22a2 Matrix.det (\u2191Matrix.of fun i j => BilinForm.bilin (traceForm K L) (\u03b6 ^ \u2191i) (\u03b6 ^ \u2191j)) =\n    \u2191((-1) ^ (\u03c6 \u21911 / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1)))", "state_after": "case zero\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ Nat.zero} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero)\nhirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K)\nh\u03b6one : \u03b6 = 1\n\u22a2 Matrix.det\n      (\u2191Matrix.of fun i j => BilinForm.bilin (traceForm K L) (\u2191(algebraMap K L) 1 ^ \u2191i) (\u2191(algebraMap K L) 1 ^ \u2191j)) =\n    \u2191((-1) ^ (\u03c6 \u21911 / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1)))"}, {"tactic": "simp only [traceMatrix, map_one, one_pow, Matrix.det_unique, traceForm_apply, mul_one]", "annotated_tactic": ["simp only [<a>traceMatrix</a>, <a>map_one</a>, <a>one_pow</a>, <a>Matrix.det_unique</a>, <a>traceForm_apply</a>, <a>mul_one</a>]", [{"full_name": "Algebra.traceMatrix", "def_path": "Mathlib/RingTheory/Trace.lean", "def_pos": [427, 19], "def_end_pos": [427, 30]}, {"full_name": "map_one", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [202, 9], "def_end_pos": [202, 16]}, {"full_name": "one_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [90, 9], "def_end_pos": [90, 16]}, {"full_name": "Matrix.det_unique", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [118, 9], "def_end_pos": [118, 19]}, {"full_name": "Algebra.traceForm_apply", "def_path": "Mathlib/RingTheory/Trace.lean", "def_pos": [205, 9], "def_end_pos": [205, 24]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "case zero\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ Nat.zero} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero)\nhirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K)\nh\u03b6one : \u03b6 = 1\n\u22a2 Matrix.det\n      (\u2191Matrix.of fun i j => BilinForm.bilin (traceForm K L) (\u2191(algebraMap K L) 1 ^ \u2191i) (\u2191(algebraMap K L) 1 ^ \u2191j)) =\n    \u2191((-1) ^ (\u03c6 \u21911 / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1)))", "state_after": "case zero\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ Nat.zero} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero)\nhirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K)\nh\u03b6one : \u03b6 = 1\n\u22a2 \u2191Matrix.of (fun i j => \u2191(trace K L) 1) default default =\n    \u2191((-1) ^ (\u03c6 \u21911 / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1)))"}, {"tactic": "rw [\u2190 (algebraMap K L).map_one, trace_algebraMap, finrank _ hirr]", "annotated_tactic": ["rw [\u2190 (<a>algebraMap</a> K L).<a>map_one</a>, <a>trace_algebraMap</a>, <a>finrank</a> _ hirr]", [{"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "RingHom.map_one", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [559, 19], "def_end_pos": [559, 26]}, {"full_name": "Algebra.trace_algebraMap", "def_path": "Mathlib/RingTheory/Trace.lean", "def_pos": [131, 9], "def_end_pos": [131, 25]}, {"full_name": "IsCyclotomicExtension.finrank", "def_path": "Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean", "def_pos": [186, 9], "def_end_pos": [186, 16]}]], "state_before": "case zero\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ Nat.zero} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero)\nhirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K)\nh\u03b6one : \u03b6 = 1\n\u22a2 \u2191Matrix.of (fun i j => \u2191(trace K L) 1) default default =\n    \u2191((-1) ^ (\u03c6 \u21911 / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1)))", "state_after": "case zero\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ Nat.zero} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero)\nhirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K)\nh\u03b6one : \u03b6 = 1\n\u22a2 \u2191Matrix.of (fun i j => \u03c6 \u2191(p ^ Nat.zero) \u2022 1) default default =\n    \u2191((-1) ^ (\u03c6 \u21911 / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1)))"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "case zero\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ Nat.zero} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero)\nhirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K)\nh\u03b6one : \u03b6 = 1\n\u22a2 \u2191Matrix.of (fun i j => \u03c6 \u2191(p ^ Nat.zero) \u2022 1) default default =\n    \u2191((-1) ^ (\u03c6 \u21911 / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1)))", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "p : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ Nat.zero} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero)\nhirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K)\n\u22a2 1 / 2 = 0", "state_after": "no goals"}, {"tactic": "simpa using h\u03b6", "annotated_tactic": ["simpa using h\u03b6", []], "state_before": "p : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ Nat.zero} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero)\nhirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K)\n\u22a2 \u03b6 = 1", "state_after": "no goals"}, {"tactic": "by_cases hk : p ^ (k + 1) = 2", "annotated_tactic": ["by_cases hk : p ^ (k + 1) = 2", []], "state_before": "case succ\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))", "state_after": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk : p ^ (k + 1) = 2\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))\n\ncase neg\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk : \u00acp ^ (k + 1) = 2\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))"}, {"tactic": "have coe_two : 2 = ((2 : \u2115+) : \u2115) := rfl", "annotated_tactic": ["have coe_two : 2 = ((2 : \u2115+) : \u2115) := <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk : p ^ (k + 1) = 2\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))", "state_after": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk : p ^ (k + 1) = 2\ncoe_two : 2 = \u21912\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))"}, {"tactic": "have hp : p = 2 := by\n  rw [\u2190 PNat.coe_inj, PNat.pow_coe, \u2190 pow_one 2] at hk\n  replace hk :=\n    eq_of_prime_pow_eq (prime_iff.1 hp.out) (prime_iff.1 Nat.prime_two) (succ_pos _) hk\n  rwa [coe_two, PNat.coe_inj] at hk", "annotated_tactic": ["have hp : p = 2 := by\n        rw [\u2190 <a>PNat.coe_inj</a>, <a>PNat.pow_coe</a>, \u2190 <a>pow_one</a> 2] at hk\n        replace hk :=\n          <a>eq_of_prime_pow_eq</a> (<a>prime_iff</a>.1 hp.out) (<a>prime_iff</a>.1 <a>Nat.prime_two</a>) (<a>succ_pos</a> _) hk\n        rwa [coe_two, <a>PNat.coe_inj</a>] at hk", [{"full_name": "PNat.coe_inj", "def_path": "Mathlib/Data/PNat/Basic.lean", "def_pos": [116, 9], "def_end_pos": [116, 16]}, {"full_name": "PNat.pow_coe", "def_path": "Mathlib/Data/PNat/Basic.lean", "def_pos": [275, 9], "def_end_pos": [275, 16]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}, {"full_name": "eq_of_prime_pow_eq", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [736, 9], "def_end_pos": [736, 27]}, {"full_name": "Nat.prime_iff", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [570, 9], "def_end_pos": [570, 18]}, {"full_name": "Nat.prime_iff", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [570, 9], "def_end_pos": [570, 18]}, {"full_name": "Nat.prime_two", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [169, 9], "def_end_pos": [169, 18]}, {"full_name": "Nat.succ_pos", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1608, 9], "def_end_pos": [1608, 21]}, {"full_name": "PNat.coe_inj", "def_path": "Mathlib/Data/PNat/Basic.lean", "def_pos": [116, 9], "def_end_pos": [116, 16]}]], "state_before": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk : p ^ (k + 1) = 2\ncoe_two : 2 = \u21912\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))", "state_after": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk : p ^ (k + 1) = 2\ncoe_two : 2 = \u21912\nhp : p = 2\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))"}, {"tactic": "rw [hp, \u2190 PNat.coe_inj, PNat.pow_coe] at hk", "annotated_tactic": ["rw [hp, \u2190 <a>PNat.coe_inj</a>, <a>PNat.pow_coe</a>] at hk", [{"full_name": "PNat.coe_inj", "def_path": "Mathlib/Data/PNat/Basic.lean", "def_pos": [116, 9], "def_end_pos": [116, 16]}, {"full_name": "PNat.pow_coe", "def_path": "Mathlib/Data/PNat/Basic.lean", "def_pos": [275, 9], "def_end_pos": [275, 16]}]], "state_before": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk : p ^ (k + 1) = 2\ncoe_two : 2 = \u21912\nhp : p = 2\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))", "state_after": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))"}, {"tactic": "nth_rw 2 [\u2190 pow_one 2] at hk", "annotated_tactic": ["nth_rw 2 [\u2190 <a>pow_one</a> 2] at hk", [{"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}]], "state_before": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))", "state_after": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\nhk : \u21912 ^ (k + 1) = \u2191(2 ^ 1)\ncoe_two : 2 = \u21912\nhp : p = 2\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))"}, {"tactic": "replace hk := Nat.pow_right_injective rfl.le hk", "annotated_tactic": ["replace hk := <a>Nat.pow_right_injective</a> rfl.le hk", [{"full_name": "Nat.pow_right_injective", "def_path": "Mathlib/Data/Nat/Pow.lean", "def_pos": [113, 9], "def_end_pos": [113, 28]}]], "state_before": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\nhk : \u21912 ^ (k + 1) = \u2191(2 ^ 1)\ncoe_two : 2 = \u21912\nhp : p = 2\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))", "state_after": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k + 1 = 1\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))"}, {"tactic": "rw [add_left_eq_self] at hk", "annotated_tactic": ["rw [<a>add_left_eq_self</a>] at hk", [{"full_name": "add_left_eq_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [209, 3], "def_end_pos": [209, 14]}]], "state_before": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k + 1 = 1\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))", "state_after": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))"}, {"tactic": "rw [hp, hk] at h\u03b6", "annotated_tactic": ["rw [hp, hk] at h\u03b6", []], "state_before": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))", "state_after": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(2 ^ succ 0)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6\u271d).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))"}, {"tactic": "norm_num at h\u03b6", "annotated_tactic": ["norm_num at h\u03b6", []], "state_before": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(2 ^ succ 0)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6\u271d).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))", "state_after": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 \u21912\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6\u271d).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))"}, {"tactic": "rw [\u2190 coe_two] at h\u03b6", "annotated_tactic": ["rw [\u2190 coe_two] at h\u03b6", []], "state_before": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 \u21912\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6\u271d).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))", "state_after": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6\u271d).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))"}, {"tactic": "rw [coe_basis, powerBasis_gen]", "annotated_tactic": ["rw [<a>coe_basis</a>, <a>powerBasis_gen</a>]", [{"full_name": "PowerBasis.coe_basis", "def_path": "Mathlib/RingTheory/PowerBasis.lean", "def_pos": [76, 9], "def_end_pos": [76, 18]}, {"full_name": "IsPrimitiveRoot.powerBasis_gen", "def_path": "Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean", "def_pos": [122, 3], "def_end_pos": [122, 9]}]], "state_before": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6\u271d).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))", "state_after": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 (discr K fun i => \u03b6 ^ \u2191i) = \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))"}, {"tactic": "simp only [hp, hk]", "annotated_tactic": ["simp only [hp, hk]", []], "state_before": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 (discr K fun i => \u03b6 ^ \u2191i) = \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))", "state_after": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 (discr K fun i => \u03b6 ^ \u2191i) = \u2191((-1) ^ (\u03c6 \u2191(2 ^ succ 0) / 2)) * \u2191\u2191(2 ^ (\u21912 ^ (succ 0 - 1) * ((\u21912 - 1) * succ 0 - 1)))"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 (discr K fun i => \u03b6 ^ \u2191i) = \u2191((-1) ^ (\u03c6 \u2191(2 ^ succ 0) / 2)) * \u2191\u2191(2 ^ (\u21912 ^ (succ 0 - 1) * ((\u21912 - 1) * succ 0 - 1)))", "state_after": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 (discr K fun i => \u03b6 ^ \u2191i) = 1"}, {"tactic": "simp_rw [h\u03b6.eq_neg_one_of_two_right, show (-1 : L) = algebraMap K L (-1) by simp]", "annotated_tactic": ["simp_rw [h\u03b6.eq_neg_one_of_two_right, show (-1 : L) = <a>algebraMap</a> K L (-1) by simp]", [{"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}]], "state_before": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 (discr K fun i => \u03b6 ^ \u2191i) = 1", "state_after": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 (discr K fun i => \u2191(algebraMap K L) (-1) ^ \u2191i) = 1"}, {"tactic": "simp only [discr, traceMatrix_apply, Matrix.det_unique, Fin.default_eq_zero, Fin.val_zero,\n  _root_.pow_zero, traceForm_apply, mul_one]", "annotated_tactic": ["simp only [<a>discr</a>, <a>traceMatrix_apply</a>, <a>Matrix.det_unique</a>, <a>Fin.default_eq_zero</a>, <a>Fin.val_zero</a>,\n        <a>_root_.pow_zero</a>, <a>traceForm_apply</a>, <a>mul_one</a>]", [{"full_name": "Algebra.discr", "def_path": "Mathlib/RingTheory/Discriminant.lean", "def_pos": [69, 19], "def_end_pos": [69, 24]}, {"full_name": "Algebra.traceMatrix_apply", "def_path": "Mathlib/RingTheory/Trace.lean", "def_pos": [433, 9], "def_end_pos": [433, 26]}, {"full_name": "Matrix.det_unique", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [118, 9], "def_end_pos": [118, 19]}, {"full_name": "Fin.default_eq_zero", "def_path": "Mathlib/Logic/Unique.lean", "def_pos": [124, 9], "def_end_pos": [124, 28]}, {"full_name": "Fin.val_zero", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [105, 17], "def_end_pos": [105, 25]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [639, 9], "def_end_pos": [639, 17]}, {"full_name": "Algebra.traceForm_apply", "def_path": "Mathlib/RingTheory/Trace.lean", "def_pos": [205, 9], "def_end_pos": [205, 24]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 (discr K fun i => \u2191(algebraMap K L) (-1) ^ \u2191i) = 1", "state_after": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 \u2191(trace K L) 1 = 1"}, {"tactic": "rw [\u2190 (algebraMap K L).map_one, trace_algebraMap, finrank _ hirr, hp, hk]", "annotated_tactic": ["rw [\u2190 (<a>algebraMap</a> K L).<a>map_one</a>, <a>trace_algebraMap</a>, <a>finrank</a> _ hirr, hp, hk]", [{"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "RingHom.map_one", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [559, 19], "def_end_pos": [559, 26]}, {"full_name": "Algebra.trace_algebraMap", "def_path": "Mathlib/RingTheory/Trace.lean", "def_pos": [131, 9], "def_end_pos": [131, 25]}, {"full_name": "IsCyclotomicExtension.finrank", "def_path": "Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean", "def_pos": [186, 9], "def_end_pos": [186, 16]}]], "state_before": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 \u2191(trace K L) 1 = 1", "state_after": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 \u03c6 \u2191(2 ^ succ 0) \u2022 1 = 1"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 \u03c6 \u2191(2 ^ succ 0) \u2022 1 = 1", "state_after": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 \u2191(\u03c6 \u21912) = 1"}, {"tactic": "simp [\u2190 coe_two]", "annotated_tactic": ["simp [\u2190 coe_two]", []], "state_before": "case pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 \u2191(\u03c6 \u21912) = 1", "state_after": "no goals"}, {"tactic": "rw [\u2190 PNat.coe_inj, PNat.pow_coe, \u2190 pow_one 2] at hk", "annotated_tactic": ["rw [\u2190 <a>PNat.coe_inj</a>, <a>PNat.pow_coe</a>, \u2190 <a>pow_one</a> 2] at hk", [{"full_name": "PNat.coe_inj", "def_path": "Mathlib/Data/PNat/Basic.lean", "def_pos": [116, 9], "def_end_pos": [116, 16]}, {"full_name": "PNat.pow_coe", "def_path": "Mathlib/Data/PNat/Basic.lean", "def_pos": [275, 9], "def_end_pos": [275, 16]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}]], "state_before": "p : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk : p ^ (k + 1) = 2\ncoe_two : 2 = \u21912\n\u22a2 p = 2", "state_after": "p : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u2191p ^ (k + 1) = \u21912\nhk : \u2191p ^ (k + 1) = \u2191(2 ^ 1)\ncoe_two : 2 = \u21912\n\u22a2 p = 2"}, {"tactic": "replace hk :=\n  eq_of_prime_pow_eq (prime_iff.1 hp.out) (prime_iff.1 Nat.prime_two) (succ_pos _) hk", "annotated_tactic": ["replace hk :=\n          <a>eq_of_prime_pow_eq</a> (<a>prime_iff</a>.1 hp.out) (<a>prime_iff</a>.1 <a>Nat.prime_two</a>) (<a>succ_pos</a> _) hk", [{"full_name": "eq_of_prime_pow_eq", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [736, 9], "def_end_pos": [736, 27]}, {"full_name": "Nat.prime_iff", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [570, 9], "def_end_pos": [570, 18]}, {"full_name": "Nat.prime_iff", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [570, 9], "def_end_pos": [570, 18]}, {"full_name": "Nat.prime_two", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [169, 9], "def_end_pos": [169, 18]}, {"full_name": "Nat.succ_pos", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1608, 9], "def_end_pos": [1608, 21]}]], "state_before": "p : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u2191p ^ (k + 1) = \u21912\nhk : \u2191p ^ (k + 1) = \u2191(2 ^ 1)\ncoe_two : 2 = \u21912\n\u22a2 p = 2", "state_after": "p : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u2191p ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhk : \u2191p = 2\n\u22a2 p = 2"}, {"tactic": "rwa [coe_two, PNat.coe_inj] at hk", "annotated_tactic": ["rwa [coe_two, <a>PNat.coe_inj</a>] at hk", [{"full_name": "PNat.coe_inj", "def_path": "Mathlib/Data/PNat/Basic.lean", "def_pos": [116, 9], "def_end_pos": [116, 16]}]], "state_before": "p : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u2191p ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhk : \u2191p = 2\n\u22a2 p = 2", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "p : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 -1 = \u2191(algebraMap K L) (-1)", "state_after": "no goals"}, {"tactic": "exact discr_prime_pow_ne_two h\u03b6 hirr hk", "annotated_tactic": ["exact <a>discr_prime_pow_ne_two</a> h\u03b6 hirr hk", [{"full_name": "IsCyclotomicExtension.discr_prime_pow_ne_two", "def_path": "Mathlib/NumberTheory/Cyclotomic/Discriminant.lean", "def_pos": [65, 9], "def_end_pos": [65, 31]}]], "state_before": "case neg\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk : \u00acp ^ (k + 1) = 2\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Constructions.lean", "full_name": "isOpen_pi_iff", "start": [1334, 1], "end": [1358, 81], "traced_tactics": [{"tactic": "rw [isOpen_iff_nhds]", "annotated_tactic": ["rw [<a>isOpen_iff_nhds</a>]", [{"full_name": "isOpen_iff_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1247, 9], "def_end_pos": [1247, 24]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\n\u22a2 IsOpen s \u2194 \u2200 (f : (a : \u03b9) \u2192 \u03c0 a), f \u2208 s \u2192 \u2203 I u, (\u2200 (a : \u03b9), a \u2208 I \u2192 IsOpen (u a) \u2227 f a \u2208 u a) \u2227 Set.pi (\u2191I) u \u2286 s", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\n\u22a2 (\u2200 (a : (a : \u03b9) \u2192 \u03c0 a), a \u2208 s \u2192 \ud835\udcdd a \u2264 \ud835\udcdf s) \u2194\n    \u2200 (f : (a : \u03b9) \u2192 \u03c0 a), f \u2208 s \u2192 \u2203 I u, (\u2200 (a : \u03b9), a \u2208 I \u2192 IsOpen (u a) \u2227 f a \u2208 u a) \u2227 Set.pi (\u2191I) u \u2286 s"}, {"tactic": "simp_rw [le_principal_iff, nhds_pi, Filter.mem_pi', mem_nhds_iff]", "annotated_tactic": ["simp_rw [<a>le_principal_iff</a>, <a>nhds_pi</a>, <a>Filter.mem_pi'</a>, <a>mem_nhds_iff</a>]", [{"full_name": "Filter.le_principal_iff", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [655, 9], "def_end_pos": [655, 25]}, {"full_name": "nhds_pi", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1227, 9], "def_end_pos": [1227, 16]}, {"full_name": "Filter.mem_pi'", "def_path": "Mathlib/Order/Filter/Pi.lean", "def_pos": [84, 9], "def_end_pos": [84, 16]}, {"full_name": "mem_nhds_iff", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [898, 9], "def_end_pos": [898, 21]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\n\u22a2 (\u2200 (a : (a : \u03b9) \u2192 \u03c0 a), a \u2208 s \u2192 \ud835\udcdd a \u2264 \ud835\udcdf s) \u2194\n    \u2200 (f : (a : \u03b9) \u2192 \u03c0 a), f \u2208 s \u2192 \u2203 I u, (\u2200 (a : \u03b9), a \u2208 I \u2192 IsOpen (u a) \u2227 f a \u2208 u a) \u2227 Set.pi (\u2191I) u \u2286 s", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\n\u22a2 (\u2200 (a : (a : \u03b9) \u2192 \u03c0 a), a \u2208 s \u2192 \u2203 I t, (\u2200 (i : \u03b9), \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1) \u2227 Set.pi (\u2191I) t \u2286 s) \u2194\n    \u2200 (f : (a : \u03b9) \u2192 \u03c0 a), f \u2208 s \u2192 \u2203 I u, (\u2200 (a : \u03b9), a \u2208 I \u2192 IsOpen (u a) \u2227 f a \u2208 u a) \u2227 Set.pi (\u2191I) u \u2286 s"}, {"tactic": "refine ball_congr fun a _ => \u27e8?_, ?_\u27e9", "annotated_tactic": ["refine <a>ball_congr</a> fun a _ => \u27e8?_, ?_\u27e9", [{"full_name": "ball_congr", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1039, 9], "def_end_pos": [1039, 19]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\n\u22a2 (\u2200 (a : (a : \u03b9) \u2192 \u03c0 a), a \u2208 s \u2192 \u2203 I t, (\u2200 (i : \u03b9), \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1) \u2227 Set.pi (\u2191I) t \u2286 s) \u2194\n    \u2200 (f : (a : \u03b9) \u2192 \u03c0 a), f \u2208 s \u2192 \u2203 I u, (\u2200 (a : \u03b9), a \u2208 I \u2192 IsOpen (u a) \u2227 f a \u2208 u a) \u2227 Set.pi (\u2191I) u \u2286 s", "state_after": "case refine_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\n\u22a2 (\u2203 I t, (\u2200 (i : \u03b9), \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1) \u2227 Set.pi (\u2191I) t \u2286 s) \u2192\n    \u2203 I u, (\u2200 (a_2 : \u03b9), a_2 \u2208 I \u2192 IsOpen (u a_2) \u2227 a a_2 \u2208 u a_2) \u2227 Set.pi (\u2191I) u \u2286 s\n\ncase refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\n\u22a2 (\u2203 I u, (\u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (u a_1) \u2227 a a_1 \u2208 u a_1) \u2227 Set.pi (\u2191I) u \u2286 s) \u2192\n    \u2203 I t, (\u2200 (i : \u03b9), \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1) \u2227 Set.pi (\u2191I) t \u2286 s"}, {"tactic": "rintro \u27e8I, t, \u27e8h1, h2\u27e9\u27e9", "annotated_tactic": ["rintro \u27e8I, t, \u27e8h1, h2\u27e9\u27e9", []], "state_before": "case refine_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\n\u22a2 (\u2203 I t, (\u2200 (i : \u03b9), \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1) \u2227 Set.pi (\u2191I) t \u2286 s) \u2192\n    \u2203 I u, (\u2200 (a_2 : \u03b9), a_2 \u2208 I \u2192 IsOpen (u a_2) \u2227 a a_2 \u2208 u a_2) \u2227 Set.pi (\u2191I) u \u2286 s", "state_after": "case refine_1.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (i : \u03b9) \u2192 Set (\u03c0 i)\nh1 : \u2200 (i : \u03b9), \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1\nh2 : Set.pi (\u2191I) t \u2286 s\n\u22a2 \u2203 I u, (\u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (u a_1) \u2227 a a_1 \u2208 u a_1) \u2227 Set.pi (\u2191I) u \u2286 s"}, {"tactic": "refine \u27e8I, fun a => eval a '' (I : Set \u03b9).pi fun a => (h1 a).choose, fun i hi => ?_, ?_\u27e9", "annotated_tactic": ["refine \u27e8I, fun a => <a>eval</a> a '' (I : <a>Set</a> \u03b9).<a>pi</a> fun a => (h1 a).<a>choose</a>, fun i hi => ?_, ?_\u27e9", [{"full_name": "Function.eval", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [29, 24], "def_end_pos": [29, 28]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set.pi", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [665, 5], "def_end_pos": [665, 7]}, {"full_name": "Exists.choose", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [442, 32], "def_end_pos": [442, 45]}]], "state_before": "case refine_1.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (i : \u03b9) \u2192 Set (\u03c0 i)\nh1 : \u2200 (i : \u03b9), \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1\nh2 : Set.pi (\u2191I) t \u2286 s\n\u22a2 \u2203 I u, (\u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (u a_1) \u2227 a a_1 \u2208 u a_1) \u2227 Set.pi (\u2191I) u \u2286 s", "state_after": "case refine_1.intro.intro.intro.refine_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (i : \u03b9) \u2192 Set (\u03c0 i)\nh1 : \u2200 (i : \u03b9), \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1\nh2 : Set.pi (\u2191I) t \u2286 s\ni : \u03b9\nhi : i \u2208 I\n\u22a2 IsOpen\n      ((fun a_1 => eval a_1 '' Set.pi \u2191I fun a_2 => Exists.choose (_ : \u2203 t_1, t_1 \u2286 t a_2 \u2227 IsOpen t_1 \u2227 a a_2 \u2208 t_1))\n        i) \u2227\n    a i \u2208\n      (fun a_1 => eval a_1 '' Set.pi \u2191I fun a_2 => Exists.choose (_ : \u2203 t_1, t_1 \u2286 t a_2 \u2227 IsOpen t_1 \u2227 a a_2 \u2208 t_1)) i\n\ncase refine_1.intro.intro.intro.refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (i : \u03b9) \u2192 Set (\u03c0 i)\nh1 : \u2200 (i : \u03b9), \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1\nh2 : Set.pi (\u2191I) t \u2286 s\n\u22a2 (Set.pi \u2191I fun a_1 =>\n      eval a_1 '' Set.pi \u2191I fun a_2 => Exists.choose (_ : \u2203 t_1, t_1 \u2286 t a_2 \u2227 IsOpen t_1 \u2227 a a_2 \u2208 t_1)) \u2286\n    s"}, {"tactic": "simp_rw [Set.eval_image_pi (Finset.mem_coe.mpr hi)\n    (pi_nonempty_iff.mpr fun i => \u27e8_, fun _ => (h1 i).choose_spec.2.2\u27e9)]", "annotated_tactic": ["simp_rw [<a>Set.eval_image_pi</a> (Finset.mem_coe.mpr hi)\n          (pi_nonempty_iff.mpr fun i => \u27e8_, fun _ => (h1 i).<a>choose_spec</a>.2.2\u27e9)]", [{"full_name": "Set.eval_image_pi", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [867, 9], "def_end_pos": [867, 22]}, {"full_name": "Exists.choose_spec", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [445, 9], "def_end_pos": [445, 27]}]], "state_before": "case refine_1.intro.intro.intro.refine_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (i : \u03b9) \u2192 Set (\u03c0 i)\nh1 : \u2200 (i : \u03b9), \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1\nh2 : Set.pi (\u2191I) t \u2286 s\ni : \u03b9\nhi : i \u2208 I\n\u22a2 IsOpen\n      ((fun a_1 => eval a_1 '' Set.pi \u2191I fun a_2 => Exists.choose (_ : \u2203 t_1, t_1 \u2286 t a_2 \u2227 IsOpen t_1 \u2227 a a_2 \u2208 t_1))\n        i) \u2227\n    a i \u2208\n      (fun a_1 => eval a_1 '' Set.pi \u2191I fun a_2 => Exists.choose (_ : \u2203 t_1, t_1 \u2286 t a_2 \u2227 IsOpen t_1 \u2227 a a_2 \u2208 t_1)) i", "state_after": "case refine_1.intro.intro.intro.refine_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (i : \u03b9) \u2192 Set (\u03c0 i)\nh1 : \u2200 (i : \u03b9), \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1\nh2 : Set.pi (\u2191I) t \u2286 s\ni : \u03b9\nhi : i \u2208 I\n\u22a2 IsOpen (Exists.choose (_ : \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1)) \u2227\n    a i \u2208 Exists.choose (_ : \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1)"}, {"tactic": "exact (h1 i).choose_spec.2", "annotated_tactic": ["exact (h1 i).<a>choose_spec</a>.2", [{"full_name": "Exists.choose_spec", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [445, 9], "def_end_pos": [445, 27]}]], "state_before": "case refine_1.intro.intro.intro.refine_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (i : \u03b9) \u2192 Set (\u03c0 i)\nh1 : \u2200 (i : \u03b9), \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1\nh2 : Set.pi (\u2191I) t \u2286 s\ni : \u03b9\nhi : i \u2208 I\n\u22a2 IsOpen (Exists.choose (_ : \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1)) \u2227\n    a i \u2208 Exists.choose (_ : \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1)", "state_after": "no goals"}, {"tactic": "exact Subset.trans\n  (Set.pi_mono fun i hi => (Set.eval_image_pi_subset hi).trans (h1 i).choose_spec.1) h2", "annotated_tactic": ["exact <a>Subset.trans</a>\n        (<a>Set.pi_mono</a> fun i hi => (<a>Set.eval_image_pi_subset</a> hi).<a>trans</a> (h1 i).<a>choose_spec</a>.1) h2", [{"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}, {"full_name": "Set.pi_mono", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [698, 9], "def_end_pos": [698, 16]}, {"full_name": "Set.eval_image_pi_subset", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [852, 9], "def_end_pos": [852, 29]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}, {"full_name": "Exists.choose_spec", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [445, 9], "def_end_pos": [445, 27]}]], "state_before": "case refine_1.intro.intro.intro.refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (i : \u03b9) \u2192 Set (\u03c0 i)\nh1 : \u2200 (i : \u03b9), \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1\nh2 : Set.pi (\u2191I) t \u2286 s\n\u22a2 (Set.pi \u2191I fun a_1 =>\n      eval a_1 '' Set.pi \u2191I fun a_2 => Exists.choose (_ : \u2203 t_1, t_1 \u2286 t a_2 \u2227 IsOpen t_1 \u2227 a a_2 \u2208 t_1)) \u2286\n    s", "state_after": "no goals"}, {"tactic": "rintro \u27e8I, t, \u27e8h1, h2\u27e9\u27e9", "annotated_tactic": ["rintro \u27e8I, t, \u27e8h1, h2\u27e9\u27e9", []], "state_before": "case refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\n\u22a2 (\u2203 I u, (\u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (u a_1) \u2227 a a_1 \u2208 u a_1) \u2227 Set.pi (\u2191I) u \u2286 s) \u2192\n    \u2203 I t, (\u2200 (i : \u03b9), \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1) \u2227 Set.pi (\u2191I) t \u2286 s", "state_after": "case refine_2.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (a : \u03b9) \u2192 Set (\u03c0 a)\nh1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1\nh2 : Set.pi (\u2191I) t \u2286 s\n\u22a2 \u2203 I t, (\u2200 (i : \u03b9), \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1) \u2227 Set.pi (\u2191I) t \u2286 s"}, {"tactic": "refine \u27e8I, fun a => ite (a \u2208 I) (t a) Set.univ, fun i => ?_, ?_\u27e9", "annotated_tactic": ["refine \u27e8I, fun a => <a>ite</a> (a \u2208 I) (t a) <a>Set.univ</a>, fun i => ?_, ?_\u27e9", [{"full_name": "ite", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [935, 21], "def_end_pos": [935, 24]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}]], "state_before": "case refine_2.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (a : \u03b9) \u2192 Set (\u03c0 a)\nh1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1\nh2 : Set.pi (\u2191I) t \u2286 s\n\u22a2 \u2203 I t, (\u2200 (i : \u03b9), \u2203 t_1, t_1 \u2286 t i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1) \u2227 Set.pi (\u2191I) t \u2286 s", "state_after": "case refine_2.intro.intro.intro.refine_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (a : \u03b9) \u2192 Set (\u03c0 a)\nh1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1\nh2 : Set.pi (\u2191I) t \u2286 s\ni : \u03b9\n\u22a2 \u2203 t_1, t_1 \u2286 (fun a => if a \u2208 I then t a else univ) i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1\n\ncase refine_2.intro.intro.intro.refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (a : \u03b9) \u2192 Set (\u03c0 a)\nh1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1\nh2 : Set.pi (\u2191I) t \u2286 s\n\u22a2 (Set.pi \u2191I fun a => if a \u2208 I then t a else univ) \u2286 s"}, {"tactic": "by_cases hi : i \u2208 I", "annotated_tactic": ["by_cases hi : i \u2208 I", []], "state_before": "case refine_2.intro.intro.intro.refine_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (a : \u03b9) \u2192 Set (\u03c0 a)\nh1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1\nh2 : Set.pi (\u2191I) t \u2286 s\ni : \u03b9\n\u22a2 \u2203 t_1, t_1 \u2286 (fun a => if a \u2208 I then t a else univ) i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1", "state_after": "case pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (a : \u03b9) \u2192 Set (\u03c0 a)\nh1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1\nh2 : Set.pi (\u2191I) t \u2286 s\ni : \u03b9\nhi : i \u2208 I\n\u22a2 \u2203 t_1, t_1 \u2286 (fun a => if a \u2208 I then t a else univ) i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1\n\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (a : \u03b9) \u2192 Set (\u03c0 a)\nh1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1\nh2 : Set.pi (\u2191I) t \u2286 s\ni : \u03b9\nhi : \u00aci \u2208 I\n\u22a2 \u2203 t_1, t_1 \u2286 (fun a => if a \u2208 I then t a else univ) i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1"}, {"tactic": "use t i", "annotated_tactic": ["use t i", []], "state_before": "case pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (a : \u03b9) \u2192 Set (\u03c0 a)\nh1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1\nh2 : Set.pi (\u2191I) t \u2286 s\ni : \u03b9\nhi : i \u2208 I\n\u22a2 \u2203 t_1, t_1 \u2286 (fun a => if a \u2208 I then t a else univ) i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1", "state_after": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (a : \u03b9) \u2192 Set (\u03c0 a)\nh1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1\nh2 : Set.pi (\u2191I) t \u2286 s\ni : \u03b9\nhi : i \u2208 I\n\u22a2 t i \u2286 (fun a => if a \u2208 I then t a else univ) i \u2227 IsOpen (t i) \u2227 a i \u2208 t i"}, {"tactic": "simp_rw [if_pos hi]", "annotated_tactic": ["simp_rw [<a>if_pos</a> hi]", [{"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}]], "state_before": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (a : \u03b9) \u2192 Set (\u03c0 a)\nh1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1\nh2 : Set.pi (\u2191I) t \u2286 s\ni : \u03b9\nhi : i \u2208 I\n\u22a2 t i \u2286 (fun a => if a \u2208 I then t a else univ) i \u2227 IsOpen (t i) \u2227 a i \u2208 t i", "state_after": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (a : \u03b9) \u2192 Set (\u03c0 a)\nh1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1\nh2 : Set.pi (\u2191I) t \u2286 s\ni : \u03b9\nhi : i \u2208 I\n\u22a2 t i \u2286 t i \u2227 IsOpen (t i) \u2227 a i \u2208 t i"}, {"tactic": "exact \u27e8Subset.rfl, (h1 i) hi\u27e9", "annotated_tactic": ["exact \u27e8<a>Subset.rfl</a>, (h1 i) hi\u27e9", [{"full_name": "Set.Subset.rfl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}]], "state_before": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (a : \u03b9) \u2192 Set (\u03c0 a)\nh1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1\nh2 : Set.pi (\u2191I) t \u2286 s\ni : \u03b9\nhi : i \u2208 I\n\u22a2 t i \u2286 t i \u2227 IsOpen (t i) \u2227 a i \u2208 t i", "state_after": "no goals"}, {"tactic": "use Set.univ", "annotated_tactic": ["use <a>Set.univ</a>", [{"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}]], "state_before": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (a : \u03b9) \u2192 Set (\u03c0 a)\nh1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1\nh2 : Set.pi (\u2191I) t \u2286 s\ni : \u03b9\nhi : \u00aci \u2208 I\n\u22a2 \u2203 t_1, t_1 \u2286 (fun a => if a \u2208 I then t a else univ) i \u2227 IsOpen t_1 \u2227 a i \u2208 t_1", "state_after": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (a : \u03b9) \u2192 Set (\u03c0 a)\nh1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1\nh2 : Set.pi (\u2191I) t \u2286 s\ni : \u03b9\nhi : \u00aci \u2208 I\n\u22a2 univ \u2286 (fun a => if a \u2208 I then t a else univ) i \u2227 IsOpen univ \u2227 a i \u2208 univ"}, {"tactic": "simp_rw [if_neg hi]", "annotated_tactic": ["simp_rw [<a>if_neg</a> hi]", [{"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}]], "state_before": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (a : \u03b9) \u2192 Set (\u03c0 a)\nh1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1\nh2 : Set.pi (\u2191I) t \u2286 s\ni : \u03b9\nhi : \u00aci \u2208 I\n\u22a2 univ \u2286 (fun a => if a \u2208 I then t a else univ) i \u2227 IsOpen univ \u2227 a i \u2208 univ", "state_after": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (a : \u03b9) \u2192 Set (\u03c0 a)\nh1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1\nh2 : Set.pi (\u2191I) t \u2286 s\ni : \u03b9\nhi : \u00aci \u2208 I\n\u22a2 univ \u2286 univ \u2227 IsOpen univ \u2227 a i \u2208 univ"}, {"tactic": "exact \u27e8Subset.rfl, isOpen_univ, mem_univ _\u27e9", "annotated_tactic": ["exact \u27e8<a>Subset.rfl</a>, <a>isOpen_univ</a>, <a>mem_univ</a> _\u27e9", [{"full_name": "Set.Subset.rfl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}, {"full_name": "isOpen_univ", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [123, 17], "def_end_pos": [123, 28]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}]], "state_before": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (a : \u03b9) \u2192 Set (\u03c0 a)\nh1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1\nh2 : Set.pi (\u2191I) t \u2286 s\ni : \u03b9\nhi : \u00aci \u2208 I\n\u22a2 univ \u2286 univ \u2227 IsOpen univ \u2227 a i \u2208 univ", "state_after": "no goals"}, {"tactic": "rw [\u2190 Set.univ_pi_ite]", "annotated_tactic": ["rw [\u2190 <a>Set.univ_pi_ite</a>]", [{"full_name": "Set.univ_pi_ite", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [923, 9], "def_end_pos": [923, 20]}]], "state_before": "case refine_2.intro.intro.intro.refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (a : \u03b9) \u2192 Set (\u03c0 a)\nh1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1\nh2 : Set.pi (\u2191I) t \u2286 s\n\u22a2 (Set.pi \u2191I fun a => if a \u2208 I then t a else univ) \u2286 s", "state_after": "case refine_2.intro.intro.intro.refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (a : \u03b9) \u2192 Set (\u03c0 a)\nh1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1\nh2 : Set.pi (\u2191I) t \u2286 s\n\u22a2 (Set.pi univ fun i => if i \u2208 \u2191I then if i \u2208 I then t i else univ else univ) \u2286 s"}, {"tactic": "simp only [\u2190 ite_and, \u2190 Finset.mem_coe, and_self_iff, Set.univ_pi_ite, h2]", "annotated_tactic": ["simp only [\u2190 <a>ite_and</a>, \u2190 <a>Finset.mem_coe</a>, <a>and_self_iff</a>, <a>Set.univ_pi_ite</a>, h2]", [{"full_name": "ite_and", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1255, 9], "def_end_pos": [1255, 16]}, {"full_name": "Finset.mem_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}, {"full_name": "and_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [155, 9], "def_end_pos": [155, 21]}, {"full_name": "Set.univ_pi_ite", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [923, 9], "def_end_pos": [923, 20]}]], "state_before": "case refine_2.intro.intro.intro.refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\n\u03b4 : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\n\u03ba : Type u_7\ninst\u271d : TopologicalSpace \u03b1\nT : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u2192 \u03c0 i\ns : Set ((a : \u03b9) \u2192 \u03c0 a)\na : (a : \u03b9) \u2192 \u03c0 a\nx\u271d : a \u2208 s\nI : Finset \u03b9\nt : (a : \u03b9) \u2192 Set (\u03c0 a)\nh1 : \u2200 (a_1 : \u03b9), a_1 \u2208 I \u2192 IsOpen (t a_1) \u2227 a a_1 \u2208 t a_1\nh2 : Set.pi (\u2191I) t \u2286 s\n\u22a2 (Set.pi univ fun i => if i \u2208 \u2191I then if i \u2208 I then t i else univ else univ) \u2286 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.finite_le_nat", "start": [913, 1], "end": [914, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "full_name": "MeasureTheory.Measure.eqOn_open_of_ae_eq", "start": [117, 1], "end": [128, 62], "traced_tactics": [{"tactic": "replace h := ae_imp_of_ae_restrict h", "annotated_tactic": ["replace h := <a>ae_imp_of_ae_restrict</a> h", [{"full_name": "MeasureTheory.ae_imp_of_ae_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2566, 9], "def_end_pos": [2566, 30]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\nm : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : T2Space Y\n\u03bc \u03bd : Measure X\ninst\u271d : IsOpenPosMeasure \u03bc\ns U F : Set X\nx : X\nf g : X \u2192 Y\nh : f =\u1da0[ae (restrict \u03bc U)] g\nhU : IsOpen U\nhf : ContinuousOn f U\nhg : ContinuousOn g U\n\u22a2 EqOn f g U", "state_after": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\nm : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : T2Space Y\n\u03bc \u03bd : Measure X\ninst\u271d : IsOpenPosMeasure \u03bc\ns U F : Set X\nx : X\nf g : X \u2192 Y\nhU : IsOpen U\nhf : ContinuousOn f U\nhg : ContinuousOn g U\nh : \u2200\u1d50 (x : X) \u2202\u03bc, x \u2208 U \u2192 f x = g x\n\u22a2 EqOn f g U"}, {"tactic": "simp only [EventuallyEq, ae_iff, not_imp] at h", "annotated_tactic": ["simp only [<a>EventuallyEq</a>, <a>ae_iff</a>, <a>not_imp</a>] at h", [{"full_name": "Filter.EventuallyEq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1438, 5], "def_end_pos": [1438, 17]}, {"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [388, 9], "def_end_pos": [388, 15]}, {"full_name": "not_imp", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [433, 9], "def_end_pos": [433, 16]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\nm : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : T2Space Y\n\u03bc \u03bd : Measure X\ninst\u271d : IsOpenPosMeasure \u03bc\ns U F : Set X\nx : X\nf g : X \u2192 Y\nhU : IsOpen U\nhf : ContinuousOn f U\nhg : ContinuousOn g U\nh : \u2200\u1d50 (x : X) \u2202\u03bc, x \u2208 U \u2192 f x = g x\n\u22a2 EqOn f g U", "state_after": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\nm : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : T2Space Y\n\u03bc \u03bd : Measure X\ninst\u271d : IsOpenPosMeasure \u03bc\ns U F : Set X\nx : X\nf g : X \u2192 Y\nhU : IsOpen U\nhf : ContinuousOn f U\nhg : ContinuousOn g U\nh : \u2191\u2191\u03bc {a | a \u2208 U \u2227 \u00acf a = g a} = 0\n\u22a2 EqOn f g U"}, {"tactic": "have : IsOpen (U \u2229 { a | f a \u2260 g a }) := by\n  refine' isOpen_iff_mem_nhds.mpr fun a ha => inter_mem (hU.mem_nhds ha.1) _\n  rcases ha with \u27e8ha : a \u2208 U, ha' : (f a, g a) \u2208 (diagonal Y)\u1d9c\u27e9\n  exact\n    (hf.rst.imntinuousAt (hU.mem_nhds ha)).prod_mk_nhds (hg.continuousAt (hU.mem_nhds ha))\n      (isClosed_diagonal.isOpen_compl.mem_nhds ha')", "annotated_tactic": ["have : <a>IsOpen</a> (U \u2229 { a | f a \u2260 g a }) := by\n    refine' isOpen_iff_mem_nhds.mpr fun a ha => <a>inter_mem</a> (hU.mem_nhds ha.1) _\n    rcases ha with \u27e8ha : a \u2208 U, ha' : (f a, g a) \u2208 (<a>diagonal</a> Y)\u1d9c\u27e9\n    exact\n      (hf.rst.imntinuousAt (hU.mem_nhds ha)).<a>prod_mk_nhds</a> (hg.continuousAt (hU.mem_nhds ha))\n        (isClosed_diagonal.isOpen_compl.mem_nhds ha')", [{"full_name": "IsOpen", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [101, 5], "def_end_pos": [101, 11]}, {"full_name": "Filter.inter_mem", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [156, 9], "def_end_pos": [156, 18]}, {"full_name": "Set.diagonal", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [495, 5], "def_end_pos": [495, 13]}, {"full_name": "Filter.Tendsto.prod_mk_nhds", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [582, 9], "def_end_pos": [582, 36]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\nm : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : T2Space Y\n\u03bc \u03bd : Measure X\ninst\u271d : IsOpenPosMeasure \u03bc\ns U F : Set X\nx : X\nf g : X \u2192 Y\nhU : IsOpen U\nhf : ContinuousOn f U\nhg : ContinuousOn g U\nh : \u2191\u2191\u03bc {a | a \u2208 U \u2227 \u00acf a = g a} = 0\n\u22a2 EqOn f g U", "state_after": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\nm : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : T2Space Y\n\u03bc \u03bd : Measure X\ninst\u271d : IsOpenPosMeasure \u03bc\ns U F : Set X\nx : X\nf g : X \u2192 Y\nhU : IsOpen U\nhf : ContinuousOn f U\nhg : ContinuousOn g U\nh : \u2191\u2191\u03bc {a | a \u2208 U \u2227 \u00acf a = g a} = 0\nthis : IsOpen (U \u2229 {a | f a \u2260 g a})\n\u22a2 EqOn f g U"}, {"tactic": "replace := (this.eq_empty_of_measure_zero h).le", "annotated_tactic": ["replace := (this.eq_empty_of_measure_zero h).<a>le</a>", [{"full_name": "Eq.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [159, 7], "def_end_pos": [159, 12]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\nm : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : T2Space Y\n\u03bc \u03bd : Measure X\ninst\u271d : IsOpenPosMeasure \u03bc\ns U F : Set X\nx : X\nf g : X \u2192 Y\nhU : IsOpen U\nhf : ContinuousOn f U\nhg : ContinuousOn g U\nh : \u2191\u2191\u03bc {a | a \u2208 U \u2227 \u00acf a = g a} = 0\nthis : IsOpen (U \u2229 {a | f a \u2260 g a})\n\u22a2 EqOn f g U", "state_after": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\nm : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : T2Space Y\n\u03bc \u03bd : Measure X\ninst\u271d : IsOpenPosMeasure \u03bc\ns U F : Set X\nx : X\nf g : X \u2192 Y\nhU : IsOpen U\nhf : ContinuousOn f U\nhg : ContinuousOn g U\nh : \u2191\u2191\u03bc {a | a \u2208 U \u2227 \u00acf a = g a} = 0\nthis : U \u2229 {a | f a \u2260 g a} \u2264 \u2205\n\u22a2 EqOn f g U"}, {"tactic": "exact fun x hx => Classical.not_not.1 fun h => this \u27e8hx, h\u27e9", "annotated_tactic": ["exact fun x hx => <a>Classical.not_not</a>.1 fun h => this \u27e8hx, h\u27e9", [{"full_name": "Classical.not_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [683, 24], "def_end_pos": [683, 31]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\nm : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : T2Space Y\n\u03bc \u03bd : Measure X\ninst\u271d : IsOpenPosMeasure \u03bc\ns U F : Set X\nx : X\nf g : X \u2192 Y\nhU : IsOpen U\nhf : ContinuousOn f U\nhg : ContinuousOn g U\nh : \u2191\u2191\u03bc {a | a \u2208 U \u2227 \u00acf a = g a} = 0\nthis : U \u2229 {a | f a \u2260 g a} \u2264 \u2205\n\u22a2 EqOn f g U", "state_after": "no goals"}, {"tactic": "refine' isOpen_iff_mem_nhds.mpr fun a ha => inter_mem (hU.mem_nhds ha.1) _", "annotated_tactic": ["refine' isOpen_iff_mem_nhds.mpr fun a ha => <a>inter_mem</a> (hU.mem_nhds ha.1) _", [{"full_name": "Filter.inter_mem", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [156, 9], "def_end_pos": [156, 18]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\nm : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : T2Space Y\n\u03bc \u03bd : Measure X\ninst\u271d : IsOpenPosMeasure \u03bc\ns U F : Set X\nx : X\nf g : X \u2192 Y\nhU : IsOpen U\nhf : ContinuousOn f U\nhg : ContinuousOn g U\nh : \u2191\u2191\u03bc {a | a \u2208 U \u2227 \u00acf a = g a} = 0\n\u22a2 IsOpen (U \u2229 {a | f a \u2260 g a})", "state_after": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\nm : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : T2Space Y\n\u03bc \u03bd : Measure X\ninst\u271d : IsOpenPosMeasure \u03bc\ns U F : Set X\nx : X\nf g : X \u2192 Y\nhU : IsOpen U\nhf : ContinuousOn f U\nhg : ContinuousOn g U\nh : \u2191\u2191\u03bc {a | a \u2208 U \u2227 \u00acf a = g a} = 0\na : X\nha : a \u2208 U \u2229 {a | f a \u2260 g a}\n\u22a2 {a | f a \u2260 g a} \u2208 \ud835\udcdd a"}, {"tactic": "rcases ha with \u27e8ha : a \u2208 U, ha' : (f a, g a) \u2208 (diagonal Y)\u1d9c\u27e9", "annotated_tactic": ["rcases ha with \u27e8ha : a \u2208 U, ha' : (f a, g a) \u2208 (<a>diagonal</a> Y)\u1d9c\u27e9", [{"full_name": "Set.diagonal", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [495, 5], "def_end_pos": [495, 13]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\nm : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : T2Space Y\n\u03bc \u03bd : Measure X\ninst\u271d : IsOpenPosMeasure \u03bc\ns U F : Set X\nx : X\nf g : X \u2192 Y\nhU : IsOpen U\nhf : ContinuousOn f U\nhg : ContinuousOn g U\nh : \u2191\u2191\u03bc {a | a \u2208 U \u2227 \u00acf a = g a} = 0\na : X\nha : a \u2208 U \u2229 {a | f a \u2260 g a}\n\u22a2 {a | f a \u2260 g a} \u2208 \ud835\udcdd a", "state_after": "case intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\nm : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : T2Space Y\n\u03bc \u03bd : Measure X\ninst\u271d : IsOpenPosMeasure \u03bc\ns U F : Set X\nx : X\nf g : X \u2192 Y\nhU : IsOpen U\nhf : ContinuousOn f U\nhg : ContinuousOn g U\nh : \u2191\u2191\u03bc {a | a \u2208 U \u2227 \u00acf a = g a} = 0\na : X\nha : a \u2208 U\nha' : (f a, g a) \u2208 (diagonal Y)\u1d9c\n\u22a2 {a | f a \u2260 g a} \u2208 \ud835\udcdd a"}, {"tactic": "exact\n  (hf.rst.imntinuousAt (hU.mem_nhds ha)).prod_mk_nhds (hg.continuousAt (hU.mem_nhds ha))\n    (isClosed_diagonal.isOpen_compl.mem_nhds ha')", "annotated_tactic": ["exact\n      (hf.rst.imntinuousAt (hU.mem_nhds ha)).<a>prod_mk_nhds</a> (hg.continuousAt (hU.mem_nhds ha))\n        (isClosed_diagonal.isOpen_compl.mem_nhds ha')", [{"full_name": "Filter.Tendsto.prod_mk_nhds", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [582, 9], "def_end_pos": [582, 36]}]], "state_before": "case intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\nm : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : T2Space Y\n\u03bc \u03bd : Measure X\ninst\u271d : IsOpenPosMeasure \u03bc\ns U F : Set X\nx : X\nf g : X \u2192 Y\nhU : IsOpen U\nhf : ContinuousOn f U\nhg : ContinuousOn g U\nh : \u2191\u2191\u03bc {a | a \u2208 U \u2227 \u00acf a = g a} = 0\na : X\nha : a \u2208 U\nha' : (f a, g a) \u2208 (diagonal Y)\u1d9c\n\u22a2 {a | f a \u2260 g a} \u2208 \ud835\udcdd a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "full_name": "LinearMap.toMatrix'_apply", "start": [337, 1], "end": [345, 30], "traced_tactics": [{"tactic": "simp only [LinearMap.toMatrix', LinearEquiv.coe_mk, of_apply]", "annotated_tactic": ["simp only [<a>LinearMap.toMatrix'</a>, <a>LinearEquiv.coe_mk</a>, <a>of_apply</a>]", [{"full_name": "LinearMap.toMatrix'", "def_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "def_pos": [284, 5], "def_end_pos": [284, 24]}, {"full_name": "LinearEquiv.coe_mk", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [184, 9], "def_end_pos": [184, 15]}, {"full_name": "Matrix.of_apply", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [119, 9], "def_end_pos": [119, 17]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ni : m\nj : n\n\u22a2 \u2191toMatrix' f i j = \u2191f (fun j' => if j' = j then 1 else 0) i", "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ni : m\nj : n\n\u22a2 \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i = \u2191f (fun j' => if j' = j then 1 else 0) i"}, {"tactic": "refine congr_fun ?_ _", "annotated_tactic": ["refine <a>congr_fun</a> ?_ _", [{"full_name": "congr_fun", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [42, 7], "def_end_pos": [42, 16]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ni : m\nj : n\n\u22a2 \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i = \u2191f (fun j' => if j' = j then 1 else 0) i", "state_after": "R : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ni : m\nj : n\n\u22a2 \u2191f (\u2191(stdBasis R (fun x => R) j) 1) = \u2191f fun j' => if j' = j then 1 else 0"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ni : m\nj : n\n\u22a2 \u2191f (\u2191(stdBasis R (fun x => R) j) 1) = \u2191f fun j' => if j' = j then 1 else 0", "state_after": "case h.e_6.h\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ni : m\nj : n\n\u22a2 \u2191(stdBasis R (fun x => R) j) 1 = fun j' => if j' = j then 1 else 0"}, {"tactic": "ext j'", "annotated_tactic": ["ext j'", []], "state_before": "case h.e_6.h\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ni : m\nj : n\n\u22a2 \u2191(stdBasis R (fun x => R) j) 1 = fun j' => if j' = j then 1 else 0", "state_after": "case h.e_6.h.h\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ni : m\nj j' : n\n\u22a2 \u2191(stdBasis R (fun x => R) j) 1 j' = if j' = j then 1 else 0"}, {"tactic": "split_ifs with h", "annotated_tactic": ["split_ifs with h", []], "state_before": "case h.e_6.h.h\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ni : m\nj j' : n\n\u22a2 \u2191(stdBasis R (fun x => R) j) 1 j' = if j' = j then 1 else 0", "state_after": "case pos\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ni : m\nj j' : n\nh : j' = j\n\u22a2 \u2191(stdBasis R (fun x => R) j) 1 j' = 1\n\ncase neg\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ni : m\nj j' : n\nh : \u00acj' = j\n\u22a2 \u2191(stdBasis R (fun x => R) j) 1 j' = 0"}, {"tactic": "apply stdBasis_ne _ _ _ _ h", "annotated_tactic": ["apply <a>stdBasis_ne</a> _ _ _ _ h", [{"full_name": "LinearMap.stdBasis_ne", "def_path": "Mathlib/LinearAlgebra/StdBasis.lean", "def_pos": [71, 9], "def_end_pos": [71, 20]}]], "state_before": "case neg\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ni : m\nj j' : n\nh : \u00acj' = j\n\u22a2 \u2191(stdBasis R (fun x => R) j) 1 j' = 0", "state_after": "no goals"}, {"tactic": "rw [h, stdBasis_same]", "annotated_tactic": ["rw [h, <a>stdBasis_same</a>]", [{"full_name": "LinearMap.stdBasis_same", "def_path": "Mathlib/LinearAlgebra/StdBasis.lean", "def_pos": [67, 9], "def_end_pos": [67, 22]}]], "state_before": "case pos\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ni : m\nj j' : n\nh : j' = j\n\u22a2 \u2191(stdBasis R (fun x => R) j) 1 j' = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Matrix/ZPow.lean", "full_name": "Matrix.inv_zpow'", "start": [145, 1], "end": [146, 28], "traced_tactics": [{"tactic": "rw [zpow_neg h, inv_zpow]", "annotated_tactic": ["rw [<a>zpow_neg</a> h, <a>inv_zpow</a>]", [{"full_name": "Matrix.zpow_neg", "def_path": "Mathlib/LinearAlgebra/Matrix/ZPow.lean", "def_pos": [137, 9], "def_end_pos": [137, 17]}, {"full_name": "Matrix.inv_zpow", "def_path": "Mathlib/LinearAlgebra/Matrix/ZPow.lean", "def_pos": [99, 9], "def_end_pos": [99, 17]}]], "state_before": "n' : Type u_1\ninst\u271d\u00b2 : DecidableEq n'\ninst\u271d\u00b9 : Fintype n'\nR : Type u_2\ninst\u271d : CommRing R\nA : M\nh : IsUnit (det A)\nn : \u2124\n\u22a2 A\u207b\u00b9 ^ n = A ^ (-n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Sym/Sym2.lean", "full_name": "Sym2.other_spec", "start": [362, 1], "end": [363, 34], "traced_tactics": [{"tactic": "erw [\u2190 Classical.choose_spec h]", "annotated_tactic": ["erw [\u2190 <a>Classical.choose_spec</a> h]", [{"full_name": "Classical.choose_spec", "def_path": "lake-packages/lean4/src/lean/Init/Classical.lean", "def_pos": [22, 9], "def_end_pos": [22, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nz : Sym2 \u03b1\nh : a \u2208 z\n\u22a2 Quotient.mk (Rel.setoid \u03b1) (a, Mem.other h) = z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Fintype/Basic.lean", "full_name": "Finset.eq_univ_of_forall", "start": [84, 1], "end": [85, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Ring/Equiv.lean", "full_name": "RingEquiv.trans_apply", "start": [323, 1], "end": [324, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/ConcreteCategory/Basic.lean", "full_name": "CategoryTheory.ConcreteCategory.surjective_of_epi_of_preservesPushout", "start": [174, 1], "end": [176, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Algebra/Equiv.lean", "full_name": "AlgEquiv.ext_iff", "start": [163, 11], "end": [164, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.coeFn_smul", "start": [253, 1], "end": [258, 12], "traced_tactics": [{"tactic": "funext", "annotated_tactic": ["funext", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2074 : SMul R \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650 \u211d\u22650\nc : R\n\u03bc : FiniteMeasure \u03a9\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(c \u2022 \u03bc) s)) = c \u2022 fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2074 : SMul R \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650 \u211d\u22650\nc : R\n\u03bc : FiniteMeasure \u03a9\nx\u271d : Set \u03a9\n\u22a2 ENNReal.toNNReal (\u2191\u2191\u2191(c \u2022 \u03bc) x\u271d) = (c \u2022 fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) x\u271d"}, {"tactic": "simp only [Pi.smul_apply, \u2190 ENNReal.coe_eq_coe, ne_eq, ennreal_coeFn_eq_coeFn_toMeasure,\n  ENNReal.coe_smul]", "annotated_tactic": ["simp only [<a>Pi.smul_apply</a>, \u2190 <a>ENNReal.coe_eq_coe</a>, <a>ne_eq</a>, <a>ennreal_coeFn_eq_coeFn_toMeasure</a>,\n    <a>ENNReal.coe_smul</a>]", [{"full_name": "Pi.smul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [116, 60], "def_end_pos": [116, 70]}, {"full_name": "ENNReal.coe_eq_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [346, 28], "def_end_pos": [346, 38]}, {"full_name": "ne_eq", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [76, 17], "def_end_pos": [76, 22]}, {"full_name": "MeasureTheory.FiniteMeasure.ennreal_coeFn_eq_coeFn_toMeasure", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [141, 9], "def_end_pos": [141, 41]}, {"full_name": "ENNReal.coe_smul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [539, 9], "def_end_pos": [539, 17]}]], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2074 : SMul R \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650 \u211d\u22650\nc : R\n\u03bc : FiniteMeasure \u03a9\nx\u271d : Set \u03a9\n\u22a2 ENNReal.toNNReal (\u2191\u2191\u2191(c \u2022 \u03bc) x\u271d) = (c \u2022 fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) x\u271d", "state_after": "case h\n\u03a9 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2074 : SMul R \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650 \u211d\u22650\nc : R\n\u03bc : FiniteMeasure \u03a9\nx\u271d : Set \u03a9\n\u22a2 \u2191\u2191\u2191(c \u2022 \u03bc) x\u271d = c \u2022 \u2191\u2191\u2191\u03bc x\u271d"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "case h\n\u03a9 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2074 : SMul R \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650 \u211d\u22650\nc : R\n\u03bc : FiniteMeasure \u03a9\nx\u271d : Set \u03a9\n\u22a2 \u2191\u2191\u2191(c \u2022 \u03bc) x\u271d = c \u2022 \u2191\u2191\u2191\u03bc x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Congruence.lean", "full_name": "RingCon.coe_nsmul", "start": [273, 1], "end": [274, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "full_name": "MvPolynomial.coeToMvPowerSeries.ringHom_apply", "start": [1210, 1], "end": [1211, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "full_name": "abs_sub_sq", "start": [108, 1], "end": [111, 37], "traced_tactics": [{"tactic": "rw [abs_mul_abs_self]", "annotated_tactic": ["rw [<a>abs_mul_abs_self</a>]", [{"full_name": "abs_mul_abs_self", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [49, 9], "def_end_pos": [49, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedCommRing \u03b1\na\u271d b\u271d c d a b : \u03b1\n\u22a2 |a - b| * |a - b| = a * a + b * b - (1 + 1) * a * b", "state_after": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedCommRing \u03b1\na\u271d b\u271d c d a b : \u03b1\n\u22a2 (a - b) * (a - b) = a * a + b * b - (1 + 1) * a * b"}, {"tactic": "simp only [mul_add, add_comm, add_left_comm, mul_comm, sub_eq_add_neg, mul_one, mul_neg,\n  neg_add_rev, neg_neg, add_assoc]", "annotated_tactic": ["simp only [<a>mul_add</a>, <a>add_comm</a>, <a>add_left_comm</a>, <a>mul_comm</a>, <a>sub_eq_add_neg</a>, <a>mul_one</a>, <a>mul_neg</a>,\n    <a>neg_add_rev</a>, <a>neg_neg</a>, <a>add_assoc</a>]", [{"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "add_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "mul_neg", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [294, 9], "def_end_pos": [294, 16]}, {"full_name": "neg_add_rev", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1049, 30], "def_end_pos": [1049, 41]}, {"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [799, 3], "def_end_pos": [799, 14]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedCommRing \u03b1\na\u271d b\u271d c d a b : \u03b1\n\u22a2 (a - b) * (a - b) = a * a + b * b - (1 + 1) * a * b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Submonoid/Pointwise.lean", "full_name": "AddSubmonoid.mul_mem_mul", "start": [533, 1], "end": [534, 50], "traced_tactics": [{"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\nG : Type u_2\nM\u271d : Type u_3\nR : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : Monoid M\u271d\ninst\u271d\u00b9 : AddMonoid A\ninst\u271d : NonUnitalNonAssocSemiring R\nM N : AddSubmonoid R\nm n : R\nhm : m \u2208 M\nhn : n \u2208 N\n\u22a2 \u2191(\u2191AddMonoidHom.mul \u2191{ val := m, property := hm }) n = m * n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Hom/Freiman.lean", "full_name": "FreimanHom.comp_id", "start": [276, 1], "end": [277, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Convex/Between.lean", "full_name": "Wbtw.trans_sbtw_right", "start": [486, 1], "end": [490, 30], "traced_tactics": [{"tactic": "rw [wbtw_comm] at *", "annotated_tactic": ["rw [<a>wbtw_comm</a>] at *", [{"full_name": "wbtw_comm", "def_path": "Mathlib/Analysis/Convex/Between.lean", "def_pos": [273, 9], "def_end_pos": [273, 18]}]], "state_before": "R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nw x y z : P\nh\u2081 : Wbtw R w x z\nh\u2082 : Sbtw R x y z\n\u22a2 Sbtw R w y z", "state_after": "R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nw x y z : P\nh\u2081 : Wbtw R z x w\nh\u2082 : Sbtw R x y z\n\u22a2 Sbtw R w y z"}, {"tactic": "rw [sbtw_comm] at *", "annotated_tactic": ["rw [<a>sbtw_comm</a>] at *", [{"full_name": "sbtw_comm", "def_path": "Mathlib/Analysis/Convex/Between.lean", "def_pos": [280, 9], "def_end_pos": [280, 18]}]], "state_before": "R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nw x y z : P\nh\u2081 : Wbtw R z x w\nh\u2082 : Sbtw R x y z\n\u22a2 Sbtw R w y z", "state_after": "R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nw x y z : P\nh\u2081 : Wbtw R z x w\nh\u2082 : Sbtw R z y x\n\u22a2 Sbtw R z y w"}, {"tactic": "exact h\u2081.trans_sbtw_left h\u2082", "annotated_tactic": ["exact h\u2081.trans_sbtw_left h\u2082", []], "state_before": "R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : OrderedRing R\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module R V\ninst\u271d\u2074 : AddTorsor V P\ninst\u271d\u00b3 : AddCommGroup V'\ninst\u271d\u00b2 : Module R V'\ninst\u271d\u00b9 : AddTorsor V' P'\ninst\u271d : NoZeroSMulDivisors R V\nw x y z : P\nh\u2081 : Wbtw R z x w\nh\u2082 : Sbtw R z y x\n\u22a2 Sbtw R z y w", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Normed/Group/AddTorsor.lean", "full_name": "ContinuousAt.vsub", "start": [286, 8], "end": [289, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Image.lean", "full_name": "Finset.image_comm", "start": [432, 1], "end": [434, 90], "traced_tactics": [{"tactic": "simp_rw [image_image, comp, h_comm]", "annotated_tactic": ["simp_rw [<a>image_image</a>, <a>comp</a>, h_comm]", [{"full_name": "Finset.image_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [428, 9], "def_end_pos": [428, 20]}, {"full_name": "Function.comp", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : DecidableEq \u03b2\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\na : \u03b1\nb c : \u03b2\n\u03b2' : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b2'\ninst\u271d : DecidableEq \u03b3\nf : \u03b2 \u2192 \u03b3\ng : \u03b1 \u2192 \u03b2\nf' : \u03b1 \u2192 \u03b2'\ng' : \u03b2' \u2192 \u03b3\nh_comm : \u2200 (a : \u03b1), f (g a) = g' (f' a)\n\u22a2 image f (image g s) = image g' (image f' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Multiset/FinsetOps.lean", "full_name": "Multiset.ndinsert_zero", "start": [40, 1], "end": [41, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "full_name": "Rat.mkRat_num_den", "start": [90, 1], "end": [92, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/Cyclotomic/Basic.lean", "full_name": "IsCyclotomicExtension.subsingleton_iff", "start": [157, 1], "end": [170, 56], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "n : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\n\u22a2 IsCyclotomicExtension S A B \u2194 S = \u2205 \u2228 S = {1}", "state_after": "case mp\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\n\u22a2 IsCyclotomicExtension S A B \u2192 S = \u2205 \u2228 S = {1}\n\ncase mpr\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\n\u22a2 S = \u2205 \u2228 S = {1} \u2192 IsCyclotomicExtension S A B"}, {"tactic": "rintro \u27e8hprim, -\u27e9", "annotated_tactic": ["rintro \u27e8hprim, -\u27e9", []], "state_before": "case mp\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\n\u22a2 IsCyclotomicExtension S A B \u2192 S = \u2205 \u2228 S = {1}", "state_after": "case mp.mk\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\nhprim : \u2200 {n : \u2115+}, n \u2208 S \u2192 \u2203 r, IsPrimitiveRoot r \u2191n\n\u22a2 S = \u2205 \u2228 S = {1}"}, {"tactic": "rw [\u2190 subset_singleton_iff_eq]", "annotated_tactic": ["rw [\u2190 <a>subset_singleton_iff_eq</a>]", [{"full_name": "Set.subset_singleton_iff_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1511, 9], "def_end_pos": [1511, 32]}]], "state_before": "case mp.mk\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\nhprim : \u2200 {n : \u2115+}, n \u2208 S \u2192 \u2203 r, IsPrimitiveRoot r \u2191n\n\u22a2 S = \u2205 \u2228 S = {1}", "state_after": "case mp.mk\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\nhprim : \u2200 {n : \u2115+}, n \u2208 S \u2192 \u2203 r, IsPrimitiveRoot r \u2191n\n\u22a2 S \u2286 {1}"}, {"tactic": "intro t ht", "annotated_tactic": ["intro t ht", []], "state_before": "case mp.mk\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\nhprim : \u2200 {n : \u2115+}, n \u2208 S \u2192 \u2203 r, IsPrimitiveRoot r \u2191n\n\u22a2 S \u2286 {1}", "state_after": "case mp.mk\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\nhprim : \u2200 {n : \u2115+}, n \u2208 S \u2192 \u2203 r, IsPrimitiveRoot r \u2191n\nt : \u2115+\nht : t \u2208 S\n\u22a2 t \u2208 {1}"}, {"tactic": "obtain \u27e8\u03b6, h\u03b6\u27e9 := hprim ht", "annotated_tactic": ["obtain \u27e8\u03b6, h\u03b6\u27e9 := hprim ht", []], "state_before": "case mp.mk\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\nhprim : \u2200 {n : \u2115+}, n \u2208 S \u2192 \u2203 r, IsPrimitiveRoot r \u2191n\nt : \u2115+\nht : t \u2208 S\n\u22a2 t \u2208 {1}", "state_after": "case mp.mk.intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\nhprim : \u2200 {n : \u2115+}, n \u2208 S \u2192 \u2203 r, IsPrimitiveRoot r \u2191n\nt : \u2115+\nht : t \u2208 S\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191t\n\u22a2 t \u2208 {1}"}, {"tactic": "rw [mem_singleton_iff, \u2190 PNat.coe_eq_one_iff]", "annotated_tactic": ["rw [<a>mem_singleton_iff</a>, \u2190 <a>PNat.coe_eq_one_iff</a>]", [{"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}, {"full_name": "PNat.coe_eq_one_iff", "def_path": "Mathlib/Data/PNat/Defs.lean", "def_pos": [200, 9], "def_end_pos": [200, 23]}]], "state_before": "case mp.mk.intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\nhprim : \u2200 {n : \u2115+}, n \u2208 S \u2192 \u2203 r, IsPrimitiveRoot r \u2191n\nt : \u2115+\nht : t \u2208 S\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191t\n\u22a2 t \u2208 {1}", "state_after": "case mp.mk.intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\nhprim : \u2200 {n : \u2115+}, n \u2208 S \u2192 \u2203 r, IsPrimitiveRoot r \u2191n\nt : \u2115+\nht : t \u2208 S\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191t\n\u22a2 \u2191t = 1"}, {"tactic": "exact_mod_cast h\u03b6.unique (IsPrimitiveRoot.of_subsingleton \u03b6)", "annotated_tactic": ["exact_mod_cast h\u03b6.unique (<a>IsPrimitiveRoot.of_subsingleton</a> \u03b6)", [{"full_name": "IsPrimitiveRoot.of_subsingleton", "def_path": "Mathlib/RingTheory/RootsOfUnity/Basic.lean", "def_pos": [353, 9], "def_end_pos": [353, 24]}]], "state_before": "case mp.mk.intro\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\nhprim : \u2200 {n : \u2115+}, n \u2208 S \u2192 \u2203 r, IsPrimitiveRoot r \u2191n\nt : \u2115+\nht : t \u2208 S\n\u03b6 : B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191t\n\u22a2 \u2191t = 1", "state_after": "no goals"}, {"tactic": "rintro (rfl | rfl)", "annotated_tactic": ["rintro (rfl | rfl)", []], "state_before": "case mpr\nn : \u2115+\nS T : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\n\u22a2 S = \u2205 \u2228 S = {1} \u2192 IsCyclotomicExtension S A B", "state_after": "case mpr.inl\nn : \u2115+\nT : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\n\u22a2 IsCyclotomicExtension \u2205 A B\n\ncase mpr.inr\nn : \u2115+\nT : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\n\u22a2 IsCyclotomicExtension {1} A B"}, {"tactic": "exact \u27e8fun h => h.elim, fun x => by convert (mem_top (R := A) : x \u2208 \u22a4)\u27e9", "annotated_tactic": ["exact \u27e8fun h => h.elim, fun x => by convert (<a>mem_top</a> (R := A) : x \u2208 \u22a4)\u27e9", [{"full_name": "Algebra.mem_top", "def_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "def_pos": [794, 9], "def_end_pos": [794, 16]}]], "state_before": "case mpr.inl\nn : \u2115+\nT : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\n\u22a2 IsCyclotomicExtension \u2205 A B", "state_after": "no goals"}, {"tactic": "convert (mem_top (R := A) : x \u2208 \u22a4)", "annotated_tactic": ["convert (<a>mem_top</a> (R := A) : x \u2208 \u22a4)", [{"full_name": "Algebra.mem_top", "def_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "def_pos": [794, 9], "def_end_pos": [794, 16]}]], "state_before": "n : \u2115+\nT : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\nx : B\n\u22a2 x \u2208 adjoin A {b | \u2203 n, n \u2208 \u2205 \u2227 b ^ \u2191n = 1}", "state_after": "no goals"}, {"tactic": "rw [iff_singleton]", "annotated_tactic": ["rw [<a>iff_singleton</a>]", [{"full_name": "IsCyclotomicExtension.iff_singleton", "def_path": "Mathlib/NumberTheory/Cyclotomic/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 22]}]], "state_before": "case mpr.inr\nn : \u2115+\nT : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\n\u22a2 IsCyclotomicExtension {1} A B", "state_after": "case mpr.inr\nn : \u2115+\nT : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\n\u22a2 (\u2203 r, IsPrimitiveRoot r \u21911) \u2227 \u2200 (x : B), x \u2208 adjoin A {b | b ^ \u21911 = 1}"}, {"tactic": "exact \u27e8\u27e80, IsPrimitiveRoot.of_subsingleton 0\u27e9,\n  fun x => by convert (mem_top (R := A) : x \u2208 \u22a4)\u27e9", "annotated_tactic": ["exact \u27e8\u27e80, <a>IsPrimitiveRoot.of_subsingleton</a> 0\u27e9,\n        fun x => by convert (<a>mem_top</a> (R := A) : x \u2208 \u22a4)\u27e9", [{"full_name": "IsPrimitiveRoot.of_subsingleton", "def_path": "Mathlib/RingTheory/RootsOfUnity/Basic.lean", "def_pos": [353, 9], "def_end_pos": [353, 24]}, {"full_name": "Algebra.mem_top", "def_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "def_pos": [794, 9], "def_end_pos": [794, 16]}]], "state_before": "case mpr.inr\nn : \u2115+\nT : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\n\u22a2 (\u2203 r, IsPrimitiveRoot r \u21911) \u2227 \u2200 (x : B), x \u2208 adjoin A {b | b ^ \u21911 = 1}", "state_after": "no goals"}, {"tactic": "convert (mem_top (R := A) : x \u2208 \u22a4)", "annotated_tactic": ["convert (<a>mem_top</a> (R := A) : x \u2208 \u22a4)", [{"full_name": "Algebra.mem_top", "def_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "def_pos": [794, 9], "def_end_pos": [794, 16]}]], "state_before": "n : \u2115+\nT : Set \u2115+\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Subsingleton B\nx : B\n\u22a2 x \u2208 adjoin A {b | b ^ \u21911 = 1}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.AEStronglyMeasurable.restrict", "start": [1252, 11], "end": [1254, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/InnerProductSpace/TwoDim.lean", "full_name": "Orientation.abs_areaForm_of_orthogonal", "start": [148, 1], "end": [156, 13], "traced_tactics": [{"tactic": "rw [o.areaForm_to_volumeForm, o.abs_volumeForm_apply_of_pairwise_orthogonal]", "annotated_tactic": ["rw [o.areaForm_to_volumeForm, o.abs_volumeForm_apply_of_pairwise_orthogonal]", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\n\u22a2 |\u2191(\u2191(areaForm o) x) y| = \u2016x\u2016 * \u2016y\u2016", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\n\u22a2 (Finset.prod Finset.univ fun i => \u2016Matrix.vecCons x ![y] i\u2016) = \u2016x\u2016 * \u2016y\u2016\n\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\n\u22a2 Pairwise fun i j => inner (Matrix.vecCons x ![y] i) (Matrix.vecCons x ![y] j) = 0"}, {"tactic": "intro i j hij", "annotated_tactic": ["intro i j hij", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\n\u22a2 Pairwise fun i j => inner (Matrix.vecCons x ![y] i) (Matrix.vecCons x ![y] j) = 0", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\ni j : Fin 2\nhij : i \u2260 j\n\u22a2 inner (Matrix.vecCons x ![y] i) (Matrix.vecCons x ![y] j) = 0"}, {"tactic": "fin_cases i <;> fin_cases j", "annotated_tactic": ["fin_cases i <;> fin_cases j", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\ni j : Fin 2\nhij : i \u2260 j\n\u22a2 inner (Matrix.vecCons x ![y] i) (Matrix.vecCons x ![y] j) = 0", "state_after": "case head.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\nhij : { val := 0, isLt := (_ : 0 < 2) } \u2260 { val := 0, isLt := (_ : 0 < 2) }\n\u22a2 inner (Matrix.vecCons x ![y] { val := 0, isLt := (_ : 0 < 2) })\n      (Matrix.vecCons x ![y] { val := 0, isLt := (_ : 0 < 2) }) =\n    0\n\ncase head.tail.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\nhij : { val := 0, isLt := (_ : 0 < 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n\u22a2 inner (Matrix.vecCons x ![y] { val := 0, isLt := (_ : 0 < 2) })\n      (Matrix.vecCons x ![y] { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =\n    0\n\ncase tail.head.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\nhij : { val := 1, isLt := (_ : (fun a => a < 2) 1) } \u2260 { val := 0, isLt := (_ : 0 < 2) }\n\u22a2 inner (Matrix.vecCons x ![y] { val := 1, isLt := (_ : (fun a => a < 2) 1) })\n      (Matrix.vecCons x ![y] { val := 0, isLt := (_ : 0 < 2) }) =\n    0\n\ncase tail.head.tail.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\nhij : { val := 1, isLt := (_ : (fun a => a < 2) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n\u22a2 inner (Matrix.vecCons x ![y] { val := 1, isLt := (_ : (fun a => a < 2) 1) })\n      (Matrix.vecCons x ![y] { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =\n    0"}, {"tactic": "simp [Fin.prod_univ_succ]", "annotated_tactic": ["simp [<a>Fin.prod_univ_succ</a>]", [{"full_name": "Fin.prod_univ_succ", "def_path": "Mathlib/Algebra/BigOperators/Fin.lean", "def_pos": [81, 9], "def_end_pos": [81, 23]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\n\u22a2 (Finset.prod Finset.univ fun i => \u2016Matrix.vecCons x ![y] i\u2016) = \u2016x\u2016 * \u2016y\u2016", "state_after": "no goals"}, {"tactic": "simp_all", "annotated_tactic": ["simp_all", []], "state_before": "case head.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\nhij : { val := 0, isLt := (_ : 0 < 2) } \u2260 { val := 0, isLt := (_ : 0 < 2) }\n\u22a2 inner (Matrix.vecCons x ![y] { val := 0, isLt := (_ : 0 < 2) })\n      (Matrix.vecCons x ![y] { val := 0, isLt := (_ : 0 < 2) }) =\n    0", "state_after": "no goals"}, {"tactic": "simpa using h", "annotated_tactic": ["simpa using h", []], "state_before": "case head.tail.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\nhij : { val := 0, isLt := (_ : 0 < 2) } \u2260 { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n\u22a2 inner (Matrix.vecCons x ![y] { val := 0, isLt := (_ : 0 < 2) })\n      (Matrix.vecCons x ![y] { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =\n    0", "state_after": "no goals"}, {"tactic": "simpa [real_inner_comm] using h", "annotated_tactic": ["simpa [<a>real_inner_comm</a>] using h", [{"full_name": "real_inner_comm", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [442, 9], "def_end_pos": [442, 24]}]], "state_before": "case tail.head.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\nhij : { val := 1, isLt := (_ : (fun a => a < 2) 1) } \u2260 { val := 0, isLt := (_ : 0 < 2) }\n\u22a2 inner (Matrix.vecCons x ![y] { val := 1, isLt := (_ : (fun a => a < 2) 1) })\n      (Matrix.vecCons x ![y] { val := 0, isLt := (_ : 0 < 2) }) =\n    0", "state_after": "no goals"}, {"tactic": "simp_all", "annotated_tactic": ["simp_all", []], "state_before": "case tail.head.tail.head\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\nh : inner x y = 0\nhij : { val := 1, isLt := (_ : (fun a => a < 2) 1) } \u2260 { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n\u22a2 inner (Matrix.vecCons x ![y] { val := 1, isLt := (_ : (fun a => a < 2) 1) })\n      (Matrix.vecCons x ![y] { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =\n    0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean", "full_name": "LinearMap.SeparatingLeft.toMatrix\u2082", "start": [733, 1], "end": [735, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.pow_cast_right", "start": [605, 1], "end": [606, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Separation.lean", "full_name": "t1Space_iff_continuous_cofinite_of", "start": [512, 1], "end": [514, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecialFunctions/Integrals.lean", "full_name": "integral_sin_pow", "start": [647, 1], "end": [655, 7], "traced_tactics": [{"tactic": "have : n + 2 \u2260 0 := by linarith", "annotated_tactic": ["have : n + 2 \u2260 0 := by linarith", []], "state_before": "a b : \u211d\nn : \u2115\n\u22a2 \u222b (x : \u211d) in a..b, sin x ^ (n + 2) =\n    (sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b) / (\u2191n + 2) + (\u2191n + 1) / (\u2191n + 2) * \u222b (x : \u211d) in a..b, sin x ^ n", "state_after": "a b : \u211d\nn : \u2115\nthis : n + 2 \u2260 0\n\u22a2 \u222b (x : \u211d) in a..b, sin x ^ (n + 2) =\n    (sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b) / (\u2191n + 2) + (\u2191n + 1) / (\u2191n + 2) * \u222b (x : \u211d) in a..b, sin x ^ n"}, {"tactic": "have : (n : \u211d) + 2 \u2260 0 := by norm_cast", "annotated_tactic": ["have : (n : \u211d) + 2 \u2260 0 := by norm_cast", []], "state_before": "a b : \u211d\nn : \u2115\nthis : n + 2 \u2260 0\n\u22a2 \u222b (x : \u211d) in a..b, sin x ^ (n + 2) =\n    (sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b) / (\u2191n + 2) + (\u2191n + 1) / (\u2191n + 2) * \u222b (x : \u211d) in a..b, sin x ^ n", "state_after": "a b : \u211d\nn : \u2115\nthis\u271d : n + 2 \u2260 0\nthis : \u2191n + 2 \u2260 0\n\u22a2 \u222b (x : \u211d) in a..b, sin x ^ (n + 2) =\n    (sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b) / (\u2191n + 2) + (\u2191n + 1) / (\u2191n + 2) * \u222b (x : \u211d) in a..b, sin x ^ n"}, {"tactic": "field_simp", "annotated_tactic": ["field_simp", []], "state_before": "a b : \u211d\nn : \u2115\nthis\u271d : n + 2 \u2260 0\nthis : \u2191n + 2 \u2260 0\n\u22a2 \u222b (x : \u211d) in a..b, sin x ^ (n + 2) =\n    (sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b) / (\u2191n + 2) + (\u2191n + 1) / (\u2191n + 2) * \u222b (x : \u211d) in a..b, sin x ^ n", "state_after": "a b : \u211d\nn : \u2115\nthis\u271d : n + 2 \u2260 0\nthis : \u2191n + 2 \u2260 0\n\u22a2 (\u222b (x : \u211d) in a..b, sin x ^ (n + 2)) * (\u2191n + 2) =\n    sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b + (\u2191n + 1) * \u222b (x : \u211d) in a..b, sin x ^ n"}, {"tactic": "convert eq_sub_iff_add_eq.mp (integral_sin_pow_aux n) using 1", "annotated_tactic": ["convert eq_sub_iff_add_eq.mp (<a>integral_sin_pow_aux</a> n) using 1", [{"full_name": "integral_sin_pow_aux", "def_path": "Mathlib/Analysis/SpecialFunctions/Integrals.lean", "def_pos": [614, 9], "def_end_pos": [614, 29]}]], "state_before": "a b : \u211d\nn : \u2115\nthis\u271d : n + 2 \u2260 0\nthis : \u2191n + 2 \u2260 0\n\u22a2 (\u222b (x : \u211d) in a..b, sin x ^ (n + 2)) * (\u2191n + 2) =\n    sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b + (\u2191n + 1) * \u222b (x : \u211d) in a..b, sin x ^ n", "state_after": "case h.e'_2\na b : \u211d\nn : \u2115\nthis\u271d : n + 2 \u2260 0\nthis : \u2191n + 2 \u2260 0\n\u22a2 (\u222b (x : \u211d) in a..b, sin x ^ (n + 2)) * (\u2191n + 2) =\n    (\u222b (x : \u211d) in a..b, sin x ^ (n + 2)) + (\u2191n + 1) * \u222b (x : \u211d) in a..b, sin x ^ (n + 2)"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case h.e'_2\na b : \u211d\nn : \u2115\nthis\u271d : n + 2 \u2260 0\nthis : \u2191n + 2 \u2260 0\n\u22a2 (\u222b (x : \u211d) in a..b, sin x ^ (n + 2)) * (\u2191n + 2) =\n    (\u222b (x : \u211d) in a..b, sin x ^ (n + 2)) + (\u2191n + 1) * \u222b (x : \u211d) in a..b, sin x ^ (n + 2)", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "a b : \u211d\nn : \u2115\n\u22a2 n + 2 \u2260 0", "state_after": "no goals"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "a b : \u211d\nn : \u2115\nthis : n + 2 \u2260 0\n\u22a2 \u2191n + 2 \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/DFinsupp/NeLocus.lean", "full_name": "DFinsupp.mapRange_neLocus_eq", "start": [110, 1], "end": [114, 47], "traced_tactics": [{"tactic": "ext a", "annotated_tactic": ["ext a", []], "state_before": "\u03b1 : Type u_1\nN : \u03b1 \u2192 Type u_2\ninst\u271d\u2075 : DecidableEq \u03b1\nM : \u03b1 \u2192 Type u_3\nP : \u03b1 \u2192 Type u_4\ninst\u271d\u2074 : (a : \u03b1) \u2192 Zero (N a)\ninst\u271d\u00b3 : (a : \u03b1) \u2192 Zero (M a)\ninst\u271d\u00b2 : (a : \u03b1) \u2192 Zero (P a)\ninst\u271d\u00b9 : (a : \u03b1) \u2192 DecidableEq (N a)\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (M a)\nf g : \u03a0\u2080 (a : \u03b1), N a\nF : (a : \u03b1) \u2192 N a \u2192 M a\nF0 : \u2200 (a : \u03b1), F a 0 = 0\nhF : \u2200 (a : \u03b1), Function.Injective (F a)\n\u22a2 neLocus (mapRange F F0 f) (mapRange F F0 g) = neLocus f g", "state_after": "case a\n\u03b1 : Type u_1\nN : \u03b1 \u2192 Type u_2\ninst\u271d\u2075 : DecidableEq \u03b1\nM : \u03b1 \u2192 Type u_3\nP : \u03b1 \u2192 Type u_4\ninst\u271d\u2074 : (a : \u03b1) \u2192 Zero (N a)\ninst\u271d\u00b3 : (a : \u03b1) \u2192 Zero (M a)\ninst\u271d\u00b2 : (a : \u03b1) \u2192 Zero (P a)\ninst\u271d\u00b9 : (a : \u03b1) \u2192 DecidableEq (N a)\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (M a)\nf g : \u03a0\u2080 (a : \u03b1), N a\nF : (a : \u03b1) \u2192 N a \u2192 M a\nF0 : \u2200 (a : \u03b1), F a 0 = 0\nhF : \u2200 (a : \u03b1), Function.Injective (F a)\na : \u03b1\n\u22a2 a \u2208 neLocus (mapRange F F0 f) (mapRange F F0 g) \u2194 a \u2208 neLocus f g"}, {"tactic": "simpa only [mem_neLocus] using (hF a).ne_iff", "annotated_tactic": ["simpa only [<a>mem_neLocus</a>] using (hF a).<a>ne_iff</a>", [{"full_name": "DFinsupp.mem_neLocus", "def_path": "Mathlib/Data/DFinsupp/NeLocus.lean", "def_pos": [41, 9], "def_end_pos": [41, 20]}, {"full_name": "Function.Injective.ne_iff", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [111, 9], "def_end_pos": [111, 25]}]], "state_before": "case a\n\u03b1 : Type u_1\nN : \u03b1 \u2192 Type u_2\ninst\u271d\u2075 : DecidableEq \u03b1\nM : \u03b1 \u2192 Type u_3\nP : \u03b1 \u2192 Type u_4\ninst\u271d\u2074 : (a : \u03b1) \u2192 Zero (N a)\ninst\u271d\u00b3 : (a : \u03b1) \u2192 Zero (M a)\ninst\u271d\u00b2 : (a : \u03b1) \u2192 Zero (P a)\ninst\u271d\u00b9 : (a : \u03b1) \u2192 DecidableEq (N a)\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (M a)\nf g : \u03a0\u2080 (a : \u03b1), N a\nF : (a : \u03b1) \u2192 N a \u2192 M a\nF0 : \u2200 (a : \u03b1), F a 0 = 0\nhF : \u2200 (a : \u03b1), Function.Injective (F a)\na : \u03b1\n\u22a2 a \u2208 neLocus (mapRange F F0 f) (mapRange F F0 g) \u2194 a \u2208 neLocus f g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.ToPartrec.Code.id_eval", "start": [183, 1], "end": [183, 57], "traced_tactics": [{"tactic": "simp [id]", "annotated_tactic": ["simp [<a>id</a>]", [{"full_name": "Turing.ToPartrec.Code.id", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [178, 5], "def_end_pos": [178, 7]}]], "state_before": "v : List \u2115\n\u22a2 eval id v = pure v", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/SuccPred/IntervalSucc.lean", "full_name": "Antitone.pairwise_disjoint_on_Ioc_pred", "start": [123, 1], "end": [125, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Lipschitz.lean", "full_name": "LipschitzWith.iterate", "start": [291, 11], "end": [293, 75], "traced_tactics": [{"tactic": "simpa only [pow_zero] using LipschitzWith.id", "annotated_tactic": ["simpa only [<a>pow_zero</a>] using <a>LipschitzWith.id</a>", [{"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [639, 9], "def_end_pos": [639, 17]}, {"full_name": "LipschitzWith.id", "def_path": "Mathlib/Topology/MetricSpace/Lipschitz.lean", "def_pos": [217, 19], "def_end_pos": [217, 21]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b1\nhf : LipschitzWith K f\n\u22a2 LipschitzWith (K ^ 0) f^[0]", "state_after": "no goals"}, {"tactic": "rw [pow_succ']", "annotated_tactic": ["rw [<a>pow_succ'</a>]", [{"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [213, 9], "def_end_pos": [213, 25]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b1\nhf : LipschitzWith K f\nn : \u2115\n\u22a2 LipschitzWith (K ^ (n + 1)) f^[n + 1]", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b1\nhf : LipschitzWith K f\nn : \u2115\n\u22a2 LipschitzWith (K ^ n * K) f^[n + 1]"}, {"tactic": "exact (LipschitzWith.iterate hf n).comp hf", "annotated_tactic": ["exact (LipschitzWith.iterate hf n).<a>comp</a> hf", [{"full_name": "LipschitzWith.comp", "def_path": "Mathlib/Topology/MetricSpace/Lipschitz.lean", "def_pos": [243, 19], "def_end_pos": [243, 23]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nx y : \u03b1\nr : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b1\nhf : LipschitzWith K f\nn : \u2115\n\u22a2 LipschitzWith (K ^ n * K) f^[n + 1]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Constructions.lean", "full_name": "continuous_quotient_mk'", "start": [1171, 1], "end": [1172, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Sigma/Basic.lean", "full_name": "Sigma.ext", "start": [68, 1], "end": [69, 46], "traced_tactics": [{"tactic": "cases x\u2080", "annotated_tactic": ["cases x\u2080", []], "state_before": "\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\nx\u2080 x\u2081 : Sigma \u03b2\nh\u2080 : x\u2080.fst = x\u2081.fst\nh\u2081 : HEq x\u2080.snd x\u2081.snd\n\u22a2 x\u2080 = x\u2081", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\nx\u2081 : Sigma \u03b2\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\nh\u2080 : { fst := fst\u271d, snd := snd\u271d }.fst = x\u2081.fst\nh\u2081 : HEq { fst := fst\u271d, snd := snd\u271d }.snd x\u2081.snd\n\u22a2 { fst := fst\u271d, snd := snd\u271d } = x\u2081"}, {"tactic": "cases x\u2081", "annotated_tactic": ["cases x\u2081", []], "state_before": "case mk\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\nx\u2081 : Sigma \u03b2\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\nh\u2080 : { fst := fst\u271d, snd := snd\u271d }.fst = x\u2081.fst\nh\u2081 : HEq { fst := fst\u271d, snd := snd\u271d }.snd x\u2081.snd\n\u22a2 { fst := fst\u271d, snd := snd\u271d } = x\u2081", "state_after": "case mk.mk\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\nfst\u271d\u00b9 : \u03b1\nsnd\u271d\u00b9 : \u03b2 fst\u271d\u00b9\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\nh\u2080 : { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.fst = { fst := fst\u271d, snd := snd\u271d }.fst\nh\u2081 : HEq { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.snd { fst := fst\u271d, snd := snd\u271d }.snd\n\u22a2 { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 } = { fst := fst\u271d, snd := snd\u271d }"}, {"tactic": "cases h\u2080", "annotated_tactic": ["cases h\u2080", []], "state_before": "case mk.mk\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\nfst\u271d\u00b9 : \u03b1\nsnd\u271d\u00b9 : \u03b2 fst\u271d\u00b9\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\nh\u2080 : { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.fst = { fst := fst\u271d, snd := snd\u271d }.fst\nh\u2081 : HEq { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.snd { fst := fst\u271d, snd := snd\u271d }.snd\n\u22a2 { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 } = { fst := fst\u271d, snd := snd\u271d }", "state_after": "case mk.mk.refl\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\nfst\u271d : \u03b1\nsnd\u271d\u00b9 snd\u271d : \u03b2 fst\u271d\nh\u2081 : HEq { fst := fst\u271d, snd := snd\u271d\u00b9 }.snd { fst := fst\u271d, snd := snd\u271d }.snd\n\u22a2 { fst := fst\u271d, snd := snd\u271d\u00b9 } = { fst := fst\u271d, snd := snd\u271d }"}, {"tactic": "cases h\u2081", "annotated_tactic": ["cases h\u2081", []], "state_before": "case mk.mk.refl\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\nfst\u271d : \u03b1\nsnd\u271d\u00b9 snd\u271d : \u03b2 fst\u271d\nh\u2081 : HEq { fst := fst\u271d, snd := snd\u271d\u00b9 }.snd { fst := fst\u271d, snd := snd\u271d }.snd\n\u22a2 { fst := fst\u271d, snd := snd\u271d\u00b9 } = { fst := fst\u271d, snd := snd\u271d }", "state_after": "case mk.mk.refl.refl\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\n\u22a2 { fst := fst\u271d, snd := snd\u271d } = { fst := fst\u271d, snd := snd\u271d }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case mk.mk.refl.refl\n\u03b1 : Type u_1\n\u03b1\u2081 : Type u_2\n\u03b1\u2082 : Type u_3\n\u03b2 : \u03b1 \u2192 Type u_4\n\u03b2\u2081 : \u03b1\u2081 \u2192 Type u_5\n\u03b2\u2082 : \u03b1\u2082 \u2192 Type u_6\nfst\u271d : \u03b1\nsnd\u271d : \u03b2 fst\u271d\n\u22a2 { fst := fst\u271d, snd := snd\u271d } = { fst := fst\u271d, snd := snd\u271d }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Ideal/Operations.lean", "full_name": "Ideal.comap_of_equiv", "start": [1787, 1], "end": [1790, 91], "traced_tactics": [{"tactic": "rw [\u2190 RingEquiv.toRingHom_eq_coe, \u2190 RingEquiv.toRingHom_eq_coe, comap_comap,\n  RingEquiv.toRingHom_eq_coe, RingEquiv.toRingHom_eq_coe, RingEquiv.symm_comp, comap_id]", "annotated_tactic": ["rw [\u2190 <a>RingEquiv.toRingHom_eq_coe</a>, \u2190 <a>RingEquiv.toRingHom_eq_coe</a>, <a>comap_comap</a>,\n    <a>RingEquiv.toRingHom_eq_coe</a>, <a>RingEquiv.toRingHom_eq_coe</a>, <a>RingEquiv.symm_comp</a>, <a>comap_id</a>]", [{"full_name": "RingEquiv.toRingHom_eq_coe", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [696, 17], "def_end_pos": [696, 33]}, {"full_name": "RingEquiv.toRingHom_eq_coe", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [696, 17], "def_end_pos": [696, 33]}, {"full_name": "Ideal.comap_comap", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [1470, 9], "def_end_pos": [1470, 20]}, {"full_name": "RingEquiv.toRingHom_eq_coe", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [696, 17], "def_end_pos": [696, 33]}, {"full_name": "RingEquiv.toRingHom_eq_coe", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [696, 17], "def_end_pos": [696, 33]}, {"full_name": "RingEquiv.symm_comp", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [580, 9], "def_end_pos": [580, 18]}, {"full_name": "Ideal.comap_id", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [1461, 9], "def_end_pos": [1461, 17]}]], "state_before": "R : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : RingHomClass F R S\nf\u271d : F\nI\u271d I : Ideal R\nf : R \u2243+* S\n\u22a2 comap (\u2191f) (comap (\u2191(RingEquiv.symm f)) I) = I", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.smul_finset_union", "start": [1659, 1], "end": [1660, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/ClassNumber/Finite.lean", "full_name": "ClassGroup.norm_lt", "start": [111, 1], "end": [134, 50], "traced_tactics": [{"tactic": "obtain \u27e8i\u27e9 := bS.index_nonempty", "annotated_tactic": ["obtain \u27e8i\u27e9 := bS.index_nonempty", []], "state_before": "R : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\n\u22a2 \u2191(\u2191abv (\u2191(Algebra.norm R) a)) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9", "state_after": "case intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\n\u22a2 \u2191(\u2191abv (\u2191(Algebra.norm R) a)) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9"}, {"tactic": "have him : (Finset.univ.image fun k => abv (bS.repr a k)).Nonempty :=\n  \u27e8_, Finset.mem_image.mpr \u27e8i, Finset.mem_univ _, rfl\u27e9\u27e9", "annotated_tactic": ["have him : (Finset.univ.image fun k => abv (bS.repr a k)).<a>Nonempty</a> :=\n    \u27e8_, Finset.mem_image.mpr \u27e8i, <a>Finset.mem_univ</a> _, <a>rfl</a>\u27e9\u27e9", [{"full_name": "Finset.Nonempty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [474, 15], "def_end_pos": [474, 23]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\n\u22a2 \u2191(\u2191abv (\u2191(Algebra.norm R) a)) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9", "state_after": "case intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\n\u22a2 \u2191(\u2191abv (\u2191(Algebra.norm R) a)) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9"}, {"tactic": "set y' : \u2124 := Finset.max' _ him with y'_def", "annotated_tactic": ["set y' : \u2124 := <a>Finset.max'</a> _ him with y'_def", [{"full_name": "Finset.max'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1409, 5], "def_end_pos": [1409, 9]}]], "state_before": "case intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\n\u22a2 \u2191(\u2191abv (\u2191(Algebra.norm R) a)) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9", "state_after": "case intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\n\u22a2 \u2191(\u2191abv (\u2191(Algebra.norm R) a)) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9"}, {"tactic": "have hy' : \u2200 k, abv (bS.repr a k) \u2264 y' := by\n  intro k\n  exact @Finset.le_max' \u2124 _ _ _ (Finset.mem_image.mpr \u27e8k, Finset.mem_univ _, rfl\u27e9)", "annotated_tactic": ["have hy' : \u2200 k, abv (bS.repr a k) \u2264 y' := by\n    intro k\n    exact @<a>Finset.le_max'</a> \u2124 _ _ _ (Finset.mem_image.mpr \u27e8k, <a>Finset.mem_univ</a> _, <a>rfl</a>\u27e9)", [{"full_name": "Finset.le_max'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1445, 9], "def_end_pos": [1445, 16]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\n\u22a2 \u2191(\u2191abv (\u2191(Algebra.norm R) a)) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9", "state_after": "case intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\n\u22a2 \u2191(\u2191abv (\u2191(Algebra.norm R) a)) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9"}, {"tactic": "have : (y' : T) < y := by\n  rw [y'_def, \u2190\n    Finset.max'_image (show Monotone (_ : \u2124 \u2192 T) from fun x y h => Int.cast_le.mpr h)]\n  apply (Finset.max'_lt_iff _ (him.image _)).mpr\n  simp only [Finset.mem_image, exists_prop]\n  rintro _ \u27e8x, \u27e8k, -, rfl\u27e9, rfl\u27e9\n  exact hy k", "annotated_tactic": ["have : (y' : T) < y := by\n    rw [y'_def, \u2190\n      <a>Finset.max'_image</a> (show <a>Monotone</a> (_ : \u2124 \u2192 T) from fun x y h => Int.cast_le.mpr h)]\n    apply (<a>Finset.max'_lt_iff</a> _ (him.image _)).<a>mpr</a>\n    simp only [<a>Finset.mem_image</a>, <a>exists_prop</a>]\n    rintro _ \u27e8x, \u27e8k, -, rfl\u27e9, rfl\u27e9\n    exact hy k", [{"full_name": "Finset.max'_image", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1589, 9], "def_end_pos": [1589, 19]}, {"full_name": "Monotone", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [77, 5], "def_end_pos": [77, 13]}, {"full_name": "Finset.max'_lt_iff", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1463, 9], "def_end_pos": [1463, 20]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}]], "state_before": "case intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\n\u22a2 \u2191(\u2191abv (\u2191(Algebra.norm R) a)) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9", "state_after": "case intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\nthis : \u2191y' < y\n\u22a2 \u2191(\u2191abv (\u2191(Algebra.norm R) a)) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9"}, {"tactic": "have y'_nonneg : 0 \u2264 y' := le_trans (abv.nonneg _) (hy' i)", "annotated_tactic": ["have y'_nonneg : 0 \u2264 y' := <a>le_trans</a> (abv.nonneg _) (hy' i)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "state_before": "case intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\nthis : \u2191y' < y\n\u22a2 \u2191(\u2191abv (\u2191(Algebra.norm R) a)) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9", "state_after": "case intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\nthis : \u2191y' < y\ny'_nonneg : 0 \u2264 y'\n\u22a2 \u2191(\u2191abv (\u2191(Algebra.norm R) a)) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9"}, {"tactic": "apply (Int.cast_le.mpr (norm_le abv bS a hy')).trans_lt", "annotated_tactic": ["apply (Int.cast_le.mpr (<a>norm_le</a> abv bS a hy')).<a>trans_lt</a>", [{"full_name": "ClassGroup.norm_le", "def_path": "Mathlib/NumberTheory/ClassNumber/Finite.lean", "def_pos": [88, 9], "def_end_pos": [88, 16]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}]], "state_before": "case intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\nthis : \u2191y' < y\ny'_nonneg : 0 \u2264 y'\n\u22a2 \u2191(\u2191abv (\u2191(Algebra.norm R) a)) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9", "state_after": "case intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\nthis : \u2191y' < y\ny'_nonneg : 0 \u2264 y'\n\u22a2 \u2191(normBound abv bS * y' ^ Fintype.card \u03b9) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9"}, {"tactic": "simp only [Int.cast_mul, Int.cast_pow]", "annotated_tactic": ["simp only [<a>Int.cast_mul</a>, <a>Int.cast_pow</a>]", [{"full_name": "Int.cast_mul", "def_path": "Mathlib/Data/Int/Basic.lean", "def_pos": [67, 9], "def_end_pos": [67, 17]}, {"full_name": "Int.cast_pow", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [624, 9], "def_end_pos": [624, 21]}]], "state_before": "case intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\nthis : \u2191y' < y\ny'_nonneg : 0 \u2264 y'\n\u22a2 \u2191(normBound abv bS * y' ^ Fintype.card \u03b9) < \u2191(normBound abv bS) * y ^ Fintype.card \u03b9", "state_after": "case intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\nthis : \u2191y' < y\ny'_nonneg : 0 \u2264 y'\n\u22a2 \u2191(normBound abv bS) *\n      \u2191(Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him) ^ Fintype.card \u03b9 <\n    \u2191(normBound abv bS) * y ^ Fintype.card \u03b9"}, {"tactic": "apply mul_lt_mul' le_rfl", "annotated_tactic": ["apply <a>mul_lt_mul'</a> <a>le_rfl</a>", [{"full_name": "mul_lt_mul'", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [525, 9], "def_end_pos": [525, 20]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\nthis : \u2191y' < y\ny'_nonneg : 0 \u2264 y'\n\u22a2 \u2191(normBound abv bS) *\n      \u2191(Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him) ^ Fintype.card \u03b9 <\n    \u2191(normBound abv bS) * y ^ Fintype.card \u03b9", "state_after": "case intro.hbd\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\nthis : \u2191y' < y\ny'_nonneg : 0 \u2264 y'\n\u22a2 \u2191(Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him) ^ Fintype.card \u03b9 < y ^ Fintype.card \u03b9\n\ncase intro.hb\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\nthis : \u2191y' < y\ny'_nonneg : 0 \u2264 y'\n\u22a2 0 \u2264 \u2191(Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him) ^ Fintype.card \u03b9\n\ncase intro.hc\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\nthis : \u2191y' < y\ny'_nonneg : 0 \u2264 y'\n\u22a2 0 < \u2191(normBound abv bS)"}, {"tactic": "intro k", "annotated_tactic": ["intro k", []], "state_before": "R : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\n\u22a2 \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'", "state_after": "R : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nk : \u03b9\n\u22a2 \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'"}, {"tactic": "exact @Finset.le_max' \u2124 _ _ _ (Finset.mem_image.mpr \u27e8k, Finset.mem_univ _, rfl\u27e9)", "annotated_tactic": ["exact @<a>Finset.le_max'</a> \u2124 _ _ _ (Finset.mem_image.mpr \u27e8k, <a>Finset.mem_univ</a> _, <a>rfl</a>\u27e9)", [{"full_name": "Finset.le_max'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1445, 9], "def_end_pos": [1445, 16]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "R : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nk : \u03b9\n\u22a2 \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'", "state_after": "no goals"}, {"tactic": "rw [y'_def, \u2190\n  Finset.max'_image (show Monotone (_ : \u2124 \u2192 T) from fun x y h => Int.cast_le.mpr h)]", "annotated_tactic": ["rw [y'_def, \u2190\n      <a>Finset.max'_image</a> (show <a>Monotone</a> (_ : \u2124 \u2192 T) from fun x y h => Int.cast_le.mpr h)]", [{"full_name": "Finset.max'_image", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1589, 9], "def_end_pos": [1589, 19]}, {"full_name": "Monotone", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [77, 5], "def_end_pos": [77, 13]}]], "state_before": "R : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\n\u22a2 \u2191y' < y", "state_after": "R : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\n\u22a2 Finset.max' (Finset.image (fun x => \u2191x) (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)) ?h < y\n\ncase h\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\n\u22a2 Finset.Nonempty (Finset.image (fun x => \u2191x) (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ))"}, {"tactic": "apply (Finset.max'_lt_iff _ (him.image _)).mpr", "annotated_tactic": ["apply (<a>Finset.max'_lt_iff</a> _ (him.image _)).<a>mpr</a>", [{"full_name": "Finset.max'_lt_iff", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1463, 9], "def_end_pos": [1463, 20]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "R : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\n\u22a2 Finset.max' (Finset.image (fun x => \u2191x) (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)) ?h < y\n\ncase h\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\n\u22a2 Finset.Nonempty (Finset.image (fun x => \u2191x) (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ))", "state_after": "R : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\n\u22a2 \u2200 (y_1 : T), y_1 \u2208 Finset.image (fun x => \u2191x) (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) \u2192 y_1 < y"}, {"tactic": "simp only [Finset.mem_image, exists_prop]", "annotated_tactic": ["simp only [<a>Finset.mem_image</a>, <a>exists_prop</a>]", [{"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}]], "state_before": "R : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\n\u22a2 \u2200 (y_1 : T), y_1 \u2208 Finset.image (fun x => \u2191x) (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) \u2192 y_1 < y", "state_after": "R : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\n\u22a2 \u2200 (y_1 : T), (\u2203 a_1, (\u2203 a_2, a_2 \u2208 Finset.univ \u2227 \u2191abv (\u2191(\u2191bS.repr a) a_2) = a_1) \u2227 \u2191a_1 = y_1) \u2192 y_1 < y"}, {"tactic": "rintro _ \u27e8x, \u27e8k, -, rfl\u27e9, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8x, \u27e8k, -, rfl\u27e9, rfl\u27e9", []], "state_before": "R : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\n\u22a2 \u2200 (y_1 : T), (\u2203 a_1, (\u2203 a_2, a_2 \u2208 Finset.univ \u2227 \u2191abv (\u2191(\u2191bS.repr a) a_2) = a_1) \u2227 \u2191a_1 = y_1) \u2192 y_1 < y", "state_after": "case intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\nk : \u03b9\n\u22a2 \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y"}, {"tactic": "exact hy k", "annotated_tactic": ["exact hy k", []], "state_before": "case intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\nk : \u03b9\n\u22a2 \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y", "state_after": "no goals"}, {"tactic": "exact pow_lt_pow_of_lt_left this (Int.cast_nonneg.mpr y'_nonneg) (Fintype.card_pos_iff.mpr \u27e8i\u27e9)", "annotated_tactic": ["exact <a>pow_lt_pow_of_lt_left</a> this (Int.cast_nonneg.mpr y'_nonneg) (Fintype.card_pos_iff.mpr \u27e8i\u27e9)", [{"full_name": "pow_lt_pow_of_lt_left", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [471, 9], "def_end_pos": [471, 30]}]], "state_before": "case intro.hbd\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\nthis : \u2191y' < y\ny'_nonneg : 0 \u2264 y'\n\u22a2 \u2191(Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him) ^ Fintype.card \u03b9 < y ^ Fintype.card \u03b9", "state_after": "no goals"}, {"tactic": "exact pow_nonneg (Int.cast_nonneg.mpr y'_nonneg) _", "annotated_tactic": ["exact <a>pow_nonneg</a> (Int.cast_nonneg.mpr y'_nonneg) _", [{"full_name": "pow_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [244, 9], "def_end_pos": [244, 19]}]], "state_before": "case intro.hb\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\nthis : \u2191y' < y\ny'_nonneg : 0 \u2264 y'\n\u22a2 0 \u2264 \u2191(Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him) ^ Fintype.card \u03b9", "state_after": "no goals"}, {"tactic": "exact Int.cast_pos.mpr (normBound_pos abv bS)", "annotated_tactic": ["exact Int.cast_pos.mpr (<a>normBound_pos</a> abv bS)", [{"full_name": "ClassGroup.normBound_pos", "def_path": "Mathlib/NumberTheory/ClassNumber/Finite.lean", "def_pos": [70, 9], "def_end_pos": [70, 22]}]], "state_before": "case intro.hc\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst\u271d\u00b9\u2075 : EuclideanDomain R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : IsDomain S\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra R K\ninst\u271d\u2079 : IsFractionRing R K\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : FiniteDimensional K L\ninst\u271d\u2076 : IsSeparable K L\nalgRL : Algebra R L\ninst\u271d\u2075 : IsScalarTower R K L\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R \u2124\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nbS : Basis \u03b9 R S\nT : Type u_6\ninst\u271d : LinearOrderedRing T\na : S\ny : T\nhy : \u2200 (k : \u03b9), \u2191(\u2191abv (\u2191(\u2191bS.repr a) k)) < y\ni : \u03b9\nhim : Finset.Nonempty (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ)\ny' : \u2124 := Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\ny'_def : y' = Finset.max' (Finset.image (fun k => \u2191abv (\u2191(\u2191bS.repr a) k)) Finset.univ) him\nhy' : \u2200 (k : \u03b9), \u2191abv (\u2191(\u2191bS.repr a) k) \u2264 y'\nthis : \u2191y' < y\ny'_nonneg : 0 \u2264 y'\n\u22a2 0 < \u2191(normBound abv bS)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "ZNum.div_to_int", "start": [1724, 1], "end": [1743, 33], "traced_tactics": [{"tactic": "simp [Int.ediv_zero]", "annotated_tactic": ["simp [<a>Int.ediv_zero</a>]", [{"full_name": "Int.ediv_zero", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [46, 33], "def_end_pos": [46, 42]}]], "state_before": "\u22a2 \u2191(0 / 0) = \u21910 / \u21910", "state_after": "no goals"}, {"tactic": "rw [\u2190 Num.to_nat_to_int]", "annotated_tactic": ["rw [\u2190 <a>Num.to_nat_to_int</a>]", [{"full_name": "Num.to_nat_to_int", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [479, 9], "def_end_pos": [479, 22]}]], "state_before": "n d : PosNum\n\u22a2 \u2191(PosNum.div' n d) = \u2191(pos n) / \u2191(pos d)", "state_after": "n d : PosNum\n\u22a2 \u2191\u2191(PosNum.div' n d) = \u2191(pos n) / \u2191(pos d)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "n d : PosNum\n\u22a2 \u2191\u2191(PosNum.div' n d) = \u2191(pos n) / \u2191(pos d)", "state_after": "no goals"}, {"tactic": "rw [\u2190 Num.to_nat_to_int]", "annotated_tactic": ["rw [\u2190 <a>Num.to_nat_to_int</a>]", [{"full_name": "Num.to_nat_to_int", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [479, 9], "def_end_pos": [479, 22]}]], "state_before": "n d : PosNum\n\u22a2 -\u2191(PosNum.div' n d) = \u2191(pos n) / \u2191(neg d)", "state_after": "n d : PosNum\n\u22a2 -\u2191\u2191(PosNum.div' n d) = \u2191(pos n) / \u2191(neg d)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "n d : PosNum\n\u22a2 -\u2191\u2191(PosNum.div' n d) = \u2191(pos n) / \u2191(neg d)", "state_after": "no goals"}, {"tactic": "rw [n.to_int_eq_succ_pred, d.to_int_eq_succ_pred, \u2190 PosNum.to_nat_to_int, Num.succ'_to_nat,\n  Num.div_to_nat]", "annotated_tactic": ["rw [n.to_int_eq_succ_pred, d.to_int_eq_succ_pred, \u2190 <a>PosNum.to_nat_to_int</a>, <a>Num.succ'_to_nat</a>,\n        <a>Num.div_to_nat</a>]", [{"full_name": "PosNum.to_nat_to_int", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [64, 9], "def_end_pos": [64, 22]}, {"full_name": "Num.succ'_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [291, 9], "def_end_pos": [291, 21]}, {"full_name": "Num.div_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1639, 9], "def_end_pos": [1639, 19]}]], "state_before": "n d : PosNum\n\u22a2 -\u2191(Num.succ' (PosNum.pred' n / Num.pos d)) = -\u2191n / \u2191d", "state_after": "n d : PosNum\n\u22a2 -\u2191(\u2191(PosNum.pred' n) / \u2191(Num.pos d) + 1) = -(\u2191\u2191(PosNum.pred' n) + 1) / (\u2191\u2191(PosNum.pred' d) + 1)"}, {"tactic": "change -[n.pred' / \u2191d+1] = -[n.pred' / (d.pred' + 1)+1]", "annotated_tactic": ["change -[n.pred' / \u2191d+1] = -[n.pred' / (d.pred' + 1)+1]", []], "state_before": "n d : PosNum\n\u22a2 -\u2191(\u2191(PosNum.pred' n) / \u2191(Num.pos d) + 1) = -(\u2191\u2191(PosNum.pred' n) + 1) / (\u2191\u2191(PosNum.pred' d) + 1)", "state_after": "n d : PosNum\n\u22a2 -[\u2191(PosNum.pred' n) / \u2191d+1] = -[\u2191(PosNum.pred' n) / (\u2191(PosNum.pred' d) + 1)+1]"}, {"tactic": "rw [d.to_nat_eq_succ_pred]", "annotated_tactic": ["rw [d.to_nat_eq_succ_pred]", []], "state_before": "n d : PosNum\n\u22a2 -[\u2191(PosNum.pred' n) / \u2191d+1] = -[\u2191(PosNum.pred' n) / (\u2191(PosNum.pred' d) + 1)+1]", "state_after": "no goals"}, {"tactic": "rw [n.to_int_eq_succ_pred, d.to_int_eq_succ_pred, \u2190 PosNum.to_nat_to_int, Num.succ'_to_nat,\n  Num.div_to_nat]", "annotated_tactic": ["rw [n.to_int_eq_succ_pred, d.to_int_eq_succ_pred, \u2190 <a>PosNum.to_nat_to_int</a>, <a>Num.succ'_to_nat</a>,\n        <a>Num.div_to_nat</a>]", [{"full_name": "PosNum.to_nat_to_int", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [64, 9], "def_end_pos": [64, 22]}, {"full_name": "Num.succ'_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [291, 9], "def_end_pos": [291, 21]}, {"full_name": "Num.div_to_nat", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1639, 9], "def_end_pos": [1639, 19]}]], "state_before": "n d : PosNum\n\u22a2 \u2191(Num.succ' (PosNum.pred' n / Num.pos d)) = -\u2191n / -\u2191d", "state_after": "n d : PosNum\n\u22a2 \u2191(\u2191(PosNum.pred' n) / \u2191(Num.pos d) + 1) = -(\u2191\u2191(PosNum.pred' n) + 1) / -(\u2191\u2191(PosNum.pred' d) + 1)"}, {"tactic": "change (Nat.succ (_ / d) : \u2124) = Nat.succ (n.pred' / (d.pred' + 1))", "annotated_tactic": ["change (<a>Nat.succ</a> (_ / d) : \u2124) = <a>Nat.succ</a> (n.pred' / (d.pred' + 1))", [{"full_name": "Nat.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}, {"full_name": "Nat.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}]], "state_before": "n d : PosNum\n\u22a2 \u2191(\u2191(PosNum.pred' n) / \u2191(Num.pos d) + 1) = -(\u2191\u2191(PosNum.pred' n) + 1) / -(\u2191\u2191(PosNum.pred' d) + 1)", "state_after": "n d : PosNum\n\u22a2 \u2191(Nat.succ (\u2191(PosNum.pred' n) / \u2191d)) = \u2191(Nat.succ (\u2191(PosNum.pred' n) / (\u2191(PosNum.pred' d) + 1)))"}, {"tactic": "rw [d.to_nat_eq_succ_pred]", "annotated_tactic": ["rw [d.to_nat_eq_succ_pred]", []], "state_before": "n d : PosNum\n\u22a2 \u2191(Nat.succ (\u2191(PosNum.pred' n) / \u2191d)) = \u2191(Nat.succ (\u2191(PosNum.pred' n) / (\u2191(PosNum.pred' d) + 1)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Category/GroupCat/Basic.lean", "full_name": "AddCommGroupCat.asHom_injective", "start": [358, 1], "end": [359, 89], "traced_tactics": [{"tactic": "convert congr_arg (fun k : AddCommGroupCat.of \u2124 \u27f6 G => (k : \u2124 \u2192 G) (1 : \u2124)) w <;> simp", "annotated_tactic": ["convert <a>congr_arg</a> (fun k : <a>AddCommGroupCat.of</a> \u2124 \u27f6 G => (k : \u2124 \u2192 G) (1 : \u2124)) w <;> simp", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "AddCommGroupCat.of", "def_path": "Mathlib/Algebra/Category/GroupCat/Basic.lean", "def_pos": [245, 3], "def_end_pos": [245, 14]}]], "state_before": "G : AddCommGroupCat\nh k : \u2191G\nw : asHom h = asHom k\n\u22a2 h = k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Monotone/Basic.lean", "full_name": "Subsingleton.antitone'", "start": [529, 1], "end": [530, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/LocalHomeomorph.lean", "full_name": "LocalHomeomorph.symm_image_target_inter_eq", "start": [295, 1], "end": [297, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/RingDivision.lean", "full_name": "Polynomial.Monic.prime_of_degree_eq_one", "start": [419, 1], "end": [421, 30], "traced_tactics": [{"tactic": "simpa [hm.leadingCoeff] using eq_X_add_C_of_degree_eq_one hp1", "annotated_tactic": ["simpa [hm.leadingCoeff] using <a>eq_X_add_C_of_degree_eq_one</a> hp1", [{"full_name": "Polynomial.eq_X_add_C_of_degree_eq_one", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [452, 9], "def_end_pos": [452, 36]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np q : R[X]\nhp1 : degree p = 1\nhm : Monic p\n\u22a2 p = X - \u2191C (-coeff p 0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Bases.lean", "full_name": "Filter.IsBasis.mem_filter_iff", "start": [216, 11], "end": [219, 30], "traced_tactics": [{"tactic": "simp only [IsBasis.filter, FilterBasis.mem_filter_iff, mem_filterBasis_iff,\n  exists_exists_and_eq_and]", "annotated_tactic": ["simp only [<a>IsBasis.filter</a>, <a>FilterBasis.mem_filter_iff</a>, <a>mem_filterBasis_iff</a>,\n    <a>exists_exists_and_eq_and</a>]", [{"full_name": "Filter.IsBasis.filter", "def_path": "Mathlib/Order/Filter/Bases.lean", "def_pos": [212, 15], "def_end_pos": [212, 21]}, {"full_name": "FilterBasis.mem_filter_iff", "def_path": "Mathlib/Order/Filter/Bases.lean", "def_pos": [175, 9], "def_end_pos": [175, 23]}, {"full_name": "Filter.IsBasis.mem_filterBasis_iff", "def_path": "Mathlib/Order/Filter/Bases.lean", "def_pos": [155, 9], "def_end_pos": [155, 28]}, {"full_name": "exists_exists_and_eq_and", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [786, 17], "def_end_pos": [786, 41]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nh : IsBasis p s\nU : Set \u03b1\n\u22a2 U \u2208 IsBasis.filter h \u2194 \u2203 i, p i \u2227 s i \u2286 U", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "full_name": "MonoidHomClass.isometry_iff_norm", "start": [883, 1], "end": [887, 20], "traced_tactics": [{"tactic": "simp only [isometry_iff_dist_eq, dist_eq_norm_div, \u2190 map_div]", "annotated_tactic": ["simp only [<a>isometry_iff_dist_eq</a>, <a>dist_eq_norm_div</a>, \u2190 <a>map_div</a>]", [{"full_name": "isometry_iff_dist_eq", "def_path": "Mathlib/Topology/MetricSpace/Isometry.lean", "def_pos": [47, 9], "def_end_pos": [47, 29]}, {"full_name": "dist_eq_norm_div", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [373, 9], "def_end_pos": [373, 25]}, {"full_name": "map_div", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [429, 9], "def_end_pos": [429, 16]}]], "state_before": "\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b3 : SeminormedGroup E\ninst\u271d\u00b2 : SeminormedGroup F\ninst\u271d\u00b9 : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : MonoidHomClass \ud835\udcd5 E F\nf : \ud835\udcd5\n\u22a2 Isometry \u2191f \u2194 \u2200 (x : E), \u2016\u2191f x\u2016 = \u2016x\u2016", "state_after": "\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b3 : SeminormedGroup E\ninst\u271d\u00b2 : SeminormedGroup F\ninst\u271d\u00b9 : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : MonoidHomClass \ud835\udcd5 E F\nf : \ud835\udcd5\n\u22a2 (\u2200 (x y : E), \u2016\u2191f (x / y)\u2016 = \u2016x / y\u2016) \u2194 \u2200 (x : E), \u2016\u2191f x\u2016 = \u2016x\u2016"}, {"tactic": "refine' \u27e8fun h x => _, fun h x y => h _\u27e9", "annotated_tactic": ["refine' \u27e8fun h x => _, fun h x y => h _\u27e9", []], "state_before": "\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b3 : SeminormedGroup E\ninst\u271d\u00b2 : SeminormedGroup F\ninst\u271d\u00b9 : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : MonoidHomClass \ud835\udcd5 E F\nf : \ud835\udcd5\n\u22a2 (\u2200 (x y : E), \u2016\u2191f (x / y)\u2016 = \u2016x / y\u2016) \u2194 \u2200 (x : E), \u2016\u2191f x\u2016 = \u2016x\u2016", "state_after": "\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b3 : SeminormedGroup E\ninst\u271d\u00b2 : SeminormedGroup F\ninst\u271d\u00b9 : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : MonoidHomClass \ud835\udcd5 E F\nf : \ud835\udcd5\nh : \u2200 (x y : E), \u2016\u2191f (x / y)\u2016 = \u2016x / y\u2016\nx : E\n\u22a2 \u2016\u2191f x\u2016 = \u2016x\u2016"}, {"tactic": "simpa using h x 1", "annotated_tactic": ["simpa using h x 1", []], "state_before": "\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b3 : SeminormedGroup E\ninst\u271d\u00b2 : SeminormedGroup F\ninst\u271d\u00b9 : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : MonoidHomClass \ud835\udcd5 E F\nf : \ud835\udcd5\nh : \u2200 (x y : E), \u2016\u2191f (x / y)\u2016 = \u2016x / y\u2016\nx : E\n\u22a2 \u2016\u2191f x\u2016 = \u2016x\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/NormedSpace/Star/Multiplier.lean", "full_name": "DoubleCentralizer.toProdMulOpposite_injective", "start": [333, 1], "end": [337, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Projection.lean", "full_name": "Submodule.mk_quotientEquivOfIsCompl_apply", "start": [92, 1], "end": [94, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.induction_on_max_value", "start": [1752, 1], "end": [1764, 66], "traced_tactics": [{"tactic": "induction' s using Finset.strongInductionOn with s ihs", "annotated_tactic": ["induction' s using <a>Finset.strongInductionOn</a> with s ihs", [{"full_name": "Finset.strongInductionOn", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [682, 5], "def_end_pos": [682, 22]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : DecidableEq \u03b9\nf : \u03b9 \u2192 \u03b1\np : Finset \u03b9 \u2192 Prop\ns : Finset \u03b9\nh0 : p \u2205\nstep : \u2200 (a : \u03b9) (s : Finset \u03b9), \u00aca \u2208 s \u2192 (\u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2264 f a) \u2192 p s \u2192 p (insert a s)\n\u22a2 p s", "state_after": "case a\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : DecidableEq \u03b9\nf : \u03b9 \u2192 \u03b1\np : Finset \u03b9 \u2192 Prop\nh0 : p \u2205\nstep : \u2200 (a : \u03b9) (s : Finset \u03b9), \u00aca \u2208 s \u2192 (\u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2264 f a) \u2192 p s \u2192 p (insert a s)\ns : Finset \u03b9\nihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t\n\u22a2 p s"}, {"tactic": "rcases (s.image f).eq_empty_or_nonempty with (hne | hne)", "annotated_tactic": ["rcases (s.image f).<a>eq_empty_or_nonempty</a> with (hne | hne)", [{"full_name": "Finset.eq_empty_or_nonempty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [613, 9], "def_end_pos": [613, 29]}]], "state_before": "case a\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : DecidableEq \u03b9\nf : \u03b9 \u2192 \u03b1\np : Finset \u03b9 \u2192 Prop\nh0 : p \u2205\nstep : \u2200 (a : \u03b9) (s : Finset \u03b9), \u00aca \u2208 s \u2192 (\u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2264 f a) \u2192 p s \u2192 p (insert a s)\ns : Finset \u03b9\nihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t\n\u22a2 p s", "state_after": "case a.inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : DecidableEq \u03b9\nf : \u03b9 \u2192 \u03b1\np : Finset \u03b9 \u2192 Prop\nh0 : p \u2205\nstep : \u2200 (a : \u03b9) (s : Finset \u03b9), \u00aca \u2208 s \u2192 (\u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2264 f a) \u2192 p s \u2192 p (insert a s)\ns : Finset \u03b9\nihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t\nhne : image f s = \u2205\n\u22a2 p s\n\ncase a.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : DecidableEq \u03b9\nf : \u03b9 \u2192 \u03b1\np : Finset \u03b9 \u2192 Prop\nh0 : p \u2205\nstep : \u2200 (a : \u03b9) (s : Finset \u03b9), \u00aca \u2208 s \u2192 (\u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2264 f a) \u2192 p s \u2192 p (insert a s)\ns : Finset \u03b9\nihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t\nhne : Finset.Nonempty (image f s)\n\u22a2 p s"}, {"tactic": "simp only [image_eq_empty] at hne", "annotated_tactic": ["simp only [<a>image_eq_empty</a>] at hne", [{"full_name": "Finset.image_eq_empty", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [543, 9], "def_end_pos": [543, 23]}]], "state_before": "case a.inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : DecidableEq \u03b9\nf : \u03b9 \u2192 \u03b1\np : Finset \u03b9 \u2192 Prop\nh0 : p \u2205\nstep : \u2200 (a : \u03b9) (s : Finset \u03b9), \u00aca \u2208 s \u2192 (\u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2264 f a) \u2192 p s \u2192 p (insert a s)\ns : Finset \u03b9\nihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t\nhne : image f s = \u2205\n\u22a2 p s", "state_after": "case a.inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : DecidableEq \u03b9\nf : \u03b9 \u2192 \u03b1\np : Finset \u03b9 \u2192 Prop\nh0 : p \u2205\nstep : \u2200 (a : \u03b9) (s : Finset \u03b9), \u00aca \u2208 s \u2192 (\u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2264 f a) \u2192 p s \u2192 p (insert a s)\ns : Finset \u03b9\nihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t\nhne : s = \u2205\n\u22a2 p s"}, {"tactic": "simp only [hne, h0]", "annotated_tactic": ["simp only [hne, h0]", []], "state_before": "case a.inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : DecidableEq \u03b9\nf : \u03b9 \u2192 \u03b1\np : Finset \u03b9 \u2192 Prop\nh0 : p \u2205\nstep : \u2200 (a : \u03b9) (s : Finset \u03b9), \u00aca \u2208 s \u2192 (\u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2264 f a) \u2192 p s \u2192 p (insert a s)\ns : Finset \u03b9\nihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t\nhne : s = \u2205\n\u22a2 p s", "state_after": "no goals"}, {"tactic": "have H : (s.image f).max' hne \u2208 s.image f := max'_mem (s.image f) hne", "annotated_tactic": ["have H : (s.image f).<a>max'</a> hne \u2208 s.image f := <a>max'_mem</a> (s.image f) hne", [{"full_name": "Finset.max'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1409, 5], "def_end_pos": [1409, 9]}, {"full_name": "Finset.max'_mem", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 17]}]], "state_before": "case a.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : DecidableEq \u03b9\nf : \u03b9 \u2192 \u03b1\np : Finset \u03b9 \u2192 Prop\nh0 : p \u2205\nstep : \u2200 (a : \u03b9) (s : Finset \u03b9), \u00aca \u2208 s \u2192 (\u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2264 f a) \u2192 p s \u2192 p (insert a s)\ns : Finset \u03b9\nihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t\nhne : Finset.Nonempty (image f s)\n\u22a2 p s", "state_after": "case a.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : DecidableEq \u03b9\nf : \u03b9 \u2192 \u03b1\np : Finset \u03b9 \u2192 Prop\nh0 : p \u2205\nstep : \u2200 (a : \u03b9) (s : Finset \u03b9), \u00aca \u2208 s \u2192 (\u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2264 f a) \u2192 p s \u2192 p (insert a s)\ns : Finset \u03b9\nihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t\nhne : Finset.Nonempty (image f s)\nH : max' (image f s) hne \u2208 image f s\n\u22a2 p s"}, {"tactic": "simp only [mem_image, exists_prop] at H", "annotated_tactic": ["simp only [<a>mem_image</a>, <a>exists_prop</a>] at H", [{"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}]], "state_before": "case a.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : DecidableEq \u03b9\nf : \u03b9 \u2192 \u03b1\np : Finset \u03b9 \u2192 Prop\nh0 : p \u2205\nstep : \u2200 (a : \u03b9) (s : Finset \u03b9), \u00aca \u2208 s \u2192 (\u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2264 f a) \u2192 p s \u2192 p (insert a s)\ns : Finset \u03b9\nihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t\nhne : Finset.Nonempty (image f s)\nH : max' (image f s) hne \u2208 image f s\n\u22a2 p s", "state_after": "case a.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : DecidableEq \u03b9\nf : \u03b9 \u2192 \u03b1\np : Finset \u03b9 \u2192 Prop\nh0 : p \u2205\nstep : \u2200 (a : \u03b9) (s : Finset \u03b9), \u00aca \u2208 s \u2192 (\u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2264 f a) \u2192 p s \u2192 p (insert a s)\ns : Finset \u03b9\nihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t\nhne : Finset.Nonempty (image f s)\nH : \u2203 a, a \u2208 s \u2227 f a = max' (image f s) hne\n\u22a2 p s"}, {"tactic": "rcases H with \u27e8a, has, hfa\u27e9", "annotated_tactic": ["rcases H with \u27e8a, has, hfa\u27e9", []], "state_before": "case a.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type 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(\u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2264 f a) \u2192 p s \u2192 p (insert a s)\ns : Finset \u03b9\nihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t\nhne : Finset.Nonempty (image f s)\na : \u03b9\nhas : a \u2208 s\nhfa : f a = max' (image f s) hne\n\u22a2 p s"}, {"tactic": "rw [\u2190 insert_erase has]", "annotated_tactic": ["rw [\u2190 <a>insert_erase</a> has]", [{"full_name": "Finset.insert_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1957, 9], "def_end_pos": [1957, 21]}]], "state_before": "case a.inr.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : DecidableEq \u03b9\nf : \u03b9 \u2192 \u03b1\np : Finset \u03b9 \u2192 Prop\nh0 : p \u2205\nstep : \u2200 (a : \u03b9) (s : Finset \u03b9), \u00aca \u2208 s \u2192 (\u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2264 f a) \u2192 p s \u2192 p (insert a s)\ns : Finset \u03b9\nihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t\nhne : Finset.Nonempty (image f s)\na : \u03b9\nhas : a \u2208 s\nhfa : f a = max' (image f s) hne\n\u22a2 p s", "state_after": "case a.inr.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : DecidableEq \u03b9\nf : \u03b9 \u2192 \u03b1\np : Finset \u03b9 \u2192 Prop\nh0 : p \u2205\nstep : \u2200 (a : \u03b9) (s : Finset \u03b9), \u00aca \u2208 s \u2192 (\u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2264 f a) \u2192 p s \u2192 p (insert a s)\ns : Finset \u03b9\nihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t\nhne : Finset.Nonempty (image f s)\na : \u03b9\nhas : a \u2208 s\nhfa : f a = max' (image f s) hne\n\u22a2 p (insert a (erase s a))"}, {"tactic": "refine' step _ _ (not_mem_erase a s) (fun x hx => _) (ihs _ <| erase_ssubset has)", "annotated_tactic": ["refine' step _ _ (<a>not_mem_erase</a> a s) (fun x hx => _) (ihs _ <| <a>erase_ssubset</a> has)", [{"full_name": "Finset.not_mem_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1891, 9], "def_end_pos": [1891, 22]}, {"full_name": "Finset.erase_ssubset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1990, 9], "def_end_pos": [1990, 22]}]], "state_before": "case a.inr.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : DecidableEq \u03b9\nf : \u03b9 \u2192 \u03b1\np : Finset \u03b9 \u2192 Prop\nh0 : p \u2205\nstep : \u2200 (a : \u03b9) (s : Finset \u03b9), \u00aca \u2208 s \u2192 (\u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2264 f a) \u2192 p s \u2192 p (insert a s)\ns : Finset \u03b9\nihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t\nhne : Finset.Nonempty (image f s)\na : \u03b9\nhas : a \u2208 s\nhfa : f a = max' (image f s) hne\n\u22a2 p (insert a (erase s a))", "state_after": "case a.inr.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\ninst\u271d : DecidableEq \u03b9\nf : \u03b9 \u2192 \u03b1\np : Finset \u03b9 \u2192 Prop\nh0 : p \u2205\nstep : \u2200 (a : \u03b9) (s : Finset \u03b9), \u00aca \u2208 s \u2192 (\u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2264 f a) \u2192 p s \u2192 p (insert a s)\ns : Finset \u03b9\nihs : \u2200 (t : Finset \u03b9), t \u2282 s \u2192 p t\nhne : Finset.Nonempty (image f s)\na : \u03b9\nhas : a \u2208 s\nhfa : f a = max' (image f s) hne\nx : \u03b9\nhx : x \u2208 erase s a\n\u22a2 f x \u2264 f a"}, {"tactic": "rw [hfa]", "annotated_tactic": ["rw [hfa]", []], "state_before": "case a.inr.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type 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"traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "full_name": "Finsupp.multiset_sum_sum_index", "start": [563, 1], "end": [568, 91], "traced_tactics": [{"tactic": "rw [Multiset.sum_cons, Multiset.map_cons, Multiset.sum_cons, sum_add_index' h\u2080 h\u2081, ih]", "annotated_tactic": ["rw [<a>Multiset.sum_cons</a>, <a>Multiset.map_cons</a>, <a>Multiset.sum_cons</a>, <a>sum_add_index'</a> h\u2080 h\u2081, ih]", [{"full_name": "Multiset.sum_cons", "def_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "Multiset.map_cons", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1176, 9], "def_end_pos": [1176, 17]}, {"full_name": "Multiset.sum_cons", "def_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "Finsupp.sum_add_index'", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [401, 3], "def_end_pos": [401, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nA : Type u_4\nB : Type u_5\nC : Type u_6\ninst\u271d\u2074 : AddCommMonoid A\ninst\u271d\u00b3 : AddCommMonoid B\ninst\u271d\u00b2 : AddCommMonoid C\nt : \u03b9 \u2192 A \u2192 C\nh0 : \u2200 (i : \u03b9), t i 0 = 0\nh1 : \u2200 (i : \u03b9) (x y : A), t i (x + y) = t i x + t i y\ns\u271d : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b9 \u2192\u2080 A\ni : \u03b9\ng : \u03b9 \u2192\u2080 A\nk : \u03b9 \u2192 A \u2192 \u03b3 \u2192 B\nx : \u03b3\n\u03b2 : Type u_7\nM : Type u_8\nM' : Type u_9\nN : Type u_10\nP : Type u_11\nG : Type u_12\nH : Type u_13\nR : Type u_14\nS : Type u_15\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : AddCommMonoid N\nf : Multiset (\u03b1 \u2192\u2080 M)\nh : \u03b1 \u2192 M \u2192 N\nh\u2080 : \u2200 (a : \u03b1), h a 0 = 0\nh\u2081 : \u2200 (a : \u03b1) (b\u2081 b\u2082 : M), h a (b\u2081 + b\u2082) = h a b\u2081 + h a b\u2082\na : \u03b1 \u2192\u2080 M\ns : Multiset (\u03b1 \u2192\u2080 M)\nih : sum (Multiset.sum s) h = Multiset.sum (Multiset.map (fun g => sum g h) s)\n\u22a2 sum (Multiset.sum (a ::\u2098 s)) h = Multiset.sum (Multiset.map (fun g => sum g h) (a ::\u2098 s))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Seminorm.lean", "full_name": "Seminorm.ext", "start": [137, 1], "end": [138, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "full_name": "List.isPrefixOf_cons\u2082", "start": [379, 1], "end": [380, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.toNat_eq_iff", "start": [1794, 1], "end": [1799, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Polynomial.lean", "full_name": "Polynomial.continuousWithinAt_aeval", "start": [88, 11], "end": [90, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/FieldTheory/RatFunc.lean", "full_name": "RatFunc.num_mul_eq_mul_denom_iff", "start": [1258, 1], "end": [1264, 45], "traced_tactics": [{"tactic": "rw [\u2190 (algebraMap_injective K).eq_iff, eq_div_iff (algebraMap_ne_zero hq)]", "annotated_tactic": ["rw [\u2190 (<a>algebraMap_injective</a> K).<a>eq_iff</a>, <a>eq_div_iff</a> (<a>algebraMap_ne_zero</a> hq)]", [{"full_name": "RatFunc.algebraMap_injective", "def_path": "Mathlib/FieldTheory/RatFunc.lean", "def_pos": [907, 9], "def_end_pos": [907, 29]}, {"full_name": "Function.Injective.eq_iff", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [95, 9], "def_end_pos": [95, 25]}, {"full_name": "eq_div_iff", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [93, 9], "def_end_pos": [93, 19]}, {"full_name": "RatFunc.algebraMap_ne_zero", "def_path": "Mathlib/FieldTheory/RatFunc.lean", "def_pos": [920, 9], "def_end_pos": [920, 27]}]], "state_before": "K : Type u\ninst\u271d : Field K\nx : RatFunc K\np q : K[X]\nhq : q \u2260 0\n\u22a2 num x * q = p * denom x \u2194 x = \u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q", "state_after": "K : Type u\ninst\u271d : Field K\nx : RatFunc K\np q : K[X]\nhq : q \u2260 0\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) (num x * q) = \u2191(algebraMap K[X] (RatFunc K)) (p * denom x) \u2194\n    x * \u2191(algebraMap K[X] (RatFunc K)) q = \u2191(algebraMap K[X] (RatFunc K)) p"}, {"tactic": "conv_rhs => rw [\u2190 num_div_denom x]", "annotated_tactic": ["conv_rhs => rw [\u2190 <a>num_div_denom</a> x]", [{"full_name": "RatFunc.num_div_denom", "def_path": "Mathlib/FieldTheory/RatFunc.lean", "def_pos": [1225, 9], "def_end_pos": [1225, 22]}]], "state_before": "K : Type u\ninst\u271d : Field K\nx : RatFunc K\np q : K[X]\nhq : q \u2260 0\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) (num x * q) = \u2191(algebraMap K[X] (RatFunc K)) (p * denom x) \u2194\n    x * \u2191(algebraMap K[X] (RatFunc K)) q = \u2191(algebraMap K[X] (RatFunc K)) p", "state_after": "K : Type u\ninst\u271d : Field K\nx : RatFunc K\np q : K[X]\nhq : q \u2260 0\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) (num x * q) = \u2191(algebraMap K[X] (RatFunc K)) (p * denom x) \u2194\n    \u2191(algebraMap K[X] (RatFunc K)) (num x) / \u2191(algebraMap K[X] (RatFunc K)) (denom x) *\n        \u2191(algebraMap K[X] (RatFunc K)) q =\n      \u2191(algebraMap K[X] (RatFunc K)) p"}, {"tactic": "rw [RingHom.map_mul, RingHom.map_mul, div_eq_mul_inv, mul_assoc, mul_comm (Inv.inv _), \u2190\n  mul_assoc, \u2190 div_eq_mul_inv, div_eq_iff]", "annotated_tactic": ["rw [<a>RingHom.map_mul</a>, <a>RingHom.map_mul</a>, <a>div_eq_mul_inv</a>, <a>mul_assoc</a>, <a>mul_comm</a> (<a>Inv.inv</a> _), \u2190\n    <a>mul_assoc</a>, \u2190 <a>div_eq_mul_inv</a>, <a>div_eq_iff</a>]", [{"full_name": "RingHom.map_mul", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [569, 19], "def_end_pos": [569, 26]}, {"full_name": "RingHom.map_mul", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [569, 19], "def_end_pos": [569, 26]}, {"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "Inv.inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [109, 3], "def_end_pos": [109, 6]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}, {"full_name": "div_eq_iff", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [88, 9], "def_end_pos": [88, 19]}]], "state_before": "K : Type u\ninst\u271d : Field K\nx : RatFunc K\np q : K[X]\nhq : q \u2260 0\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) (num x * q) = \u2191(algebraMap K[X] (RatFunc K)) (p * denom x) \u2194\n    \u2191(algebraMap K[X] (RatFunc K)) (num x) / \u2191(algebraMap K[X] (RatFunc K)) (denom x) *\n        \u2191(algebraMap K[X] (RatFunc K)) q =\n      \u2191(algebraMap K[X] (RatFunc K)) p", "state_after": "K : Type u\ninst\u271d : Field K\nx : RatFunc K\np q : K[X]\nhq : q \u2260 0\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) (denom x) \u2260 0"}, {"tactic": "exact algebraMap_ne_zero (denom_ne_zero x)", "annotated_tactic": ["exact <a>algebraMap_ne_zero</a> (<a>denom_ne_zero</a> x)", [{"full_name": "RatFunc.algebraMap_ne_zero", "def_path": "Mathlib/FieldTheory/RatFunc.lean", "def_pos": [920, 9], "def_end_pos": [920, 27]}, {"full_name": "RatFunc.denom_ne_zero", "def_path": "Mathlib/FieldTheory/RatFunc.lean", "def_pos": [1196, 9], "def_end_pos": [1196, 22]}]], "state_before": "K : Type u\ninst\u271d : Field K\nx : RatFunc K\np q : K[X]\nhq : q \u2260 0\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) (denom x) \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Directed.lean", "full_name": "Directed.mono", "start": [94, 1], "end": [97, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean", "full_name": "NonUnitalSubsemiring.mk'_toAddSubmonoid", "start": [201, 1], "end": [203, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "Filter.Tendsto.atBot_add", "start": [1978, 1], "end": [1981, 24], "traced_tactics": [{"tactic": "conv in _ + _ => rw [add_comm]", "annotated_tactic": ["conv in _ + _ => rw [<a>add_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup \u03b1\ninst\u271d : OrderTopology \u03b1\nl : Filter \u03b2\nf g : \u03b2 \u2192 \u03b1\nC : \u03b1\nhf : Tendsto f l atBot\nhg : Tendsto g l (\ud835\udcdd C)\n\u22a2 Tendsto (fun x => f x + g x) l atBot", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup \u03b1\ninst\u271d : OrderTopology \u03b1\nl : Filter \u03b2\nf g : \u03b2 \u2192 \u03b1\nC : \u03b1\nhf : Tendsto f l atBot\nhg : Tendsto g l (\ud835\udcdd C)\n\u22a2 Tendsto (fun x => g x + f x) l atBot"}, {"tactic": "exact hg.add_atBot hf", "annotated_tactic": ["exact hg.add_atBot hf", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup \u03b1\ninst\u271d : OrderTopology \u03b1\nl : Filter \u03b2\nf g : \u03b2 \u2192 \u03b1\nC : \u03b1\nhf : Tendsto f l atBot\nhg : Tendsto g l (\ud835\udcdd C)\n\u22a2 Tendsto (fun x => g x + f x) l atBot", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Bases.lean", "full_name": "Filter.inf_principal_neBot_iff", "start": [680, 1], "end": [681, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Lie/Solvable.lean", "full_name": "LieIdeal.derivedSeries_map_eq", "start": [188, 1], "end": [194, 85], "traced_tactics": [{"tactic": "induction' k with k ih", "annotated_tactic": ["induction' k with k ih", []], "state_before": "R : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\ninst\u271d\u00b9 : LieRing L'\ninst\u271d : LieAlgebra R L'\nI J : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nk : \u2115\nh : Function.Surjective \u2191f\n\u22a2 map f (derivedSeries R L' k) = derivedSeries R L k", "state_after": "case zero\nR : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\ninst\u271d\u00b9 : LieRing L'\ninst\u271d : LieAlgebra R L'\nI J : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nh : Function.Surjective \u2191f\n\u22a2 map f (derivedSeries R L' Nat.zero) = derivedSeries R L Nat.zero\n\ncase succ\nR : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\ninst\u271d\u00b9 : LieRing L'\ninst\u271d : LieAlgebra R L'\nI J : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nh : Function.Surjective \u2191f\nk : \u2115\nih : map f (derivedSeries R L' k) = derivedSeries R L k\n\u22a2 map f (derivedSeries R L' (Nat.succ k)) = derivedSeries R L (Nat.succ k)"}, {"tactic": "change (\u22a4 : LieIdeal R L').map f = \u22a4", "annotated_tactic": ["change (\u22a4 : <a>LieIdeal</a> R L').<a>map</a> f = \u22a4", [{"full_name": "LieIdeal", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [228, 8], "def_end_pos": [228, 16]}, {"full_name": "LieIdeal.map", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [921, 5], "def_end_pos": [921, 8]}]], "state_before": "case zero\nR : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\ninst\u271d\u00b9 : LieRing L'\ninst\u271d : LieAlgebra R L'\nI J : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nh : Function.Surjective \u2191f\n\u22a2 map f (derivedSeries R L' Nat.zero) = derivedSeries R L Nat.zero", "state_after": "case zero\nR : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\ninst\u271d\u00b9 : LieRing L'\ninst\u271d : LieAlgebra R L'\nI J : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nh : Function.Surjective \u2191f\n\u22a2 map f \u22a4 = \u22a4"}, {"tactic": "rw [\u2190 f.idealRange_eq_map]", "annotated_tactic": ["rw [\u2190 f.idealRange_eq_map]", []], "state_before": "case zero\nR : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\ninst\u271d\u00b9 : LieRing L'\ninst\u271d : LieAlgebra R L'\nI J : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nh : Function.Surjective \u2191f\n\u22a2 map f \u22a4 = \u22a4", "state_after": "case zero\nR : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\ninst\u271d\u00b9 : LieRing L'\ninst\u271d : LieAlgebra R L'\nI J : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nh : Function.Surjective \u2191f\n\u22a2 LieHom.idealRange f = \u22a4"}, {"tactic": "exact f.idealRange_eq_top_of_surjective h", "annotated_tactic": ["exact f.idealRange_eq_top_of_surjective h", []], "state_before": "case zero\nR : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\ninst\u271d\u00b9 : LieRing L'\ninst\u271d : LieAlgebra R L'\nI J : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nh : Function.Surjective \u2191f\n\u22a2 LieHom.idealRange f = \u22a4", "state_after": "no goals"}, {"tactic": "simp only [derivedSeries_def, map_bracket_eq f h, ih, derivedSeriesOfIdeal_succ]", "annotated_tactic": ["simp only [<a>derivedSeries_def</a>, <a>map_bracket_eq</a> f h, ih, <a>derivedSeriesOfIdeal_succ</a>]", [{"full_name": "LieAlgebra.derivedSeries_def", "def_path": "Mathlib/Algebra/Lie/Solvable.lean", "def_pos": [76, 9], "def_end_pos": [76, 26]}, {"full_name": "LieIdeal.map_bracket_eq", "def_path": "Mathlib/Algebra/Lie/IdealOperations.lean", "def_pos": [277, 9], "def_end_pos": [277, 23]}, {"full_name": "LieAlgebra.derivedSeriesOfIdeal_succ", "def_path": "Mathlib/Algebra/Lie/Solvable.lean", "def_pos": [65, 9], "def_end_pos": [65, 34]}]], "state_before": "case succ\nR : Type u\nL : Type v\nM : Type w\nL' : Type w\u2081\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\ninst\u271d\u00b9 : LieRing L'\ninst\u271d : LieAlgebra R L'\nI J : LieIdeal R L\nf : L' \u2192\u2097\u2045R\u2046 L\nh : Function.Surjective \u2191f\nk : \u2115\nih : map f (derivedSeries R L' k) = derivedSeries R L k\n\u22a2 map f (derivedSeries R L' (Nat.succ k)) = derivedSeries R L (Nat.succ k)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/Degree/TrailingDegree.lean", "full_name": "Polynomial.natTrailingDegree_neg", "start": [466, 1], "end": [467, 27], "traced_tactics": [{"tactic": "simp [natTrailingDegree]", "annotated_tactic": ["simp [<a>natTrailingDegree</a>]", [{"full_name": "Polynomial.natTrailingDegree", "def_path": "Mathlib/Data/Polynomial/Degree/TrailingDegree.lean", "def_pos": [56, 5], "def_end_pos": [56, 22]}]], "state_before": "R : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Ring R\np : R[X]\n\u22a2 natTrailingDegree (-p) = natTrailingDegree p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.eventually_of_mem", "start": [1098, 1], "end": [1100, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/InitialSeg.lean", "full_name": "wellFounded_iff_wellFounded_subrel", "start": [466, 1], "end": [473, 40], "traced_tactics": [{"tactic": "refine'\n  \u27e8fun wf b => \u27e8fun b' => ((PrincipalSeg.ofElement _ b).acc b').mpr (wf.apply b')\u27e9, fun wf =>\n    \u27e8fun b => Acc.intro _ fun b' hb' => _\u27e9\u27e9", "annotated_tactic": ["refine'\n    \u27e8fun wf b => \u27e8fun b' => ((<a>PrincipalSeg.ofElement</a> _ b).<a>acc</a> b').<a>mpr</a> (wf.apply b')\u27e9, fun wf =>\n      \u27e8fun b => <a>Acc.intro</a> _ fun b' hb' => _\u27e9\u27e9", [{"full_name": "PrincipalSeg.ofElement", "def_path": "Mathlib/Order/InitialSeg.lean", "def_pos": [381, 5], "def_end_pos": [381, 14]}, {"full_name": "PrincipalSeg.acc", "def_path": "Mathlib/Order/InitialSeg.lean", "def_pos": [455, 19], "def_end_pos": [455, 22]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}, {"full_name": "Acc.intro", "def_path": "lake-packages/lean4/src/lean/Init/WF.lean", "def_pos": [13, 5], "def_end_pos": [13, 10]}]], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b2 : Type u_4\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsTrans \u03b2 s\n\u22a2 WellFounded s \u2194 \u2200 (b : \u03b2), WellFounded (Subrel s {b' | s b' b})", "state_after": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b2 : Type u_4\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsTrans \u03b2 s\nwf : \u2200 (b : \u03b2), WellFounded (Subrel s {b' | s b' b})\nb b' : \u03b2\nhb' : s b' b\n\u22a2 Acc s b'"}, {"tactic": "let f := PrincipalSeg.ofElement s b", "annotated_tactic": ["let f := <a>PrincipalSeg.ofElement</a> s b", [{"full_name": "PrincipalSeg.ofElement", "def_path": "Mathlib/Order/InitialSeg.lean", "def_pos": [381, 5], "def_end_pos": [381, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b2 : Type u_4\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsTrans \u03b2 s\nwf : \u2200 (b : \u03b2), WellFounded (Subrel s {b' | s b' b})\nb b' : \u03b2\nhb' : s b' b\n\u22a2 Acc s b'", "state_after": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b2 : Type u_4\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsTrans \u03b2 s\nwf : \u2200 (b : \u03b2), WellFounded (Subrel s {b' | s b' b})\nb b' : \u03b2\nhb' : s b' b\nf : Subrel s {b_1 | s b_1 b} \u227ai s := PrincipalSeg.ofElement s b\n\u22a2 Acc s b'"}, {"tactic": "obtain \u27e8b', rfl\u27e9 := f.down.mp ((PrincipalSeg.ofElement_top s b).symm \u25b8 hb' : s b' f.top)", "annotated_tactic": ["obtain \u27e8b', rfl\u27e9 := f.down.mp ((<a>PrincipalSeg.ofElement_top</a> s b).<a>symm</a> \u25b8 hb' : s b' f.top)", [{"full_name": "PrincipalSeg.ofElement_top", "def_path": "Mathlib/Order/InitialSeg.lean", "def_pos": [392, 9], "def_end_pos": [392, 22]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b2 : Type u_4\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsTrans \u03b2 s\nwf : \u2200 (b : \u03b2), WellFounded (Subrel s {b' | s b' b})\nb b' : \u03b2\nhb' : s b' b\nf : Subrel s {b_1 | s b_1 b} \u227ai s := PrincipalSeg.ofElement s b\n\u22a2 Acc s b'", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b2 : Type u_4\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsTrans \u03b2 s\nwf : \u2200 (b : \u03b2), WellFounded (Subrel s {b' | s b' b})\nb : \u03b2\nf : Subrel s {b_1 | s b_1 b} \u227ai s := PrincipalSeg.ofElement s b\nb' : \u2191{b_1 | s b_1 b}\nhb' : s (\u2191f.toRelEmbedding b') b\n\u22a2 Acc s (\u2191f.toRelEmbedding b')"}, {"tactic": "exact (f.acc b').mp ((wf b).apply b')", "annotated_tactic": ["exact (f.acc b').<a>mp</a> ((wf b).<a>apply</a> b')", [{"full_name": "Iff.mp", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [90, 3], "def_end_pos": [90, 5]}, {"full_name": "WellFounded.apply", "def_path": "lake-packages/lean4/src/lean/Init/WF.lean", "def_pos": [42, 5], "def_end_pos": [42, 10]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2\u271d \u2192 \u03b2\u271d \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b2 : Type u_4\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\ninst\u271d : IsTrans \u03b2 s\nwf : \u2200 (b : \u03b2), WellFounded (Subrel s {b' | s b' b})\nb : \u03b2\nf : Subrel s {b_1 | s b_1 b} \u227ai s := PrincipalSeg.ofElement s b\nb' : \u2191{b_1 | s b_1 b}\nhb' : s (\u2191f.toRelEmbedding b') b\n\u22a2 Acc s (\u2191f.toRelEmbedding b')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Logic/Relation.lean", "full_name": "Relation.TransGen.head", "start": [385, 1], "end": [386, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.eval\u2082_zero_apply", "start": [1530, 1], "end": [1532, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/UpperLower.lean", "full_name": "LowerSet.coe_div", "start": [212, 1], "end": [213, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.mem_uIcc_of_le", "start": [962, 1], "end": [963, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "full_name": "ofAdd_smul", "start": [1227, 1], "end": [1228, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/TypeVec.lean", "full_name": "TypeVec.append1_drop_last", "start": [127, 1], "end": [128, 37], "traced_tactics": [{"tactic": "cases i <;> rfl", 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"Mathlib/Combinatorics/SetFamily/Compression/Down.lean", "def_pos": [61, 9], "def_end_pos": [61, 28]}, {"full_name": "Finset.mem_nonMemberSubfamily", "def_path": "Mathlib/Combinatorics/SetFamily/Compression/Down.lean", "def_pos": [56, 9], "def_end_pos": [56, 31]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na : \u03b1\nh : IsLowerSet \u2191\ud835\udc9c\ns : Finset \u03b1\n\u22a2 s \u2208 Finset.memberSubfamily a \ud835\udc9c \u2192 s \u2208 Finset.nonMemberSubfamily a \ud835\udc9c", "state_after": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na : \u03b1\nh : IsLowerSet \u2191\ud835\udc9c\ns : Finset \u03b1\n\u22a2 insert a s \u2208 \ud835\udc9c \u2227 \u00aca \u2208 s \u2192 s \u2208 \ud835\udc9c \u2227 \u00aca \u2208 s"}, {"tactic": "exact And.imp_left (h <| subset_insert _ _)", "annotated_tactic": ["exact <a>And.imp_left</a> (h <| <a>subset_insert</a> _ _)", [{"full_name": "And.imp_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [152, 9], "def_end_pos": [152, 21]}, {"full_name": "Finset.subset_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1194, 9], "def_end_pos": [1194, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c \u212c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na : \u03b1\nh : IsLowerSet \u2191\ud835\udc9c\ns : Finset \u03b1\n\u22a2 insert a s \u2208 \ud835\udc9c \u2227 \u00aca \u2208 s \u2192 s \u2208 \ud835\udc9c \u2227 \u00aca \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.next_of_valid'", "start": [265, 1], "end": [267, 38], "traced_tactics": [{"tactic": "simp only [next, get_of_valid]", "annotated_tactic": ["simp only [<a>next</a>, <a>get_of_valid</a>]", [{"full_name": "String.next", "def_path": "lake-packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "String.get_of_valid", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [223, 9], "def_end_pos": [223, 21]}]], "state_before": "cs cs' : List Char\n\u22a2 next { data := cs ++ cs' } { byteIdx := utf8Len cs } = { byteIdx := utf8Len cs + csize (List.headD cs' default) }", "state_after": "cs cs' : List Char\n\u22a2 { byteIdx := utf8Len cs } + List.headD cs' default = { byteIdx := utf8Len cs + csize (List.headD cs' default) }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "cs cs' : List Char\n\u22a2 { byteIdx := utf8Len cs } + List.headD cs' default = { byteIdx := utf8Len cs + csize (List.headD cs' default) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": 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Type u_1\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhy : x \u2260 y\nhz : x \u2260 z\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh : x \u2208 y :: z :: head\u271d :: tail\u271d\n\u22a2 prev (y :: z :: head\u271d :: tail\u271d) x h = prev (z :: head\u271d :: tail\u271d) x (_ : x \u2208 z :: head\u271d :: tail\u271d)"}, {"tactic": "simp [hy, hz] at h", "annotated_tactic": ["simp [hy, hz] at h", []], "state_before": "case nil\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhy : x \u2260 y\nhz : x \u2260 z\nh : x \u2208 [y, z]\n\u22a2 prev [y, z] x h = prev [z] x (_ : x \u2208 [z])", "state_after": "no goals"}, {"tactic": "rw [prev, dif_neg hy, if_neg hz]", "annotated_tactic": ["rw [<a>prev</a>, <a>dif_neg</a> hy, <a>if_neg</a> hz]", [{"full_name": "List.prev", "def_path": "Mathlib/Data/List/Cycle.lean", "def_pos": [135, 5], "def_end_pos": [135, 9]}, {"full_name": "dif_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [812, 9], "def_end_pos": [812, 16]}, {"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}]], "state_before": "case cons\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nx y z : \u03b1\nhy : x \u2260 y\nhz : x \u2260 z\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nh : x \u2208 y :: z :: head\u271d :: tail\u271d\n\u22a2 prev (y :: z :: head\u271d :: tail\u271d) x h = prev (z :: head\u271d :: tail\u271d) x (_ : x \u2208 z :: head\u271d :: tail\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.eventuallyEq_bind", "start": [2800, 1], "end": [2802, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Finsupp.lean", "full_name": "mem_span_finset", "start": [1196, 1], "end": [1203, 76], "traced_tactics": [{"tactic": "rwa [Set.image_id]", "annotated_tactic": ["rwa [<a>Set.image_id</a>]", [{"full_name": "Set.image_id", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [401, 9], "def_end_pos": [401, 17]}]], "state_before": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : Module R N\ns : Finset M\nx : M\nhx : x \u2208 span R \u2191s\n\u22a2 x \u2208 span R (_root_.id '' \u2191s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean", "full_name": "List.Nat.mem_antidiagonalTuple", "start": [81, 1], "end": [94, 60], "traced_tactics": [{"tactic": "cases n", "annotated_tactic": ["cases n", []], "state_before": "case h0\nk n : \u2115\n\u22a2 Fin.elim0 \u2208 antidiagonalTuple 0 n \u2194 \u2211 i : Fin 0, Fin.elim0 i = n", "state_after": "case h0.zero\nk : \u2115\n\u22a2 Fin.elim0 \u2208 antidiagonalTuple 0 Nat.zero \u2194 \u2211 i : Fin 0, Fin.elim0 i = Nat.zero\n\ncase h0.succ\nk n\u271d : \u2115\n\u22a2 Fin.elim0 \u2208 antidiagonalTuple 0 (Nat.succ n\u271d) \u2194 \u2211 i : Fin 0, Fin.elim0 i = Nat.succ n\u271d"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h0.zero\nk : \u2115\n\u22a2 Fin.elim0 \u2208 antidiagonalTuple 0 Nat.zero \u2194 \u2211 i : Fin 0, Fin.elim0 i = Nat.zero", "state_after": "no goals"}, {"tactic": "simp [eq_comm]", "annotated_tactic": ["simp [<a>eq_comm</a>]", [{"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "case h0.succ\nk n\u271d : \u2115\n\u22a2 Fin.elim0 \u2208 antidiagonalTuple 0 (Nat.succ n\u271d) \u2194 \u2211 i : Fin 0, Fin.elim0 i = Nat.succ n\u271d", "state_after": "no goals"}, {"tactic": "simp_rw [Fin.sum_cons]", "annotated_tactic": ["simp_rw [<a>Fin.sum_cons</a>]", [{"full_name": "Fin.sum_cons", "def_path": "Mathlib/Algebra/BigOperators/Fin.lean", "def_pos": [97, 3], "def_end_pos": [97, 14]}]], "state_before": "case h\nk n\u271d x\u2080 : \u2115\nx : Fin n\u271d \u2192 \u2115\nih : \u2200 {n : \u2115}, x \u2208 antidiagonalTuple n\u271d n \u2194 \u2211 i : Fin n\u271d, x i = n\nn : \u2115\n\u22a2 Fin.cons x\u2080 x \u2208 antidiagonalTuple (n\u271d + 1) n \u2194 \u2211 i : Fin (n\u271d + 1), Fin.cons x\u2080 x i = n", "state_after": "case h\nk n\u271d x\u2080 : \u2115\nx : Fin n\u271d \u2192 \u2115\nih : \u2200 {n : \u2115}, x \u2208 antidiagonalTuple n\u271d n \u2194 \u2211 i : Fin n\u271d, x i = n\nn : \u2115\n\u22a2 Fin.cons x\u2080 x \u2208 antidiagonalTuple (n\u271d + 1) n \u2194 x\u2080 + \u2211 i : Fin n\u271d, x i = n"}, {"tactic": "rw [antidiagonalTuple]", "annotated_tactic": ["rw [<a>antidiagonalTuple</a>]", [{"full_name": "List.Nat.antidiagonalTuple", "def_path": "Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean", "def_pos": [63, 5], "def_end_pos": [63, 22]}]], "state_before": "case h\nk n\u271d x\u2080 : \u2115\nx : Fin n\u271d \u2192 \u2115\nih : \u2200 {n : \u2115}, x \u2208 antidiagonalTuple n\u271d n \u2194 \u2211 i : Fin n\u271d, x i = n\nn : \u2115\n\u22a2 Fin.cons x\u2080 x \u2208 antidiagonalTuple (n\u271d + 1) n \u2194 x\u2080 + \u2211 i : Fin n\u271d, x i = n", "state_after": "case h\nk n\u271d x\u2080 : \u2115\nx : Fin n\u271d \u2192 \u2115\nih : \u2200 {n : \u2115}, x \u2208 antidiagonalTuple n\u271d n \u2194 \u2211 i : Fin n\u271d, x i = n\nn : \u2115\n\u22a2 (Fin.cons x\u2080 x \u2208 List.bind (antidiagonal n) fun ni => map (fun x => Fin.cons ni.1 x) (antidiagonalTuple n\u271d ni.2)) \u2194\n    x\u2080 + \u2211 i : Fin n\u271d, x i = n"}, {"tactic": "simp_rw [List.mem_bind, List.mem_map,\n  List.Nat.mem_antidiagonal, Fin.cons_eq_cons, exists_eq_right_right, ih,\n  @eq_comm _ _ (Prod.snd _), and_comm (a := Prod.snd _ = _),\n  \u2190Prod.mk.inj_iff (a\u2081 := Prod.fst _), exists_eq_right]", "annotated_tactic": ["simp_rw [<a>List.mem_bind</a>, <a>List.mem_map</a>,\n      <a>List.Nat.mem_antidiagonal</a>, <a>Fin.cons_eq_cons</a>, <a>exists_eq_right_right</a>, ih,\n      @<a>eq_comm</a> _ _ (<a>Prod.snd</a> _), <a>and_comm</a> (a := <a>Prod.snd</a> _ = _),\n      \u2190<a>Prod.mk.inj_iff</a> (a\u2081 := <a>Prod.fst</a> _), <a>exists_eq_right</a>]", [{"full_name": "List.mem_bind", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [233, 9], "def_end_pos": [233, 17]}, {"full_name": "List.mem_map", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [159, 17], "def_end_pos": [159, 24]}, {"full_name": "List.Nat.mem_antidiagonal", "def_path": "Mathlib/Data/List/NatAntidiagonal.lean", "def_pos": [38, 9], "def_end_pos": [38, 25]}, {"full_name": "Fin.cons_eq_cons", "def_path": "Mathlib/Data/Fin/Tuple/Basic.lean", "def_pos": [109, 9], "def_end_pos": [109, 21]}, {"full_name": "exists_eq_right_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [479, 17], "def_end_pos": [479, 38]}, {"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}, {"full_name": "Prod.snd", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [470, 3], "def_end_pos": [470, 6]}, {"full_name": "and_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [159, 9], "def_end_pos": [159, 17]}, {"full_name": "Prod.snd", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [470, 3], "def_end_pos": [470, 6]}, {"full_name": "Prod.mk.inj_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 19]}, {"full_name": "Prod.fst", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [468, 3], "def_end_pos": [468, 6]}, {"full_name": "exists_eq_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [462, 17], "def_end_pos": [462, 32]}]], "state_before": "case h\nk n\u271d x\u2080 : \u2115\nx : Fin n\u271d \u2192 \u2115\nih : \u2200 {n : \u2115}, x \u2208 antidiagonalTuple n\u271d n \u2194 \u2211 i : Fin n\u271d, x i = n\nn : \u2115\n\u22a2 (Fin.cons x\u2080 x \u2208 List.bind (antidiagonal n) fun ni => map (fun x => Fin.cons ni.1 x) (antidiagonalTuple n\u271d ni.2)) \u2194\n    x\u2080 + \u2211 i : Fin n\u271d, x i = n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Quandle.lean", "full_name": "Rack.self_distrib_inv", "start": [239, 1], "end": [241, 25], "traced_tactics": [{"tactic": "rw [\u2190 left_cancel (x \u25c3\u207b\u00b9 y), right_inv, \u2190 left_cancel x, right_inv, self_distrib]", "annotated_tactic": ["rw [\u2190 <a>left_cancel</a> (x \u25c3\u207b\u00b9 y), <a>right_inv</a>, \u2190 <a>left_cancel</a> x, <a>right_inv</a>, <a>self_distrib</a>]", [{"full_name": "Rack.left_cancel", "def_path": "Mathlib/Algebra/Quandle.lean", "def_pos": [225, 9], "def_end_pos": [225, 20]}, {"full_name": "Rack.right_inv", "def_path": "Mathlib/Algebra/Quandle.lean", "def_pos": [137, 3], "def_end_pos": [137, 12]}, {"full_name": "Rack.left_cancel", "def_path": "Mathlib/Algebra/Quandle.lean", "def_pos": [225, 9], "def_end_pos": [225, 20]}, {"full_name": "Rack.right_inv", "def_path": "Mathlib/Algebra/Quandle.lean", "def_pos": [137, 3], "def_end_pos": [137, 12]}, {"full_name": "Shelf.self_distrib", "def_path": "Mathlib/Algebra/Quandle.lean", "def_pos": [99, 3], "def_end_pos": [99, 15]}]], "state_before": "R : Type u_1\ninst\u271d : Rack R\nx y z : R\n\u22a2 x \u25c3\u207b\u00b9 y \u25c3\u207b\u00b9 z = (x \u25c3\u207b\u00b9 y) \u25c3\u207b\u00b9 x \u25c3\u207b\u00b9 z", "state_after": "R : Type u_1\ninst\u271d : Rack R\nx y z : R\n\u22a2 (x \u25c3 x \u25c3\u207b\u00b9 y) \u25c3 x \u25c3 x \u25c3\u207b\u00b9 y \u25c3\u207b\u00b9 z = z"}, {"tactic": "repeat' rw [right_inv]", "annotated_tactic": ["repeat' rw [<a>right_inv</a>]", [{"full_name": "Rack.right_inv", "def_path": "Mathlib/Algebra/Quandle.lean", "def_pos": [137, 3], "def_end_pos": [137, 12]}]], "state_before": "R : Type u_1\ninst\u271d : Rack R\nx y z : R\n\u22a2 (x \u25c3 x \u25c3\u207b\u00b9 y) \u25c3 x \u25c3 x \u25c3\u207b\u00b9 y \u25c3\u207b\u00b9 z = z", "state_after": "no goals"}, {"tactic": "rw [right_inv]", "annotated_tactic": ["rw [<a>right_inv</a>]", [{"full_name": "Rack.right_inv", "def_path": "Mathlib/Algebra/Quandle.lean", "def_pos": [137, 3], "def_end_pos": [137, 12]}]], "state_before": "R : Type u_1\ninst\u271d : Rack R\nx y z : R\n\u22a2 y \u25c3 y \u25c3\u207b\u00b9 z = z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/BoxIntegral/Partition/Measure.lean", "full_name": "BoxIntegral.BoxAdditiveMap.volume_apply", "start": [147, 1], "end": [150, 46], "traced_tactics": [{"tactic": "rw [BoxAdditiveMap.volume, toSMul_apply]", "annotated_tactic": ["rw [<a>BoxAdditiveMap.volume</a>, <a>toSMul_apply</a>]", [{"full_name": "BoxIntegral.BoxAdditiveMap.volume", "def_path": "Mathlib/Analysis/BoxIntegral/Partition/Measure.lean", "def_pos": [143, 15], "def_end_pos": [143, 21]}, {"full_name": "BoxIntegral.BoxAdditiveMap.toSMul_apply", "def_path": "Mathlib/Analysis/BoxIntegral/Partition/Additive.lean", "def_pos": [183, 9], "def_end_pos": [183, 21]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b9\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nI : Box \u03b9\nx : E\n\u22a2 \u2191(\u2191BoxAdditiveMap.volume I) x = (\u220f j : \u03b9, (upper I j - lower I j)) \u2022 x", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b2 : Fintype \u03b9\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nI : Box \u03b9\nx : E\n\u22a2 \u2191(Measure.toBoxAdditive volume) I \u2022 x = (\u220f j : \u03b9, (upper I j - lower I j)) \u2022 x"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "full_name": "zsmul_eq_mul'", "start": [595, 1], "end": [596, 43], "traced_tactics": [{"tactic": "rw [zsmul_eq_mul, (n.cast_commute a).eq]", "annotated_tactic": ["rw [<a>zsmul_eq_mul</a>, (n.cast_commute a).<a>eq</a>]", [{"full_name": "zsmul_eq_mul", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [590, 9], "def_end_pos": [590, 21]}, {"full_name": "Commute.eq", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [47, 19], "def_end_pos": [47, 21]}]], "state_before": "\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : Ring R\na : R\nn : \u2124\n\u22a2 n \u2022 a = a * \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Tactic/Positivity/Basic.lean", "full_name": "Mathlib.Meta.Positivity.pow_zero_pos", "start": [312, 9], "end": [313, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean", "full_name": "CategoryTheory.Subgroupoid.isThin_iff", "start": [606, 8], "end": [606, 97], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Action.lean", "full_name": "CategoryTheory.ActionCategory.\u03c0_map", "start": [64, 1], "end": [65, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Stream/Init.lean", "full_name": "Stream'.get_odd", "start": [508, 1], "end": [509, 29], "traced_tactics": [{"tactic": "rw [odd_eq, get_even]", "annotated_tactic": ["rw [<a>odd_eq</a>, <a>get_even</a>]", [{"full_name": "Stream'.odd_eq", "def_path": "Mathlib/Data/Stream/Init.lean", "def_pos": [458, 9], "def_end_pos": [458, 15]}, {"full_name": "Stream'.get_even", "def_path": "Mathlib/Data/Stream/Init.lean", "def_pos": [501, 9], "def_end_pos": [501, 17]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\ns : Stream' \u03b1\n\u22a2 get (odd s) n = get s (2 * n + 1)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\ns : Stream' \u03b1\n\u22a2 get (tail s) (2 * n) = get s (2 * n + 1)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b4 : Type w\nn : \u2115\ns : Stream' \u03b1\n\u22a2 get (tail s) (2 * n) = get s (2 * n + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/Digits.lean", "full_name": "Nat.modEq_digits_sum", "start": [653, 1], "end": [660, 15], "traced_tactics": [{"tactic": "rw [\u2190 ofDigits_one]", "annotated_tactic": ["rw [\u2190 <a>ofDigits_one</a>]", [{"full_name": "Nat.ofDigits_one", "def_path": "Mathlib/Data/Nat/Digits.lean", "def_pos": [283, 9], "def_end_pos": [283, 21]}]], "state_before": "n\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n\u22a2 n \u2261 List.sum (digits b' n) [MOD b]", "state_after": "n\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n\u22a2 n \u2261 ofDigits 1 (digits b' n) [MOD b]"}, {"tactic": "convert ofDigits_modEq b' b (digits b' n)", "annotated_tactic": ["convert <a>ofDigits_modEq</a> b' b (<a>digits</a> b' n)", [{"full_name": "Nat.ofDigits_modEq", "def_path": "Mathlib/Data/Nat/Digits.lean", "def_pos": [627, 9], "def_end_pos": [627, 23]}, {"full_name": "Nat.digits", "def_path": "Mathlib/Data/Nat/Digits.lean", "def_pos": [82, 5], "def_end_pos": [82, 11]}]], "state_before": "n\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n\u22a2 ofDigits b' (digits b' n) \u2261 ofDigits 1 (digits b' n) [MOD b]", "state_after": "case h.e'_3.h.e'_3\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n\u22a2 1 = b' % b"}, {"tactic": "exact h.symm", "annotated_tactic": ["exact h.symm", []], "state_before": "case h.e'_3.h.e'_3\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n\u22a2 1 = b' % b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean", "full_name": "IsPrimitiveRoot.sub_one_norm_prime", "start": [412, 1], "end": [418, 50], "traced_tactics": [{"tactic": "replace hirr : Irreducible (cyclotomic (\u2191(p ^ (0 + 1)) : \u2115) K) := by simp [hirr]", "annotated_tactic": ["replace hirr : <a>Irreducible</a> (<a>cyclotomic</a> (\u2191(p ^ (0 + 1)) : \u2115) K) := by simp [hirr]", [{"full_name": "Irreducible", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [170, 11], "def_end_pos": [170, 22]}, {"full_name": "Polynomial.cyclotomic", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [255, 5], "def_end_pos": [255, 15]}]], "state_before": "p n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nhpri : Fact (Nat.Prime \u2191p)\nhcyc : IsCyclotomicExtension {p} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191p\nhirr : Irreducible (cyclotomic (\u2191p) K)\nh : p \u2260 2\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191\u2191p", "state_after": "p n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nhpri : Fact (Nat.Prime \u2191p)\nhcyc : IsCyclotomicExtension {p} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191p\nh : p \u2260 2\nhirr : Irreducible (cyclotomic (\u2191(p ^ (0 + 1))) K)\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191\u2191p"}, {"tactic": "replace h\u03b6 : IsPrimitiveRoot \u03b6 (\u2191(p ^ (0 + 1)) : \u2115) := by simp [h\u03b6]", "annotated_tactic": ["replace h\u03b6 : <a>IsPrimitiveRoot</a> \u03b6 (\u2191(p ^ (0 + 1)) : \u2115) := by simp [h\u03b6]", [{"full_name": "IsPrimitiveRoot", "def_path": "Mathlib/RingTheory/RootsOfUnity/Basic.lean", "def_pos": [290, 11], "def_end_pos": [290, 26]}]], "state_before": "p n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nhpri : Fact (Nat.Prime \u2191p)\nhcyc : IsCyclotomicExtension {p} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191p\nh : p \u2260 2\nhirr : Irreducible (cyclotomic (\u2191(p ^ (0 + 1))) K)\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191\u2191p", "state_after": "p n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nhpri : Fact (Nat.Prime \u2191p)\nhcyc : IsCyclotomicExtension {p} K L\nh : p \u2260 2\nhirr : Irreducible (cyclotomic (\u2191(p ^ (0 + 1))) K)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (0 + 1))\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191\u2191p"}, {"tactic": "haveI : IsCyclotomicExtension {p ^ (0 + 1)} K L := by simp [hcyc]", "annotated_tactic": ["haveI : <a>IsCyclotomicExtension</a> {p ^ (0 + 1)} K L := by simp [hcyc]", [{"full_name": "IsCyclotomicExtension", "def_path": "Mathlib/NumberTheory/Cyclotomic/Basic.lean", "def_pos": [82, 7], "def_end_pos": [82, 28]}]], "state_before": "p n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nhpri : Fact (Nat.Prime \u2191p)\nhcyc : IsCyclotomicExtension {p} K L\nh : p \u2260 2\nhirr : Irreducible (cyclotomic (\u2191(p ^ (0 + 1))) K)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (0 + 1))\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191\u2191p", "state_after": "p n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nhpri : Fact (Nat.Prime \u2191p)\nhcyc : IsCyclotomicExtension {p} K L\nh : p \u2260 2\nhirr : Irreducible (cyclotomic (\u2191(p ^ (0 + 1))) K)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (0 + 1))\nthis : IsCyclotomicExtension {p ^ (0 + 1)} K L\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191\u2191p"}, {"tactic": "simpa using sub_one_norm_prime_ne_two h\u03b6 hirr h", "annotated_tactic": ["simpa using <a>sub_one_norm_prime_ne_two</a> h\u03b6 hirr h", [{"full_name": "IsPrimitiveRoot.sub_one_norm_prime_ne_two", "def_path": "Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean", "def_pos": [404, 9], "def_end_pos": [404, 34]}]], "state_before": "p n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nhpri : Fact (Nat.Prime \u2191p)\nhcyc : IsCyclotomicExtension {p} K L\nh : p \u2260 2\nhirr : Irreducible (cyclotomic (\u2191(p ^ (0 + 1))) K)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (0 + 1))\nthis : IsCyclotomicExtension {p ^ (0 + 1)} K L\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191\u2191p", "state_after": "no goals"}, {"tactic": "simp [hirr]", "annotated_tactic": ["simp [hirr]", []], "state_before": "p n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nhpri : Fact (Nat.Prime \u2191p)\nhcyc : IsCyclotomicExtension {p} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191p\nhirr : Irreducible (cyclotomic (\u2191p) K)\nh : p \u2260 2\n\u22a2 Irreducible (cyclotomic (\u2191(p ^ (0 + 1))) K)", "state_after": "no goals"}, {"tactic": "simp [h\u03b6]", "annotated_tactic": ["simp [h\u03b6]", []], "state_before": "p n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nhpri : Fact (Nat.Prime \u2191p)\nhcyc : IsCyclotomicExtension {p} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191p\nh : p \u2260 2\nhirr : Irreducible (cyclotomic (\u2191(p ^ (0 + 1))) K)\n\u22a2 IsPrimitiveRoot \u03b6 \u2191(p ^ (0 + 1))", "state_after": "no goals"}, {"tactic": "simp [hcyc]", "annotated_tactic": ["simp [hcyc]", []], "state_before": "p n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nhpri : Fact (Nat.Prime \u2191p)\nhcyc : IsCyclotomicExtension {p} K L\nh : p \u2260 2\nhirr : Irreducible (cyclotomic (\u2191(p ^ (0 + 1))) K)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (0 + 1))\n\u22a2 IsCyclotomicExtension {p ^ (0 + 1)} K L", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Multiset/Sum.lean", "full_name": "Multiset.Nodup.disjSum", "start": [104, 11], "end": [109, 21], "traced_tactics": [{"tactic": "refine' ((hs.map inl_injective).add_iff <| ht.map inr_injective).2 fun x hs ht => _", "annotated_tactic": ["refine' ((hs.map <a>inl_injective</a>).<a>add_iff</a> <| ht.map <a>inr_injective</a>).2 fun x hs ht => _", [{"full_name": "Sum.inl_injective", "def_path": "Mathlib/Data/Sum/Basic.lean", "def_pos": [31, 9], "def_end_pos": [31, 22]}, {"full_name": "Multiset.Nodup.add_iff", "def_path": "Mathlib/Data/Multiset/Nodup.lean", "def_pos": [123, 9], "def_end_pos": [123, 22]}, {"full_name": "Sum.inr_injective", "def_path": "Mathlib/Data/Sum/Basic.lean", "def_pos": [34, 9], "def_end_pos": [34, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Multiset \u03b1\nt : Multiset \u03b2\ns\u2081 s\u2082 : Multiset \u03b1\nt\u2081 t\u2082 : Multiset \u03b2\na : \u03b1\nb : \u03b2\nx : \u03b1 \u2295 \u03b2\nhs : Nodup s\nht : Nodup t\n\u22a2 Nodup (disjSum s t)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Multiset \u03b1\nt : Multiset \u03b2\ns\u2081 s\u2082 : Multiset \u03b1\nt\u2081 t\u2082 : Multiset \u03b2\na : \u03b1\nb : \u03b2\nx\u271d : \u03b1 \u2295 \u03b2\nhs\u271d : Nodup s\nht\u271d : Nodup t\nx : \u03b1 \u2295 \u03b2\nhs : x \u2208 Multiset.map inl s\nht : x \u2208 Multiset.map inr t\n\u22a2 False"}, {"tactic": "rw [Multiset.mem_map] at hs ht", "annotated_tactic": ["rw [<a>Multiset.mem_map</a>] at hs ht", [{"full_name": "Multiset.mem_map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1228, 9], "def_end_pos": [1228, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Multiset \u03b1\nt : Multiset \u03b2\ns\u2081 s\u2082 : Multiset \u03b1\nt\u2081 t\u2082 : Multiset \u03b2\na : \u03b1\nb : \u03b2\nx\u271d : \u03b1 \u2295 \u03b2\nhs\u271d : Nodup s\nht\u271d : Nodup t\nx : \u03b1 \u2295 \u03b2\nhs : x \u2208 Multiset.map inl s\nht : x \u2208 Multiset.map inr t\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Multiset \u03b1\nt : Multiset \u03b2\ns\u2081 s\u2082 : Multiset \u03b1\nt\u2081 t\u2082 : Multiset \u03b2\na : \u03b1\nb : \u03b2\nx\u271d : \u03b1 \u2295 \u03b2\nhs\u271d : Nodup s\nht\u271d : Nodup t\nx : \u03b1 \u2295 \u03b2\nhs : \u2203 a, a \u2208 s \u2227 inl a = x\nht : \u2203 a, a \u2208 t \u2227 inr a = x\n\u22a2 False"}, {"tactic": "obtain \u27e8a, _, rfl\u27e9 := hs", "annotated_tactic": ["obtain \u27e8a, _, rfl\u27e9 := hs", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Multiset \u03b1\nt : Multiset \u03b2\ns\u2081 s\u2082 : Multiset \u03b1\nt\u2081 t\u2082 : Multiset \u03b2\na : \u03b1\nb : \u03b2\nx\u271d : \u03b1 \u2295 \u03b2\nhs\u271d : Nodup s\nht\u271d : Nodup t\nx : \u03b1 \u2295 \u03b2\nhs : \u2203 a, a \u2208 s \u2227 inl a = x\nht : \u2203 a, a \u2208 t \u2227 inr a = x\n\u22a2 False", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Multiset \u03b1\nt : Multiset \u03b2\ns\u2081 s\u2082 : Multiset \u03b1\nt\u2081 t\u2082 : Multiset \u03b2\na\u271d : \u03b1\nb : \u03b2\nx : \u03b1 \u2295 \u03b2\nhs : Nodup s\nht\u271d : Nodup t\na : \u03b1\nleft\u271d : a \u2208 s\nht : \u2203 a_1, a_1 \u2208 t \u2227 inr a_1 = inl a\n\u22a2 False"}, {"tactic": "obtain \u27e8b, _, h\u27e9 := ht", "annotated_tactic": ["obtain \u27e8b, _, h\u27e9 := ht", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Multiset \u03b1\nt : Multiset \u03b2\ns\u2081 s\u2082 : Multiset \u03b1\nt\u2081 t\u2082 : Multiset \u03b2\na\u271d : \u03b1\nb : \u03b2\nx : \u03b1 \u2295 \u03b2\nhs : Nodup s\nht\u271d : Nodup t\na : \u03b1\nleft\u271d : a \u2208 s\nht : \u2203 a_1, a_1 \u2208 t \u2227 inr a_1 = inl a\n\u22a2 False", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Multiset \u03b1\nt : Multiset \u03b2\ns\u2081 s\u2082 : Multiset \u03b1\nt\u2081 t\u2082 : Multiset \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\nx : \u03b1 \u2295 \u03b2\nhs : Nodup s\nht : Nodup t\na : \u03b1\nleft\u271d\u00b9 : a \u2208 s\nb : \u03b2\nleft\u271d : b \u2208 t\nh : inr b = inl a\n\u22a2 False"}, {"tactic": "exact inr_ne_inl h", "annotated_tactic": ["exact <a>inr_ne_inl</a> h", [{"full_name": "Sum.inr_ne_inl", "def_path": "lake-packages/std/Std/Data/Sum/Lemmas.lean", "def_pos": [95, 9], "def_end_pos": [95, 19]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Multiset \u03b1\nt : Multiset \u03b2\ns\u2081 s\u2082 : Multiset \u03b1\nt\u2081 t\u2082 : Multiset \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\nx : \u03b1 \u2295 \u03b2\nhs : Nodup s\nht : Nodup t\na : \u03b1\nleft\u271d\u00b9 : a \u2208 s\nb : \u03b2\nleft\u271d : b \u2208 t\nh : inr b = inl a\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "full_name": "lt_mul_left", "start": [563, 1], "end": [565, 15], "traced_tactics": [{"tactic": "convert mul_lt_mul_of_pos_right hm hn", "annotated_tactic": ["convert <a>mul_lt_mul_of_pos_right</a> hm hn", [{"full_name": "mul_lt_mul_of_pos_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [164, 9], "def_end_pos": [164, 32]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : StrictOrderedSemiring \u03b1\na b c d : \u03b1\nhn : 0 < a\nhm : 1 < b\n\u22a2 a < b * a", "state_after": "case h.e'_3\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : StrictOrderedSemiring \u03b1\na b c d : \u03b1\nhn : 0 < a\nhm : 1 < b\n\u22a2 a = 1 * a"}, {"tactic": "rw [one_mul]", "annotated_tactic": ["rw [<a>one_mul</a>]", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "case h.e'_3\n\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : StrictOrderedSemiring \u03b1\na b c d : \u03b1\nhn : 0 < a\nhm : 1 < b\n\u22a2 a = 1 * a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "full_name": "coe_basisOfLinearIndependentOfCardEqFinrank", "start": [1207, 1], "end": [1210, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/ContDiff.lean", "full_name": "ContinuousLinearEquiv.comp_contDiffAt_iff", "start": [333, 1], "end": [335, 67], "traced_tactics": [{"tactic": "simp only [\u2190 contDiffWithinAt_univ, e.comp_contDiffWithinAt_iff]", "annotated_tactic": ["simp only [\u2190 <a>contDiffWithinAt_univ</a>, e.comp_contDiffWithinAt_iff]", [{"full_name": "contDiffWithinAt_univ", "def_path": "Mathlib/Analysis/Calculus/ContDiffDef.lean", "def_pos": [1335, 9], "def_end_pos": [1335, 30]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ne : F \u2243L[\ud835\udd5c] G\n\u22a2 ContDiffAt \ud835\udd5c n (\u2191e \u2218 f) x \u2194 ContDiffAt \ud835\udd5c n f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Prod.lean", "full_name": "fderivWithin_pi", "start": [451, 1], "end": [454, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/Deriv/Add.lean", "full_name": "HasDerivWithinAt.const_sub", "start": [352, 8], "end": [354, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/WithBot.lean", "full_name": "WithBot.toDual_lt_iff", "start": [1044, 1], "end": [1046, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "le_iSup'", "start": [819, 1], "end": [820, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Archimedean.lean", "full_name": "AddSubgroup.cyclic_of_min", "start": [39, 1], "end": [53, 66], "traced_tactics": [{"tactic": "obtain \u27e8\u27e8a_in, a_pos\u27e9, a_min\u27e9 := ha", "annotated_tactic": ["obtain \u27e8\u27e8a_in, a_pos\u27e9, a_min\u27e9 := ha", []], "state_before": "G : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\nha : IsLeast {g | g \u2208 H \u2227 0 < g} a\n\u22a2 H = closure {a}", "state_after": "case intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\n\u22a2 H = closure {a}"}, {"tactic": "refine' le_antisymm _ (H.closure_le.mpr <| by simp [a_in])", "annotated_tactic": ["refine' <a>le_antisymm</a> _ (H.closure_le.mpr <| by simp [a_in])", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "case intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\n\u22a2 H = closure {a}", "state_after": "case intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\n\u22a2 H \u2264 closure {a}"}, {"tactic": "intro g g_in", "annotated_tactic": ["intro g g_in", []], "state_before": "case intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\n\u22a2 H \u2264 closure {a}", "state_after": "case intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\n\u22a2 g \u2208 closure {a}"}, {"tactic": "obtain \u27e8k, \u27e8nonneg, lt\u27e9, _\u27e9 := existsUnique_zsmul_near_of_pos' a_pos g", "annotated_tactic": ["obtain \u27e8k, \u27e8nonneg, lt\u27e9, _\u27e9 := <a>existsUnique_zsmul_near_of_pos'</a> a_pos g", [{"full_name": "existsUnique_zsmul_near_of_pos'", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [82, 9], "def_end_pos": [82, 40]}]], "state_before": "case intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\n\u22a2 g \u2208 closure {a}", "state_after": "case intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\nk : \u2124\nright\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k\nnonneg : 0 \u2264 g - k \u2022 a\nlt : g - k \u2022 a < a\n\u22a2 g \u2208 closure {a}"}, {"tactic": "simp [sub_eq_zero.mp h_zero, AddSubgroup.mem_closure_singleton]", "annotated_tactic": ["simp [sub_eq_zero.mp h_zero, <a>AddSubgroup.mem_closure_singleton</a>]", [{"full_name": "AddSubgroup.mem_closure_singleton", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [1292, 3], "def_end_pos": [1292, 14]}]], "state_before": "case intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\nk : \u2124\nright\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k\nnonneg : 0 \u2264 g - k \u2022 a\nlt : g - k \u2022 a < a\nh_zero : g - k \u2022 a = 0\n\u22a2 g \u2208 closure {a}", "state_after": "no goals"}, {"tactic": "simp [a_in]", "annotated_tactic": ["simp [a_in]", []], "state_before": "G : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\n\u22a2 {a} \u2286 \u2191H", "state_after": "no goals"}, {"tactic": "by_contra h", "annotated_tactic": ["by_contra h", []], "state_before": "G : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\nk : \u2124\nright\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k\nnonneg : 0 \u2264 g - k \u2022 a\nlt : g - k \u2022 a < a\n\u22a2 g - k \u2022 a = 0", "state_after": "G : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\nk : \u2124\nright\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k\nnonneg : 0 \u2264 g - k \u2022 a\nlt : g - k \u2022 a < a\nh : \u00acg - k \u2022 a = 0\n\u22a2 False"}, {"tactic": "have h' : \u00aca \u2264 g - k \u2022 a := not_le.mpr lt", "annotated_tactic": ["have h' : \u00aca \u2264 g - k \u2022 a := not_le.mpr lt", []], "state_before": "G : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\nk : \u2124\nright\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k\nnonneg : 0 \u2264 g - k \u2022 a\nlt : g - k \u2022 a < a\nh\u271d : \u00acg - k \u2022 a = 0\nh : a \u2264 g - k \u2022 a\n\u22a2 False", "state_after": "G : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\nk : \u2124\nright\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k\nnonneg : 0 \u2264 g - k \u2022 a\nlt : g - k \u2022 a < a\nh\u271d : \u00acg - k \u2022 a = 0\nh : a \u2264 g - k \u2022 a\nh' : \u00aca \u2264 g - k \u2022 a\n\u22a2 False"}, {"tactic": "contradiction", "annotated_tactic": ["contradiction", []], "state_before": "G : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\nk : \u2124\nright\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k\nnonneg : 0 \u2264 g - k \u2022 a\nlt : g - k \u2022 a < a\nh\u271d : \u00acg - k \u2022 a = 0\nh : a \u2264 g - k \u2022 a\nh' : \u00aca \u2264 g - k \u2022 a\n\u22a2 False", "state_after": "no goals"}, {"tactic": "refine' a_min \u27e8_, _\u27e9", "annotated_tactic": ["refine' a_min \u27e8_, _\u27e9", []], "state_before": "G : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\nk : \u2124\nright\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k\nnonneg : 0 \u2264 g - k \u2022 a\nlt : g - k \u2022 a < a\nh : \u00acg - k \u2022 a = 0\n\u22a2 a \u2264 g - k \u2022 a", "state_after": "case refine'_1\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\nk : \u2124\nright\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k\nnonneg : 0 \u2264 g - k \u2022 a\nlt : g - k \u2022 a < a\nh : \u00acg - k \u2022 a = 0\n\u22a2 g - k \u2022 a \u2208 H\n\ncase refine'_2\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\nk : \u2124\nright\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k\nnonneg : 0 \u2264 g - k \u2022 a\nlt : g - k \u2022 a < a\nh : \u00acg - k \u2022 a = 0\n\u22a2 0 < g - k \u2022 a"}, {"tactic": "exact AddSubgroup.sub_mem H g_in (AddSubgroup.zsmul_mem H a_in k)", "annotated_tactic": ["exact <a>AddSubgroup.sub_mem</a> H g_in (<a>AddSubgroup.zsmul_mem</a> H a_in k)", [{"full_name": "AddSubgroup.sub_mem", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [586, 3], "def_end_pos": [586, 14]}, {"full_name": "AddSubgroup.zsmul_mem", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [629, 3], "def_end_pos": [629, 14]}]], "state_before": "case refine'_1\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\nk : \u2124\nright\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k\nnonneg : 0 \u2264 g - k \u2022 a\nlt : g - k \u2022 a < a\nh : \u00acg - k \u2022 a = 0\n\u22a2 g - k \u2022 a \u2208 H", "state_after": "no goals"}, {"tactic": "exact lt_of_le_of_ne nonneg (Ne.symm h)", "annotated_tactic": ["exact <a>lt_of_le_of_ne</a> nonneg (<a>Ne.symm</a> h)", [{"full_name": "lt_of_le_of_ne", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [196, 9], "def_end_pos": [196, 23]}, {"full_name": "Ne.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [575, 9], "def_end_pos": [575, 16]}]], "state_before": "case refine'_2\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\na : G\na_min : a \u2208 lowerBounds {g | g \u2208 H \u2227 0 < g}\na_in : a \u2208 H\na_pos : 0 < a\ng : G\ng_in : g \u2208 H\nk : \u2124\nright\u271d : \u2200 (y : \u2124), (fun k => 0 \u2264 g - k \u2022 a \u2227 g - k \u2022 a < a) y \u2192 y = k\nnonneg : 0 \u2264 g - k \u2022 a\nlt : g - k \u2022 a < a\nh : \u00acg - k \u2022 a = 0\n\u22a2 0 < g - k \u2022 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/LocalHomeomorph.lean", "full_name": "LocalHomeomorph.secondCountableTopology_source", "start": [1236, 1], "end": [1238, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/SetFamily/HarrisKleitman.lean", "full_name": "IsLowerSet.le_card_inter_finset", "start": [97, 1], "end": [99, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Irreducible.lean", "full_name": "supIrred_ofDual", "start": [242, 1], "end": [243, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "Monotone.map_iInf_of_continuousAt'", "start": [2732, 1], "end": [2736, 6], "traced_tactics": [{"tactic": "rw [iInf, Monotone.map_sInf_of_continuousAt' Cf Mf (range_nonempty g) bdd, \u2190 range_comp, iInf]", "annotated_tactic": ["rw [<a>iInf</a>, <a>Monotone.map_sInf_of_continuousAt'</a> Cf Mf (<a>range_nonempty</a> g) bdd, \u2190 <a>range_comp</a>, <a>iInf</a>]", [{"full_name": "iInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [83, 5], "def_end_pos": [83, 9]}, {"full_name": "Monotone.map_sInf_of_continuousAt'", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [2724, 9], "def_end_pos": [2724, 43]}, {"full_name": "Set.range_nonempty", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [771, 9], "def_end_pos": [771, 23]}, {"full_name": "Set.range_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [744, 9], "def_end_pos": [744, 19]}, {"full_name": "iInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [83, 5], "def_end_pos": [83, 9]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2077 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : OrderTopology \u03b1\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : OrderClosedTopology \u03b2\ninst\u271d\u00b9 : Nonempty \u03b3\n\u03b9 : Sort u_1\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\ng : \u03b9 \u2192 \u03b1\nCf : ContinuousAt f (iInf g)\nMf : Monotone f\nbdd : autoParam (BddBelow (range g)) _auto\u271d\n\u22a2 f (\u2a05 i, g i) = \u2a05 i, f (g i)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2077 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : OrderTopology \u03b1\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : OrderClosedTopology \u03b2\ninst\u271d\u00b9 : Nonempty \u03b3\n\u03b9 : Sort u_1\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\ng : \u03b9 \u2192 \u03b1\nCf : ContinuousAt f (iInf g)\nMf : Monotone f\nbdd : autoParam (BddBelow (range g)) _auto\u271d\n\u22a2 sInf (range (f \u2218 g)) = sInf (range fun i => f (g i))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2077 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b1\ninst\u271d\u2075 : OrderTopology \u03b1\ninst\u271d\u2074 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : OrderClosedTopology \u03b2\ninst\u271d\u00b9 : Nonempty \u03b3\n\u03b9 : Sort u_1\ninst\u271d : Nonempty \u03b9\nf : \u03b1 \u2192 \u03b2\ng : \u03b9 \u2192 \u03b1\nCf : ContinuousAt f (iInf g)\nMf : Monotone f\nbdd : autoParam (BddBelow (range g)) _auto\u271d\n\u22a2 sInf (range (f \u2218 g)) = sInf (range fun i => f (g i))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Subgroup/ZPowers.lean", "full_name": "Subgroup.center_eq_infi'", "start": [250, 1], "end": [252, 48], "traced_tactics": [{"tactic": "rw [center_eq_iInf S hS, \u2190 iInf_subtype'']", "annotated_tactic": ["rw [<a>center_eq_iInf</a> S hS, \u2190 <a>iInf_subtype''</a>]", [{"full_name": "Subgroup.center_eq_iInf", "def_path": "Mathlib/GroupTheory/Subgroup/ZPowers.lean", "def_pos": [243, 9], "def_end_pos": [243, 23]}, {"full_name": "iInf_subtype''", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1280, 9], "def_end_pos": [1280, 23]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\nA : Type u_2\ninst\u271d\u00b9 : AddGroup A\nN : Type u_3\ninst\u271d : Group N\ns : Set G\ng : G\nS : Set G\nhS : closure S = \u22a4\n\u22a2 center G = \u2a05 g, centralizer \u2191(zpowers \u2191g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/NormedSpace/Multilinear.lean", "full_name": "ContinuousMultilinearMap.nnnorm_ofSubsingleton", "start": [523, 1], "end": [525, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.rel_zero_right", "start": [2723, 1], "end": [2723, 85], "traced_tactics": [{"tactic": "rw [Rel_iff]", "annotated_tactic": ["rw [<a>Rel_iff</a>]", [{"full_name": "Multiset.Rel_iff", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2661, 3], "def_end_pos": [2661, 9]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na : Multiset \u03b1\n\u22a2 Rel r a 0 \u2194 a = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na : Multiset \u03b1\n\u22a2 (a = 0 \u2227 0 = 0 \u2228 \u2203 a_1 b as bs, r a_1 b \u2227 Rel r as bs \u2227 a = a_1 ::\u2098 as \u2227 0 = b ::\u2098 bs) \u2194 a = 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\na : Multiset \u03b1\n\u22a2 (a = 0 \u2227 0 = 0 \u2228 \u2203 a_1 b as bs, r a_1 b \u2227 Rel r as bs \u2227 a = a_1 ::\u2098 as \u2227 0 = b ::\u2098 bs) \u2194 a = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Real/Hyperreal.lean", "full_name": "Hyperreal.coe_ne_zero", "start": [55, 1], "end": [56, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Game/PGame.lean", "full_name": "SetTheory.PGame.zero_lt_one", "start": [1914, 11], "end": [1915, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Order/Compact.lean", "full_name": "IsCompact.isLUB_sSup", "start": [405, 1], "end": [407, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Perm/Support.lean", "full_name": "Equiv.Perm.disjoint_conj", "start": [118, 1], "end": [119, 71], "traced_tactics": [{"tactic": "simp only [mul_apply, eq_inv_iff_eq]", "annotated_tactic": ["simp only [<a>mul_apply</a>, <a>eq_inv_iff_eq</a>]", [{"full_name": "Equiv.Perm.mul_apply", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [67, 9], "def_end_pos": [67, 18]}, {"full_name": "Equiv.Perm.eq_inv_iff_eq", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [110, 9], "def_end_pos": [110, 22]}]], "state_before": "\u03b1 : Type u_1\nf g h\u271d h : Perm \u03b1\nx\u271d : \u03b1\n\u22a2 \u2191(h * f * h\u207b\u00b9) x\u271d = x\u271d \u2228 \u2191(h * g * h\u207b\u00b9) x\u271d = x\u271d \u2194 \u2191f (\u2191h\u207b\u00b9 x\u271d) = \u2191h\u207b\u00b9 x\u271d \u2228 \u2191g (\u2191h\u207b\u00b9 x\u271d) = \u2191h\u207b\u00b9 x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Bilinear.lean", "full_name": "IsBoundedBilinearMap.differentiableOn", "start": [120, 1], "end": [122, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "tendsto_atTop_of_eventually_const", "start": [1053, 1], "end": [1055, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/AlgebraTower.lean", "full_name": "Basis.algebraMap_injective", "start": [175, 1], "end": [178, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/ConstMulAction.lean", "full_name": "isClosedMap_smul\u2080", "start": [373, 1], "end": [379, 54], "traced_tactics": [{"tactic": "rcases eq_or_ne c 0 with (rfl | hne)", "annotated_tactic": ["rcases <a>eq_or_ne</a> c 0 with (rfl | hne)", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "M\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nG\u2080 : Type u_4\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b1\ninst\u271d\u2079 : GroupWithZero G\u2080\ninst\u271d\u2078 : MulAction G\u2080 \u03b1\ninst\u271d\u2077 : ContinuousConstSMul G\u2080 \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nc\u271d : G\u2080\ns : Set \u03b2\n\ud835\udd5c : Type u_5\nM : Type u_6\ninst\u271d\u2075 : DivisionRing \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : T1Space M\ninst\u271d\u00b9 : Module \ud835\udd5c M\ninst\u271d : ContinuousConstSMul \ud835\udd5c M\nc : \ud835\udd5c\n\u22a2 IsClosedMap fun x => c \u2022 x", "state_after": "case inl\nM\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nG\u2080 : Type u_4\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b1\ninst\u271d\u2079 : GroupWithZero G\u2080\ninst\u271d\u2078 : MulAction G\u2080 \u03b1\ninst\u271d\u2077 : ContinuousConstSMul G\u2080 \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nc : G\u2080\ns : Set \u03b2\n\ud835\udd5c : Type u_5\nM : Type u_6\ninst\u271d\u2075 : DivisionRing \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : T1Space M\ninst\u271d\u00b9 : Module \ud835\udd5c M\ninst\u271d : ContinuousConstSMul \ud835\udd5c M\n\u22a2 IsClosedMap fun x => 0 \u2022 x\n\ncase inr\nM\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nG\u2080 : Type u_4\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b1\ninst\u271d\u2079 : GroupWithZero G\u2080\ninst\u271d\u2078 : MulAction G\u2080 \u03b1\ninst\u271d\u2077 : ContinuousConstSMul G\u2080 \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nc\u271d : G\u2080\ns : Set \u03b2\n\ud835\udd5c : Type u_5\nM : Type u_6\ninst\u271d\u2075 : DivisionRing \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : T1Space M\ninst\u271d\u00b9 : Module \ud835\udd5c M\ninst\u271d : ContinuousConstSMul \ud835\udd5c M\nc : \ud835\udd5c\nhne : c \u2260 0\n\u22a2 IsClosedMap fun x => c \u2022 x"}, {"tactic": "simp only [zero_smul]", "annotated_tactic": ["simp only [<a>zero_smul</a>]", [{"full_name": "zero_smul", "def_path": "Mathlib/Algebra/SMulWithZero.lean", "def_pos": [70, 9], "def_end_pos": [70, 18]}]], "state_before": "case inl\nM\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nG\u2080 : Type u_4\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b1\ninst\u271d\u2079 : GroupWithZero G\u2080\ninst\u271d\u2078 : MulAction G\u2080 \u03b1\ninst\u271d\u2077 : ContinuousConstSMul G\u2080 \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nc : G\u2080\ns : Set \u03b2\n\ud835\udd5c : Type u_5\nM : Type u_6\ninst\u271d\u2075 : DivisionRing \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : T1Space M\ninst\u271d\u00b9 : Module \ud835\udd5c M\ninst\u271d : ContinuousConstSMul \ud835\udd5c M\n\u22a2 IsClosedMap fun x => 0 \u2022 x", "state_after": "case inl\nM\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nG\u2080 : Type u_4\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b1\ninst\u271d\u2079 : GroupWithZero G\u2080\ninst\u271d\u2078 : MulAction G\u2080 \u03b1\ninst\u271d\u2077 : ContinuousConstSMul G\u2080 \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nc : G\u2080\ns : Set \u03b2\n\ud835\udd5c : Type u_5\nM : Type u_6\ninst\u271d\u2075 : DivisionRing \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : T1Space M\ninst\u271d\u00b9 : Module \ud835\udd5c M\ninst\u271d : ContinuousConstSMul \ud835\udd5c M\n\u22a2 IsClosedMap fun x => 0"}, {"tactic": "exact isClosedMap_const", "annotated_tactic": ["exact <a>isClosedMap_const</a>", [{"full_name": "isClosedMap_const", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [656, 9], "def_end_pos": [656, 26]}]], "state_before": "case inl\nM\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nG\u2080 : Type u_4\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b1\ninst\u271d\u2079 : GroupWithZero G\u2080\ninst\u271d\u2078 : MulAction G\u2080 \u03b1\ninst\u271d\u2077 : ContinuousConstSMul G\u2080 \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nc : G\u2080\ns : Set \u03b2\n\ud835\udd5c : Type u_5\nM : Type u_6\ninst\u271d\u2075 : DivisionRing \ud835\udd5c\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : T1Space M\ninst\u271d\u00b9 : Module \ud835\udd5c M\ninst\u271d : ContinuousConstSMul \ud835\udd5c M\n\u22a2 IsClosedMap fun x => 0", "state_after": "no goals"}, {"tactic": "exact (Homeomorph.smulOfNeZero c hne).isClosedMap", "annotated_tactic": ["exact (<a>Homeomorph.smulOfNeZero</a> c hne).<a>isClosedMap</a>", [{"full_name": "Homeomorph.smulOfNeZero", "def_path": "Mathlib/Topology/Algebra/ConstMulAction.lean", "def_pos": [325, 15], "def_end_pos": [325, 38]}, {"full_name": "Homeomorph.isClosedMap", "def_path": "Mathlib/Topology/Homeomorph.lean", "def_pos": [367, 19], "def_end_pos": [367, 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NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nc : F\nh : Differentiable \ud835\udd5c fun y => c + f y\n\u22a2 Differentiable \ud835\udd5c f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "full_name": "Polynomial.monic_of_natDegree_le_of_coeff_eq_one", "start": [875, 1], "end": [880, 38], "traced_tactics": [{"tactic": "unfold Monic", "annotated_tactic": ["unfold <a>Monic</a>", [{"full_name": "Polynomial.Monic", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [75, 5], "def_end_pos": [75, 10]}]], "state_before": "R : Type u\nS : Type v\na b c d : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u03b9 : Type u_1\nn : \u2115\npn : natDegree p \u2264 n\np1 : coeff p n = 1\n\u22a2 Monic p", "state_after": "R : Type u\nS : Type v\na b c d : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u03b9 : Type u_1\nn : \u2115\npn : natDegree p \u2264 n\np1 : coeff p n = 1\n\u22a2 leadingCoeff p = 1"}, {"tactic": "nontriviality", "annotated_tactic": ["nontriviality", []], "state_before": "R : Type u\nS : Type v\na b c d : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u03b9 : Type u_1\nn : \u2115\npn : natDegree p \u2264 n\np1 : coeff p n = 1\n\u22a2 leadingCoeff p = 1", "state_after": "R : Type u\nS : Type v\na b c d : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u03b9 : Type u_1\nn : \u2115\npn : natDegree p \u2264 n\np1 : coeff p n = 1\n\u271d : Nontrivial R\n\u22a2 leadingCoeff p = 1"}, {"tactic": "refine' (congr_arg _ <| natDegree_eq_of_le_of_coeff_ne_zero pn _).trans p1", "annotated_tactic": ["refine' (<a>congr_arg</a> _ <| <a>natDegree_eq_of_le_of_coeff_ne_zero</a> pn _).<a>trans</a> p1", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "Polynomial.natDegree_eq_of_le_of_coeff_ne_zero", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [194, 9], "def_end_pos": [194, 44]}, {"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}]], "state_before": "R : Type u\nS : Type v\na b c d : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u03b9 : Type u_1\nn : \u2115\npn : natDegree p \u2264 n\np1 : coeff p n = 1\n\u271d : Nontrivial R\n\u22a2 leadingCoeff p = 1", "state_after": "R : Type u\nS : Type v\na b c d : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u03b9 : Type u_1\nn : \u2115\npn : natDegree p \u2264 n\np1 : coeff p n = 1\n\u271d : Nontrivial R\n\u22a2 coeff p n \u2260 0"}, {"tactic": "exact ne_of_eq_of_ne p1 one_ne_zero", "annotated_tactic": ["exact <a>ne_of_eq_of_ne</a> p1 <a>one_ne_zero</a>", [{"full_name": "ne_of_eq_of_ne", "def_path": "Mathlib/Init/CCLemmas.lean", "def_pos": [118, 9], "def_end_pos": [118, 23]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}]], "state_before": "R : Type u\nS : Type v\na b c d : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u03b9 : Type u_1\nn : \u2115\npn : natDegree p \u2264 n\np1 : coeff p n = 1\n\u271d : Nontrivial R\n\u22a2 coeff p n \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Heyting/Hom.lean", "full_name": "HeytingHom.cancel_left", "start": [361, 1], "end": [362, 92], "traced_tactics": [{"tactic": "rw [\u2190 comp_apply, h, comp_apply]", "annotated_tactic": ["rw [\u2190 <a>comp_apply</a>, h, <a>comp_apply</a>]", [{"full_name": "HeytingHom.comp_apply", "def_path": "Mathlib/Order/Heyting/Hom.lean", "def_pos": [335, 9], "def_end_pos": [335, 19]}, {"full_name": "HeytingHom.comp_apply", "def_path": "Mathlib/Order/Heyting/Hom.lean", "def_pos": [335, 9], "def_end_pos": [335, 19]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : HeytingAlgebra \u03b1\ninst\u271d\u00b2 : HeytingAlgebra \u03b2\ninst\u271d\u00b9 : HeytingAlgebra \u03b3\ninst\u271d : HeytingAlgebra \u03b4\nf f\u2081 f\u2082 : HeytingHom \u03b1 \u03b2\ng g\u2081 g\u2082 : HeytingHom \u03b2 \u03b3\nhg : Injective \u2191g\nh : comp g f\u2081 = comp g f\u2082\na : \u03b1\n\u22a2 \u2191g (\u2191f\u2081 a) = \u2191g (\u2191f\u2082 a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": 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{"full_name": "RingHom.comp_apply", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [672, 9], "def_end_pos": [672, 19]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\nx\u271d\u00b2 : NonAssocSemiring \u03b1\nx\u271d\u00b9 : NonAssocSemiring \u03b2\nx\u271d : NonAssocSemiring \u03b3\ng : \u03b2 \u2192+* \u03b3\nf\u2081 f\u2082 : \u03b1 \u2192+* \u03b2\nhg : Injective \u2191g\nh : comp g f\u2081 = comp g f\u2082\nx : \u03b1\n\u22a2 \u2191g (\u2191f\u2081 x) = \u2191g (\u2191f\u2082 x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Logic/Equiv/Basic.lean", "full_name": "Equiv.sumArrowEquivProdArrow_apply_fst", "start": [950, 1], "end": [952, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.iUnion_vsub_left_image", "start": [710, 1], "end": [711, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Group/Units.lean", "full_name": "Units.isUnit_units_mul", "start": [680, 1], "end": [686, 17], "traced_tactics": [{"tactic": "have : IsUnit (\u2191u\u207b\u00b9 * (\u2191u * a)) := by exists u\u207b\u00b9 * v; rw [\u2190 hv, Units.val_mul]", "annotated_tactic": ["have : <a>IsUnit</a> (\u2191u\u207b\u00b9 * (\u2191u * a)) := by exists u\u207b\u00b9 * v; rw [\u2190 hv, <a>Units.val_mul</a>]", [{"full_name": "IsUnit", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [592, 5], "def_end_pos": [592, 11]}, {"full_name": "Units.val_mul", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [229, 9], "def_end_pos": [229, 16]}]], "state_before": "\u03b1 : Type u\nM\u271d : Type u_1\nN : Type u_2\nM : Type u_3\ninst\u271d : Monoid M\nu : M\u02e3\na : M\nx\u271d : IsUnit (\u2191u * a)\nv : M\u02e3\nhv : \u2191v = \u2191u * a\n\u22a2 IsUnit a", "state_after": "\u03b1 : Type u\nM\u271d : Type u_1\nN : Type u_2\nM : Type u_3\ninst\u271d : Monoid M\nu : M\u02e3\na : M\nx\u271d : IsUnit (\u2191u * a)\nv : M\u02e3\nhv : \u2191v = \u2191u * a\nthis : IsUnit (\u2191u\u207b\u00b9 * (\u2191u * a))\n\u22a2 IsUnit a"}, {"tactic": "rwa [\u2190 mul_assoc, Units.inv_mul, one_mul] at this", "annotated_tactic": ["rwa [\u2190 <a>mul_assoc</a>, <a>Units.inv_mul</a>, <a>one_mul</a>] at this", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "Units.inv_mul", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [262, 9], "def_end_pos": [262, 16]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "\u03b1 : Type u\nM\u271d : Type u_1\nN : Type u_2\nM : Type u_3\ninst\u271d : Monoid M\nu : M\u02e3\na : M\nx\u271d : IsUnit (\u2191u * a)\nv : M\u02e3\nhv : \u2191v = \u2191u * a\nthis : IsUnit (\u2191u\u207b\u00b9 * (\u2191u * a))\n\u22a2 IsUnit a", "state_after": "no goals"}, {"tactic": "exists u\u207b\u00b9 * v", "annotated_tactic": ["exists u\u207b\u00b9 * v", []], "state_before": "\u03b1 : Type u\nM\u271d : Type u_1\nN : Type u_2\nM : Type u_3\ninst\u271d : Monoid M\nu : M\u02e3\na : M\nx\u271d : IsUnit (\u2191u * a)\nv : M\u02e3\nhv : \u2191v = \u2191u * a\n\u22a2 IsUnit (\u2191u\u207b\u00b9 * (\u2191u * a))", "state_after": "\u03b1 : Type u\nM\u271d : Type u_1\nN : Type u_2\nM : Type u_3\ninst\u271d : Monoid M\nu : M\u02e3\na : M\nx\u271d : IsUnit (\u2191u * a)\nv : M\u02e3\nhv : \u2191v = \u2191u * a\n\u22a2 \u2191(u\u207b\u00b9 * v) = \u2191u\u207b\u00b9 * (\u2191u * a)"}, {"tactic": "rw [\u2190 hv, Units.val_mul]", "annotated_tactic": ["rw [\u2190 hv, <a>Units.val_mul</a>]", [{"full_name": 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{"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Hom/Lattice.lean", "full_name": "BoundedLatticeHom.coe_id", "start": [1261, 1], "end": [1262, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "full_name": "OrderIso.map_csInf", "start": [1513, 1], "end": [1515, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Basic.lean", "full_name": "Set.Ioc_union_Ioc_union_Ioc_cycle", "start": [1879, 1], "end": [1889, 25], "traced_tactics": [{"tactic": "rw [Ioc_union_Ioc, Ioc_union_Ioc] <;>\nsimp [min_le_of_left_le, min_le_of_right_le, le_max_of_le_left, le_max_of_le_right, le_refl,\n  min_assoc, max_comm]", "annotated_tactic": ["rw [<a>Ioc_union_Ioc</a>, <a>Ioc_union_Ioc</a>] <;>\n  -- Porting note: mathlib3 proof finished from here as follows:\n  -- (It can probably be restored after https://github.com/leanprover-community/mathlib4/pull/856)\n  -- ac_rfl\n  -- all_goals\n  --   solve_by_elim (config := { max_depth := 5 }) [min_le_of_left_le, min_le_of_right_le,\n  --     le_max_of_le_left, le_max_of_le_right, le_refl]\n  simp [<a>min_le_of_left_le</a>, <a>min_le_of_right_le</a>, <a>le_max_of_le_left</a>, <a>le_max_of_le_right</a>, <a>le_refl</a>,\n    <a>min_assoc</a>, <a>max_comm</a>]", [{"full_name": "Set.Ioc_union_Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1635, 9], "def_end_pos": [1635, 22]}, {"full_name": "Set.Ioc_union_Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1635, 9], "def_end_pos": [1635, 22]}, {"full_name": "min_le_of_left_le", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [106, 9], "def_end_pos": [106, 26]}, {"full_name": "min_le_of_right_le", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [110, 9], "def_end_pos": [110, 27]}, {"full_name": "le_max_of_le_left", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [90, 9], "def_end_pos": [90, 26]}, {"full_name": "le_max_of_le_right", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [94, 9], "def_end_pos": [94, 27]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}, {"full_name": "min_assoc", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [84, 9], "def_end_pos": [84, 18]}, {"full_name": "max_comm", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [113, 9], "def_end_pos": [113, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\na a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1\n\u22a2 Ioc a b \u222a Ioc b c \u222a Ioc c a = Ioc (min a (min b c)) (max a (max b c))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/GroupPower/Basic.lean", "full_name": "Commute.mul_zpow", "start": [372, 11], "end": [374, 71], "traced_tactics": [{"tactic": "simp [zpow_ofNat, h.mul_pow n]", "annotated_tactic": ["simp [<a>zpow_ofNat</a>, h.mul_pow n]", [{"full_name": "zpow_ofNat", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [948, 9], "def_end_pos": [948, 19]}]], "state_before": "\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na b : \u03b1\nh : Commute a b\nn : \u2115\n\u22a2 (a * b) ^ \u2191n = a ^ \u2191n * b ^ \u2191n", "state_after": "no goals"}, {"tactic": "simp [h.mul_pow, (h.pow_pow _ _).eq, mul_inv_rev]", "annotated_tactic": ["simp [h.mul_pow, (h.pow_pow _ _).<a>eq</a>, <a>mul_inv_rev</a>]", [{"full_name": "Commute.eq", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [47, 19], "def_end_pos": [47, 21]}, {"full_name": "mul_inv_rev", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1050, 9], "def_end_pos": [1050, 20]}]], "state_before": "\u03b1 : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d : DivisionMonoid \u03b1\na b : \u03b1\nh : Commute a b\nn : \u2115\n\u22a2 (a * b) ^ Int.negSucc n = a ^ Int.negSucc n * b ^ Int.negSucc n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Lagrange.lean", "full_name": "Lagrange.nodal_insert_eq_nodal", "start": [569, 1], "end": [571, 34], "traced_tactics": [{"tactic": "simp_rw [nodal, prod_insert hi]", "annotated_tactic": ["simp_rw [<a>nodal</a>, <a>prod_insert</a> hi]", [{"full_name": "Lagrange.nodal", "def_path": "Mathlib/LinearAlgebra/Lagrange.lean", "def_pos": [511, 5], "def_end_pos": [511, 10]}, {"full_name": "Finset.prod_insert", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [317, 9], "def_end_pos": [317, 20]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 R\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nhi : \u00aci \u2208 s\n\u22a2 nodal (insert i s) v = (X - \u2191C (v i)) * nodal s v", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean", "full_name": "Complex.betaIntegral_convergent", "start": [81, 1], "end": [91, 13], "traced_tactics": [{"tactic": "refine' (betaIntegral_convergent_left hu v).trans _", "annotated_tactic": ["refine' (<a>betaIntegral_convergent_left</a> hu v).<a>trans</a> _", [{"full_name": "Complex.betaIntegral_convergent_left", "def_path": "Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean", "def_pos": [64, 9], "def_end_pos": [64, 37]}, {"full_name": "IntervalIntegrable.trans", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [148, 9], "def_end_pos": [148, 14]}]], "state_before": "u v : \u2102\nhu : 0 < u.re\nhv : 0 < v.re\n\u22a2 IntervalIntegrable (fun x => \u2191x ^ (u - 1) * (1 - \u2191x) ^ (v - 1)) volume 0 1", "state_after": "u v : \u2102\nhu : 0 < u.re\nhv : 0 < v.re\n\u22a2 IntervalIntegrable (fun x => \u2191x ^ (u - 1) * (1 - \u2191x) ^ (v - 1)) volume (1 / 2) 1"}, {"tactic": "rw [IntervalIntegrable.iff_comp_neg]", "annotated_tactic": ["rw [<a>IntervalIntegrable.iff_comp_neg</a>]", [{"full_name": "IntervalIntegrable.iff_comp_neg", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [338, 9], "def_end_pos": [338, 21]}]], "state_before": "u v : \u2102\nhu : 0 < u.re\nhv : 0 < v.re\n\u22a2 IntervalIntegrable (fun x => \u2191x ^ (u - 1) * (1 - \u2191x) ^ (v - 1)) volume (1 / 2) 1", "state_after": "u v : \u2102\nhu : 0 < u.re\nhv : 0 < v.re\n\u22a2 IntervalIntegrable (fun x => \u2191(-x) ^ (u - 1) * (1 - \u2191(-x)) ^ (v - 1)) volume (-(1 / 2)) (-1)"}, {"tactic": "convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1", "annotated_tactic": ["convert ((<a>betaIntegral_convergent_left</a> hv u).<a>comp_add_right</a> 1).<a>symm</a> using 1", [{"full_name": "Complex.betaIntegral_convergent_left", "def_path": "Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean", "def_pos": [64, 9], "def_end_pos": [64, 37]}, {"full_name": "IntervalIntegrable.comp_add_right", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [316, 9], "def_end_pos": [316, 23]}, {"full_name": "IntervalIntegrable.symm", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [139, 16], "def_end_pos": [139, 20]}]], "state_before": "u v : \u2102\nhu : 0 < u.re\nhv : 0 < v.re\n\u22a2 IntervalIntegrable (fun x => \u2191(-x) ^ (u - 1) * (1 - \u2191(-x)) ^ (v - 1)) volume (-(1 / 2)) (-1)", "state_after": "case h.e'_3\nu v : \u2102\nhu : 0 < u.re\nhv : 0 < v.re\n\u22a2 (fun x => \u2191(-x) ^ (u - 1) * (1 - \u2191(-x)) ^ (v - 1)) = fun x => \u2191(x + 1) ^ (v - 1) * (1 - \u2191(x + 1)) ^ (u - 1)\n\ncase h.e'_5\nu v : \u2102\nhu : 0 < u.re\nhv : 0 < v.re\n\u22a2 -(1 / 2) = 1 / 2 - 1\n\ncase h.e'_6\nu v : \u2102\nhu : 0 < u.re\nhv : 0 < v.re\n\u22a2 -1 = 0 - 1"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "case h.e'_3\nu v : \u2102\nhu : 0 < u.re\nhv : 0 < v.re\n\u22a2 (fun x => \u2191(-x) ^ (u - 1) * (1 - \u2191(-x)) ^ (v - 1)) = fun x => \u2191(x + 1) ^ (v - 1) * (1 - \u2191(x + 1)) ^ (u - 1)", "state_after": "case h.e'_3.h\nu v : \u2102\nhu : 0 < u.re\nhv : 0 < v.re\nx : \u211d\n\u22a2 \u2191(-x) ^ (u - 1) * (1 - \u2191(-x)) ^ (v - 1) = \u2191(x + 1) ^ (v - 1) * (1 - \u2191(x + 1)) ^ (u - 1)"}, {"tactic": "conv_lhs => rw [mul_comm]", "annotated_tactic": ["conv_lhs => rw [<a>mul_comm</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "case h.e'_3.h\nu v : \u2102\nhu : 0 < u.re\nhv : 0 < v.re\nx : \u211d\n\u22a2 \u2191(-x) ^ (u - 1) * (1 - \u2191(-x)) ^ (v - 1) = \u2191(x + 1) ^ (v - 1) * (1 - \u2191(x + 1)) ^ (u - 1)", "state_after": "case h.e'_3.h\nu v : \u2102\nhu : 0 < u.re\nhv : 0 < v.re\nx : \u211d\n\u22a2 (1 - \u2191(-x)) ^ (v - 1) * \u2191(-x) ^ (u - 1) = \u2191(x + 1) ^ (v - 1) * (1 - \u2191(x + 1)) ^ (u - 1)"}, {"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "state_before": "case h.e'_3.h.e_a.e_a\nu v : \u2102\nhu : 0 < u.re\nhv : 0 < v.re\nx : \u211d\n\u22a2 \u2191(-x) = 1 - \u2191(x + 1)", "state_after": "case h.e'_3.h.e_a.e_a\nu v : \u2102\nhu : 0 < u.re\nhv : 0 < v.re\nx : \u211d\n\u22a2 -\u2191x = 1 - (\u2191x + 1)"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case h.e'_3.h.e_a.e_a\nu v : \u2102\nhu : 0 < u.re\nhv : 0 < v.re\nx : \u211d\n\u22a2 -\u2191x = 1 - (\u2191x + 1)", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "case h.e'_5\nu v : \u2102\nhu : 0 < u.re\nhv : 0 < v.re\n\u22a2 -(1 / 2) = 1 / 2 - 1", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "case h.e'_6\nu v : \u2102\nhu : 0 < u.re\nhv : 0 < v.re\n\u22a2 -1 = 0 - 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean", "full_name": "Matrix.represents_iff'", "start": [102, 1], "end": [113, 12], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nb : \u03b9 \u2192 M\nhb : Submodule.span R (Set.range b) = \u22a4\ninst\u271d : DecidableEq \u03b9\nA : Matrix \u03b9 \u03b9 R\nf : Module.End R M\n\u22a2 Represents b A f \u2194 \u2200 (j : \u03b9), \u2211 i : \u03b9, A i j \u2022 b i = \u2191f (b j)", "state_after": "case mp\n\u03b9 : Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nb : \u03b9 \u2192 M\nhb : Submodule.span R (Set.range b) = \u22a4\ninst\u271d : DecidableEq \u03b9\nA : Matrix \u03b9 \u03b9 R\nf : Module.End R M\n\u22a2 Represents b A f \u2192 \u2200 (j : \u03b9), \u2211 i : \u03b9, A i j \u2022 b i = \u2191f (b j)\n\ncase mpr\n\u03b9 : Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nb : \u03b9 \u2192 M\nhb : Submodule.span R (Set.range b) = \u22a4\ninst\u271d : DecidableEq \u03b9\nA : Matrix \u03b9 \u03b9 R\nf : Module.End R M\n\u22a2 (\u2200 (j : \u03b9), \u2211 i : \u03b9, A i j \u2022 b i = \u2191f (b j)) \u2192 Represents b A f"}, {"tactic": "intro h i", "annotated_tactic": ["intro h i", []], "state_before": "case mp\n\u03b9 : Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nb : \u03b9 \u2192 M\nhb : Submodule.span R (Set.range b) = \u22a4\ninst\u271d : DecidableEq \u03b9\nA : Matrix \u03b9 \u03b9 R\nf : Module.End R M\n\u22a2 Represents b A f \u2192 \u2200 (j : \u03b9), \u2211 i : \u03b9, A i j \u2022 b i = \u2191f (b j)", "state_after": "case mp\n\u03b9 : Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nb : \u03b9 \u2192 M\nhb : Submodule.span R (Set.range b) = \u22a4\ninst\u271d : DecidableEq \u03b9\nA : Matrix \u03b9 \u03b9 R\nf : Module.End R M\nh : Represents b A f\ni : \u03b9\n\u22a2 \u2211 i_1 : \u03b9, A i_1 i \u2022 b i_1 = \u2191f (b i)"}, {"tactic": "have := LinearMap.congr_fun h (Pi.single i 1)", "annotated_tactic": ["have := <a>LinearMap.congr_fun</a> h (<a>Pi.single</a> i 1)", [{"full_name": "LinearMap.congr_fun", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [320, 19], "def_end_pos": [320, 28]}, {"full_name": "Pi.single", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [231, 3], "def_end_pos": [231, 14]}]], "state_before": "case mp\n\u03b9 : Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nb : \u03b9 \u2192 M\nhb : Submodule.span R (Set.range b) = \u22a4\ninst\u271d : DecidableEq \u03b9\nA : Matrix \u03b9 \u03b9 R\nf : Module.End R M\nh : Represents b A f\ni : \u03b9\n\u22a2 \u2211 i_1 : \u03b9, A i_1 i \u2022 b i_1 = \u2191f (b i)", "state_after": "case mp\n\u03b9 : Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nb : \u03b9 \u2192 M\nhb : Submodule.span R (Set.range b) = \u22a4\ninst\u271d : DecidableEq \u03b9\nA : Matrix \u03b9 \u03b9 R\nf : Module.End R M\nh : Represents b A f\ni : \u03b9\nthis : \u2191(\u2191(PiToModule.fromMatrix R b) A) (Pi.single i 1) = \u2191(\u2191(PiToModule.fromEnd R b) f) (Pi.single i 1)\n\u22a2 \u2211 i_1 : \u03b9, A i_1 i \u2022 b i_1 = \u2191f (b i)"}, {"tactic": "rwa [PiToModule.fromEnd_apply_single_one, PiToModule.fromMatrix_apply_single_one] at this", "annotated_tactic": ["rwa [<a>PiToModule.fromEnd_apply_single_one</a>, <a>PiToModule.fromMatrix_apply_single_one</a>] at this", [{"full_name": "PiToModule.fromEnd_apply_single_one", "def_path": "Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean", "def_pos": [62, 9], "def_end_pos": [62, 44]}, {"full_name": "PiToModule.fromMatrix_apply_single_one", "def_path": "Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean", "def_pos": [45, 9], "def_end_pos": [45, 47]}]], "state_before": "case mp\n\u03b9 : Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nb : \u03b9 \u2192 M\nhb : Submodule.span R (Set.range b) = \u22a4\ninst\u271d : DecidableEq \u03b9\nA : Matrix \u03b9 \u03b9 R\nf : Module.End R M\nh : Represents b A f\ni : \u03b9\nthis : \u2191(\u2191(PiToModule.fromMatrix R b) A) (Pi.single i 1) = \u2191(\u2191(PiToModule.fromEnd R b) f) (Pi.single i 1)\n\u22a2 \u2211 i_1 : \u03b9, A i_1 i \u2022 b i_1 = \u2191f (b i)", "state_after": "no goals"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mpr\n\u03b9 : Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nb : \u03b9 \u2192 M\nhb : Submodule.span R (Set.range b) = \u22a4\ninst\u271d : DecidableEq \u03b9\nA : Matrix \u03b9 \u03b9 R\nf : Module.End R M\n\u22a2 (\u2200 (j : \u03b9), \u2211 i : \u03b9, A i j \u2022 b i = \u2191f (b j)) \u2192 Represents b A f", "state_after": "case mpr\n\u03b9 : Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nb : \u03b9 \u2192 M\nhb : Submodule.span R (Set.range b) = \u22a4\ninst\u271d : DecidableEq \u03b9\nA : Matrix \u03b9 \u03b9 R\nf : Module.End R M\nh : \u2200 (j : \u03b9), \u2211 i : \u03b9, A i j \u2022 b i = \u2191f (b j)\n\u22a2 Represents b A f"}, {"tactic": "refine LinearMap.pi_ext' (fun i => LinearMap.ext_ring ?_)", "annotated_tactic": ["refine <a>LinearMap.pi_ext'</a> (fun i => <a>LinearMap.ext_ring</a> ?_)", [{"full_name": "LinearMap.pi_ext'", "def_path": "Mathlib/LinearAlgebra/Pi.lean", "def_pos": [187, 9], "def_end_pos": [187, 16]}, {"full_name": "LinearMap.ext_ring", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [496, 9], "def_end_pos": [496, 17]}]], "state_before": "case mpr\n\u03b9 : Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nb : \u03b9 \u2192 M\nhb : Submodule.span R (Set.range b) = \u22a4\ninst\u271d : DecidableEq \u03b9\nA : Matrix \u03b9 \u03b9 R\nf : Module.End R M\nh : \u2200 (j : \u03b9), \u2211 i : \u03b9, A i j \u2022 b i = \u2191f (b j)\n\u22a2 Represents b A f", "state_after": "case mpr\n\u03b9 : Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nb : \u03b9 \u2192 M\nhb : Submodule.span R (Set.range b) = \u22a4\ninst\u271d : DecidableEq \u03b9\nA : Matrix \u03b9 \u03b9 R\nf : Module.End R M\nh : \u2200 (j : \u03b9), \u2211 i : \u03b9, A i j \u2022 b i = \u2191f (b j)\ni : \u03b9\n\u22a2 \u2191(LinearMap.comp (\u2191(PiToModule.fromMatrix R b) A) (LinearMap.single i)) 1 =\n    \u2191(LinearMap.comp (\u2191(PiToModule.fromEnd R b) f) (LinearMap.single i)) 1"}, {"tactic": "simp_rw [LinearMap.comp_apply, LinearMap.coe_single, PiToModule.fromEnd_apply_single_one,\n  PiToModule.fromMatrix_apply_single_one]", "annotated_tactic": ["simp_rw [<a>LinearMap.comp_apply</a>, <a>LinearMap.coe_single</a>, <a>PiToModule.fromEnd_apply_single_one</a>,\n      <a>PiToModule.fromMatrix_apply_single_one</a>]", [{"full_name": "LinearMap.comp_apply", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [549, 9], "def_end_pos": [549, 19]}, {"full_name": "LinearMap.coe_single", "def_path": "Mathlib/LinearAlgebra/Pi.lean", "def_pos": [132, 9], "def_end_pos": [132, 19]}, {"full_name": "PiToModule.fromEnd_apply_single_one", "def_path": "Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean", "def_pos": [62, 9], "def_end_pos": [62, 44]}, {"full_name": "PiToModule.fromMatrix_apply_single_one", "def_path": "Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean", "def_pos": [45, 9], "def_end_pos": [45, 47]}]], "state_before": "case mpr\n\u03b9 : Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nb : \u03b9 \u2192 M\nhb : Submodule.span R (Set.range b) = \u22a4\ninst\u271d : DecidableEq \u03b9\nA : Matrix \u03b9 \u03b9 R\nf : Module.End R M\nh : \u2200 (j : \u03b9), \u2211 i : \u03b9, A i j \u2022 b i = \u2191f (b j)\ni : \u03b9\n\u22a2 \u2191(LinearMap.comp (\u2191(PiToModule.fromMatrix R b) A) (LinearMap.single i)) 1 =\n    \u2191(LinearMap.comp (\u2191(PiToModule.fromEnd R b) f) (LinearMap.single i)) 1", "state_after": "case mpr\n\u03b9 : Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nb : \u03b9 \u2192 M\nhb : Submodule.span R (Set.range b) = \u22a4\ninst\u271d : DecidableEq \u03b9\nA : Matrix \u03b9 \u03b9 R\nf : Module.End R M\nh : \u2200 (j : \u03b9), \u2211 i : \u03b9, A i j \u2022 b i = \u2191f (b j)\ni : \u03b9\n\u22a2 \u2211 i_1 : \u03b9, A i_1 i \u2022 b i_1 = \u2191f (b i)"}, {"tactic": "apply h", "annotated_tactic": ["apply h", []], "state_before": "case mpr\n\u03b9 : Type u_1\ninst\u271d\u2074 : Fintype \u03b9\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nb : \u03b9 \u2192 M\nhb : Submodule.span R (Set.range b) = \u22a4\ninst\u271d : DecidableEq \u03b9\nA : Matrix \u03b9 \u03b9 R\nf : Module.End R M\nh : \u2200 (j : \u03b9), \u2211 i : \u03b9, A i j \u2022 b i = \u2191f (b j)\ni : \u03b9\n\u22a2 \u2211 i_1 : \u03b9, A i_1 i \u2022 b i_1 = \u2191f (b i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Semiring.lean", "full_name": "SetSemiring.add_def", "start": [116, 1], "end": [117, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Fintype/Basic.lean", "full_name": "Finset.univ_map_equiv_to_embedding", "start": [308, 1], "end": [310, 78], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Fintype \u03b1\u271d\ns t : Finset \u03b1\u271d\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u03b2\ne : \u03b1 \u2243 \u03b2\nb : \u03b2\n\u22a2 \u2191(Equiv.toEmbedding e) (\u2191e.symm b) = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "full_name": "Metric.isOpen_thickening", "start": [922, 1], "end": [923, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "IsClosed.compl_mem_nhds", "start": [936, 1], "end": [937, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Module/Basic.lean", "full_name": "Nat.noZeroSMulDivisors", "start": [635, 1], "end": [639, 10], "traced_tactics": [{"tactic": "intro c x", "annotated_tactic": ["intro c x", []], "state_before": "\u03b1 : Type u_1\nR : Type u_2\nk : Type u_3\nS : Type u_4\nM : Type u_5\nM\u2082 : Type u_6\nM\u2083 : Type u_7\n\u03b9 : Type u_8\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroSMulDivisors R M\ninst\u271d : CharZero R\n\u22a2 \u2200 {c : \u2115} {x : M}, c \u2022 x = 0 \u2192 c = 0 \u2228 x = 0", "state_after": "\u03b1 : Type u_1\nR : Type u_2\nk : Type u_3\nS : Type u_4\nM : Type u_5\nM\u2082 : Type u_6\nM\u2083 : Type u_7\n\u03b9 : Type u_8\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroSMulDivisors R M\ninst\u271d : CharZero R\nc : \u2115\nx : M\n\u22a2 c \u2022 x = 0 \u2192 c = 0 \u2228 x = 0"}, {"tactic": "rw [nsmul_eq_smul_cast R, smul_eq_zero]", "annotated_tactic": ["rw [<a>nsmul_eq_smul_cast</a> R, <a>smul_eq_zero</a>]", [{"full_name": "nsmul_eq_smul_cast", "def_path": "Mathlib/Algebra/Module/Basic.lean", "def_pos": [390, 9], "def_end_pos": [390, 27]}, {"full_name": "smul_eq_zero", "def_path": "Mathlib/Algebra/Module/Basic.lean", "def_pos": [614, 9], "def_end_pos": [614, 21]}]], "state_before": "\u03b1 : Type u_1\nR : Type u_2\nk : Type u_3\nS : Type u_4\nM : Type u_5\nM\u2082 : Type u_6\nM\u2083 : Type u_7\n\u03b9 : Type u_8\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroSMulDivisors R M\ninst\u271d : CharZero R\nc : \u2115\nx : M\n\u22a2 c \u2022 x = 0 \u2192 c = 0 \u2228 x = 0", "state_after": "\u03b1 : Type u_1\nR : Type u_2\nk : Type u_3\nS : Type u_4\nM : Type u_5\nM\u2082 : Type u_6\nM\u2083 : Type u_7\n\u03b9 : Type u_8\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroSMulDivisors R M\ninst\u271d : CharZero R\nc : \u2115\nx : M\n\u22a2 \u2191c = 0 \u2228 x = 0 \u2192 c = 0 \u2228 x = 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\nR : Type u_2\nk : Type u_3\nS : Type u_4\nM : Type u_5\nM\u2082 : Type u_6\nM\u2083 : Type u_7\n\u03b9 : Type u_8\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroSMulDivisors R M\ninst\u271d : CharZero R\nc : \u2115\nx : M\n\u22a2 \u2191c = 0 \u2228 x = 0 \u2192 c = 0 \u2228 x = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "isOpen_Ioo", "start": [336, 1], "end": [337, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finite/Card.lean", "full_name": "Finite.one_lt_card_iff_nontrivial", "start": [83, 1], "end": [85, 75], "traced_tactics": [{"tactic": "haveI := Fintype.ofFinite \u03b1", "annotated_tactic": ["haveI := <a>Fintype.ofFinite</a> \u03b1", [{"full_name": "Fintype.ofFinite", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [448, 19], "def_end_pos": [448, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\n\u22a2 1 < Nat.card \u03b1 \u2194 Nontrivial \u03b1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Finite \u03b1\nthis : Fintype \u03b1\n\u22a2 1 < Nat.card \u03b1 \u2194 Nontrivial \u03b1"}, {"tactic": "simp only [Nat.card_eq_fintype_card, Fintype.one_lt_card_iff_nontrivial]", "annotated_tactic": ["simp only [<a>Nat.card_eq_fintype_card</a>, <a>Fintype.one_lt_card_iff_nontrivial</a>]", [{"full_name": 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36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/UpperLower/Basic.lean", "full_name": "LowerSet.symm_map", "start": [1086, 1], "end": [1087, 90], "traced_tactics": [{"tactic": "convert Set.preimage_equiv_eq_image_symm s f.toEquiv", "annotated_tactic": ["convert <a>Set.preimage_equiv_eq_image_symm</a> s f.toEquiv", [{"full_name": "Set.preimage_equiv_eq_image_symm", "def_path": "Mathlib/Logic/Equiv/Set.lean", "def_pos": [57, 9], "def_end_pos": [57, 48]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03ba : \u03b9 \u2192 Sort u_5\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf\u271d : \u03b1 \u2243o \u03b2\ns\u271d t : LowerSet \u03b1\na : \u03b1\nb : \u03b2\nf : \u03b1 \u2243o \u03b2\ns : LowerSet \u03b2\n\u22a2 \u2191(\u2191(OrderIso.symm (map f)) s) = \u2191(\u2191(map (OrderIso.symm f)) s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Bitwise.lean", "full_name": "Int.bit_negSucc", "start": [159, 1], "end": [161, 18], "traced_tactics": [{"tactic": "rw [bit_val, Nat.bit_val]", "annotated_tactic": ["rw [<a>bit_val</a>, <a>Nat.bit_val</a>]", [{"full_name": "Int.bit_val", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [130, 9], "def_end_pos": [130, 16]}, {"full_name": "Nat.bit_val", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [163, 9], "def_end_pos": [163, 16]}]], "state_before": "b : Bool\nn : \u2115\n\u22a2 bit b -[n+1] = -[Nat.bit (!b) n+1]", "state_after": "b : Bool\nn : \u2115\n\u22a2 (2 * -[n+1] + bif b then 1 else 0) = -[2 * n + bif !b then 1 else 0+1]"}, {"tactic": "cases b <;> rfl", "annotated_tactic": ["cases b <;> rfl", []], "state_before": "b : Bool\nn : \u2115\n\u22a2 (2 * -[n+1] + bif b then 1 else 0) = -[2 * n + bif !b then 1 else 0+1]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/NonUnitalSubring/Basic.lean", "full_name": "NonUnitalSubring.mem_carrier", "start": [160, 1], "end": [161, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/CubicDiscriminant.lean", "full_name": "Cubic.of_d_eq_zero", "start": [161, 1], "end": [163, 38], "traced_tactics": [{"tactic": "rw [of_c_eq_zero ha hb hc, hd, C_0]", "annotated_tactic": ["rw [<a>of_c_eq_zero</a> ha hb hc, hd, <a>C_0</a>]", [{"full_name": "Cubic.of_c_eq_zero", "def_path": "Mathlib/Algebra/CubicDiscriminant.lean", "def_pos": [153, 9], "def_end_pos": [153, 21]}, {"full_name": "Polynomial.C_0", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": 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"Mathlib/Geometry/Manifold/LocalInvariantProperties.lean", "full_name": "StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_source_aux", "start": [253, 1], "end": [268, 12], "traced_tactics": [{"tactic": "rw [\u2190 hG.right_invariance (compatible_of_mem_maximalAtlas he he')]", "annotated_tactic": ["rw [\u2190 hG.right_invariance (<a>compatible_of_mem_maximalAtlas</a> he he')]", [{"full_name": "StructureGroupoid.compatible_of_mem_maximalAtlas", "def_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "def_pos": [1016, 9], "def_end_pos": [1016, 57]}]], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 P (g \u2218 \u2191(LocalHomeomorph.symm e)) (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) (\u2191e x) \u2194\n    P (g \u2218 \u2191(LocalHomeomorph.symm e')) (\u2191(LocalHomeomorph.symm e') \u207b\u00b9' s) (\u2191e' x)", "state_after": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 P ((g \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')))\n      (\u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s))\n      (\u2191(LocalHomeomorph.symm e \u226b\u2095 e') (\u2191e x)) \u2194\n    P (g \u2218 \u2191(LocalHomeomorph.symm e')) (\u2191(LocalHomeomorph.symm e') \u207b\u00b9' s) (\u2191e' x)\n\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 \u2191e x \u2208 (LocalHomeomorph.symm e \u226b\u2095 e').toLocalEquiv.source"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 P ((g \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')))\n      (\u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s))\n      (\u2191(LocalHomeomorph.symm e \u226b\u2095 e') (\u2191e x)) \u2194\n    P (g \u2218 \u2191(LocalHomeomorph.symm e')) (\u2191(LocalHomeomorph.symm e') \u207b\u00b9' s) (\u2191e' x)\n\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 \u2191e x \u2208 (LocalHomeomorph.symm e \u226b\u2095 e').toLocalEquiv.source", "state_after": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 \u2191e x \u2208 (LocalHomeomorph.symm e \u226b\u2095 e').toLocalEquiv.source\n\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 P ((g \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')))\n      (\u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s))\n      (\u2191(LocalHomeomorph.symm e \u226b\u2095 e') (\u2191e x)) \u2194\n    P (g \u2218 \u2191(LocalHomeomorph.symm e')) (\u2191(LocalHomeomorph.symm e') \u207b\u00b9' s) (\u2191e' x)"}, {"tactic": "simp_rw [LocalHomeomorph.trans_apply, e.left_inv xe]", "annotated_tactic": ["simp_rw [<a>LocalHomeomorph.trans_apply</a>, e.left_inv xe]", [{"full_name": "LocalHomeomorph.trans_apply", "def_path": "Mathlib/Topology/LocalHomeomorph.lean", "def_pos": [816, 9], "def_end_pos": [816, 20]}]], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 P ((g \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')))\n      (\u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s))\n      (\u2191(LocalHomeomorph.symm e \u226b\u2095 e') (\u2191e x)) \u2194\n    P (g \u2218 \u2191(LocalHomeomorph.symm e')) (\u2191(LocalHomeomorph.symm e') \u207b\u00b9' s) (\u2191e' x)", "state_after": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 P ((g \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')))\n      (\u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s)) (\u2191e' x) \u2194\n    P (g \u2218 \u2191(LocalHomeomorph.symm e')) (\u2191(LocalHomeomorph.symm e') \u207b\u00b9' s) (\u2191e' x)"}, {"tactic": "rw [hG.congr_iff]", "annotated_tactic": ["rw [hG.congr_iff]", []], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 P ((g \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')))\n      (\u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s)) (\u2191e' x) \u2194\n    P (g \u2218 \u2191(LocalHomeomorph.symm e')) (\u2191(LocalHomeomorph.symm e') \u207b\u00b9' s) (\u2191e' x)", "state_after": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 P ?m.18148 (\u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s)) (\u2191e' x) \u2194\n    P (g \u2218 \u2191(LocalHomeomorph.symm e')) (\u2191(LocalHomeomorph.symm e') \u207b\u00b9' s) (\u2191e' x)\n\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 (g \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')) =\u1da0[\ud835\udcdd (\u2191e' x)] ?m.18148\n\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 H \u2192 H'"}, {"tactic": "simp only [xe, xe', mfld_simps]", "annotated_tactic": ["simp only [xe, xe', mfld_simps]", []], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 \u2191e x \u2208 (LocalHomeomorph.symm e \u226b\u2095 e').toLocalEquiv.source", "state_after": "no goals"}, {"tactic": "refine' hG.congr_set _", "annotated_tactic": ["refine' hG.congr_set _", []], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 P ?m.18148 (\u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s)) (\u2191e' x) \u2194\n    P (g \u2218 \u2191(LocalHomeomorph.symm e')) (\u2191(LocalHomeomorph.symm e') \u207b\u00b9' s) (\u2191e' x)", "state_after": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) =\u1da0[\ud835\udcdd (\u2191e' x)]\n    \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s"}, {"tactic": "refine' (eventually_of_mem _ fun y (hy : y \u2208 e'.symm \u207b\u00b9' e.source) \u21a6 _).set_eq", "annotated_tactic": ["refine' (<a>eventually_of_mem</a> _ fun y (hy : y \u2208 e'.symm \u207b\u00b9' e.source) \u21a6 _).<a>set_eq</a>", [{"full_name": "Filter.eventually_of_mem", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1098, 9], "def_end_pos": [1098, 26]}, {"full_name": "Filter.Eventually.set_eq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1459, 30], "def_end_pos": [1459, 47]}]], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) =\u1da0[\ud835\udcdd (\u2191e' x)]\n    \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s", "state_after": "case refine'_1\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' e.source \u2208 \ud835\udcdd (\u2191e' x)\n\ncase refine'_2\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\ny : H\nhy : y \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' e.source\n\u22a2 y \u2208 \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) \u2194\n    y \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s"}, {"tactic": "simp_rw [mem_preimage, LocalHomeomorph.coe_trans_symm, LocalHomeomorph.symm_symm,\n  Function.comp_apply, e.left_inv hy]", "annotated_tactic": ["simp_rw [<a>mem_preimage</a>, <a>LocalHomeomorph.coe_trans_symm</a>, <a>LocalHomeomorph.symm_symm</a>,\n      <a>Function.comp_apply</a>, e.left_inv hy]", [{"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "LocalHomeomorph.coe_trans_symm", "def_path": "Mathlib/Topology/LocalHomeomorph.lean", "def_pos": [812, 9], "def_end_pos": [812, 23]}, {"full_name": "LocalHomeomorph.symm_symm", "def_path": "Mathlib/Topology/LocalHomeomorph.lean", "def_pos": [354, 29], "def_end_pos": [354, 38]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}]], "state_before": "case refine'_2\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\ny : H\nhy : y \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' e.source\n\u22a2 y \u2208 \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) \u2194\n    y \u2208 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' s", "state_after": "no goals"}, {"tactic": "refine' (e'.symm.continuousAt <| e'.mapsTo xe').preimage_mem_nhds (e.open_source.mem_nhds _)", "annotated_tactic": ["refine' (e'.symm.continuousAt <| e'.mapsTo xe').<a>preimage_mem_nhds</a> (e.open_source.mem_nhds _)", [{"full_name": "ContinuousAt.preimage_mem_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1643, 9], "def_end_pos": [1643, 39]}]], "state_before": "case refine'_1\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 \u2191(LocalHomeomorph.symm e') \u207b\u00b9' e.source \u2208 \ud835\udcdd (\u2191e' x)", "state_after": "case refine'_1\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 \u2191(LocalHomeomorph.symm e') (\u2191e' x) \u2208 e.source"}, {"tactic": "simp_rw [e'.left_inv xe', xe]", "annotated_tactic": ["simp_rw [e'.left_inv xe', xe]", []], "state_before": "case refine'_1\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 \u2191(LocalHomeomorph.symm e') (\u2191e' x) \u2208 e.source", "state_after": "no goals"}, {"tactic": "refine' ((e'.eventually_nhds' _ xe').mpr <| e.eventually_left_inverse xe).mono fun y hy \u21a6 _", "annotated_tactic": ["refine' ((e'.eventually_nhds' _ xe').<a>mpr</a> <| e.eventually_left_inverse xe).<a>mono</a> fun y hy \u21a6 _", [{"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}]], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 (g \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e')) =\u1da0[\ud835\udcdd (\u2191e' x)]\n    g \u2218 \u2191(LocalHomeomorph.symm e')", "state_after": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\ny : H\nhy : \u2191(LocalHomeomorph.symm e) (\u2191e (\u2191(LocalHomeomorph.symm e') y)) = \u2191(LocalHomeomorph.symm e') y\n\u22a2 ((g \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e'))) y =\n    (g \u2218 \u2191(LocalHomeomorph.symm e')) y"}, {"tactic": "simp only [mfld_simps]", "annotated_tactic": ["simp only [mfld_simps]", []], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\ny : H\nhy : \u2191(LocalHomeomorph.symm e) (\u2191e (\u2191(LocalHomeomorph.symm e') y)) = \u2191(LocalHomeomorph.symm e') y\n\u22a2 ((g \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e \u226b\u2095 e'))) y =\n    (g \u2218 \u2191(LocalHomeomorph.symm e')) y", "state_after": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\ny : H\nhy : \u2191(LocalHomeomorph.symm e) (\u2191e (\u2191(LocalHomeomorph.symm e') y)) = \u2191(LocalHomeomorph.symm e') y\n\u22a2 g (\u2191(LocalHomeomorph.symm e) (\u2191e (\u2191(LocalHomeomorph.symm e') y))) = g (\u2191(LocalHomeomorph.symm e') y)"}, {"tactic": "rw [hy]", "annotated_tactic": ["rw [hy]", []], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf f' : LocalHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : LocalInvariantProp G G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\ny : H\nhy : \u2191(LocalHomeomorph.symm e) (\u2191e (\u2191(LocalHomeomorph.symm e') y)) = \u2191(LocalHomeomorph.symm e') y\n\u22a2 g (\u2191(LocalHomeomorph.symm e) (\u2191e (\u2191(LocalHomeomorph.symm e') y))) = g (\u2191(LocalHomeomorph.symm e') y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/RelClasses.lean", "full_name": "WellFoundedLT.fix_eq", "start": [421, 1], "end": [423, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "full_name": "Set.pow_mem_pow", "start": [943, 1], "end": [949, 44], "traced_tactics": [{"tactic": "rw [pow_zero]", "annotated_tactic": ["rw [<a>pow_zero</a>]", [{"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [639, 9], "def_end_pos": [639, 17]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : Monoid \u03b1\ns t : Set \u03b1\na : \u03b1\nm n : \u2115\nha : a \u2208 s\n\u22a2 a ^ 0 \u2208 s ^ 0", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : Monoid \u03b1\ns t : Set \u03b1\na : \u03b1\nm n : \u2115\nha : a \u2208 s\n\u22a2 1 \u2208 s ^ 0"}, {"tactic": "exact one_mem_one", "annotated_tactic": ["exact <a>one_mem_one</a>", [{"full_name": "Set.one_mem_one", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [106, 9], "def_end_pos": [106, 20]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : Monoid \u03b1\ns t : Set \u03b1\na : \u03b1\nm n : \u2115\nha : a \u2208 s\n\u22a2 1 \u2208 s ^ 0", "state_after": "no goals"}, {"tactic": "rw [pow_succ]", "annotated_tactic": ["rw [<a>pow_succ</a>]", [{"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [645, 9], "def_end_pos": [645, 17]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : Monoid \u03b1\ns t : Set \u03b1\na : \u03b1\nm n\u271d : \u2115\nha : a \u2208 s\nn : \u2115\n\u22a2 a ^ (n + 1) \u2208 s ^ (n + 1)", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : Monoid \u03b1\ns t : Set \u03b1\na : \u03b1\nm n\u271d : \u2115\nha : a \u2208 s\nn : \u2115\n\u22a2 a * a ^ n \u2208 s ^ (n + 1)"}, {"tactic": "exact mul_mem_mul ha (pow_mem_pow ha _)", "annotated_tactic": ["exact <a>mul_mem_mul</a> ha (pow_mem_pow ha _)", [{"full_name": "Set.mul_mem_mul", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [342, 9], "def_end_pos": [342, 20]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : Monoid \u03b1\ns t : Set \u03b1\na : \u03b1\nm n\u271d : \u2115\nha : a \u2208 s\nn : \u2115\n\u22a2 a * a ^ n \u2208 s ^ (n + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/Cyclotomic/Basic.lean", "full_name": "IsCyclotomicExtension.finiteDimensional", "start": [360, 1], "end": [362, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/CountableSeparatingOn.lean", "full_name": "Filter.exists_subset_subsingleton_mem_of_forall_separating", "start": [145, 1], "end": [159, 64], "traced_tactics": [{"tactic": "rcases h.1 with \u27e8S, hSc, hSp, hS\u27e9", "annotated_tactic": ["rcases h.1 with \u27e8S, hSc, hSp, hS\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Sort ?u.5035\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\nf g : \u03b1 \u2192 \u03b2\np : Set \u03b1 \u2192 Prop\ns : Set \u03b1\nh : HasCountableSeparatingOn \u03b1 p s\nhs : s \u2208 l\nhl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l\n\u22a2 \u2203 t, t \u2286 s \u2227 Set.Subsingleton t \u2227 t \u2208 l", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.5035\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\nf g : \u03b1 \u2192 \u03b2\np : Set \u03b1 \u2192 Prop\ns : Set \u03b1\nh : HasCountableSeparatingOn \u03b1 p s\nhs : s \u2208 l\nhl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nhSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s\nhS : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 (\u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)) \u2192 x = y\n\u22a2 \u2203 t, t \u2286 s \u2227 Set.Subsingleton t \u2227 t \u2208 l"}, {"tactic": "refine \u27e8s \u2229 \u22c2\u2080 (S \u2229 l.sets) \u2229 \u22c2 (U \u2208 S) (_ : U\u1d9c \u2208 l), U\u1d9c, ?_, ?_, ?_\u27e9", "annotated_tactic": ["refine \u27e8s \u2229 \u22c2\u2080 (S \u2229 l.sets) \u2229 \u22c2 (U \u2208 S) (_ : U\u1d9c \u2208 l), U\u1d9c, ?_, ?_, ?_\u27e9", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.5035\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\nf g : \u03b1 \u2192 \u03b2\np : Set \u03b1 \u2192 Prop\ns : Set \u03b1\nh : HasCountableSeparatingOn \u03b1 p s\nhs : s \u2208 l\nhl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nhSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s\nhS : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 (\u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)) \u2192 x = y\n\u22a2 \u2203 t, t \u2286 s \u2227 Set.Subsingleton t \u2227 t \u2208 l", "state_after": "case intro.intro.intro.refine_1\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.5035\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\nf g : \u03b1 \u2192 \u03b2\np : Set \u03b1 \u2192 Prop\ns : Set \u03b1\nh : HasCountableSeparatingOn \u03b1 p s\nhs : s \u2208 l\nhl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nhSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s\nhS : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 (\u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)) \u2192 x = y\n\u22a2 s \u2229 \u22c2\u2080 (S \u2229 l.sets) \u2229 \u22c2 U \u2208 S, \u22c2 (_ : U\u1d9c \u2208 l), U\u1d9c \u2286 s\n\ncase intro.intro.intro.refine_2\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.5035\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\nf g : \u03b1 \u2192 \u03b2\np : Set \u03b1 \u2192 Prop\ns : Set \u03b1\nh : HasCountableSeparatingOn \u03b1 p s\nhs : s \u2208 l\nhl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nhSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s\nhS : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 (\u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)) \u2192 x = y\n\u22a2 Set.Subsingleton (s \u2229 \u22c2\u2080 (S \u2229 l.sets) \u2229 \u22c2 U \u2208 S, \u22c2 (_ : U\u1d9c \u2208 l), U\u1d9c)\n\ncase intro.intro.intro.refine_3\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.5035\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\nf g : \u03b1 \u2192 \u03b2\np : Set \u03b1 \u2192 Prop\ns : Set \u03b1\nh : HasCountableSeparatingOn \u03b1 p s\nhs : s \u2208 l\nhl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nhSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s\nhS : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 (\u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)) \u2192 x = y\n\u22a2 s \u2229 \u22c2\u2080 (S \u2229 l.sets) \u2229 \u22c2 U \u2208 S, \u22c2 (_ : U\u1d9c \u2208 l), U\u1d9c \u2208 l"}, {"tactic": "exact fun _ h \u21a6 h.1.1", "annotated_tactic": ["exact fun _ h \u21a6 h.1.1", []], "state_before": "case intro.intro.intro.refine_1\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.5035\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\nf g : \u03b1 \u2192 \u03b2\np : Set \u03b1 \u2192 Prop\ns : Set \u03b1\nh : HasCountableSeparatingOn \u03b1 p s\nhs : s \u2208 l\nhl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nhSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s\nhS : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 (\u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)) \u2192 x = y\n\u22a2 s \u2229 \u22c2\u2080 (S \u2229 l.sets) \u2229 \u22c2 U \u2208 S, \u22c2 (_ : U\u1d9c \u2208 l), U\u1d9c \u2286 s", "state_after": "no goals"}, {"tactic": "intro x hx y hy", "annotated_tactic": ["intro x hx y hy", []], "state_before": "case intro.intro.intro.refine_2\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.5035\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\nf g : \u03b1 \u2192 \u03b2\np : Set \u03b1 \u2192 Prop\ns : Set \u03b1\nh : HasCountableSeparatingOn \u03b1 p s\nhs : s \u2208 l\nhl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nhSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s\nhS : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 (\u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)) \u2192 x = y\n\u22a2 Set.Subsingleton (s \u2229 \u22c2\u2080 (S \u2229 l.sets) \u2229 \u22c2 U \u2208 S, \u22c2 (_ : U\u1d9c \u2208 l), U\u1d9c)", "state_after": "case intro.intro.intro.refine_2\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.5035\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\nf g : \u03b1 \u2192 \u03b2\np : Set \u03b1 \u2192 Prop\ns : Set \u03b1\nh : HasCountableSeparatingOn \u03b1 p s\nhs : s \u2208 l\nhl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nhSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s\nhS : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 (\u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)) \u2192 x = y\nx : \u03b1\nhx : x \u2208 s \u2229 \u22c2\u2080 (S \u2229 l.sets) \u2229 \u22c2 U \u2208 S, \u22c2 (_ : U\u1d9c \u2208 l), U\u1d9c\ny : \u03b1\nhy : y \u2208 s \u2229 \u22c2\u2080 (S \u2229 l.sets) \u2229 \u22c2 U \u2208 S, \u22c2 (_ : U\u1d9c \u2208 l), U\u1d9c\n\u22a2 x = y"}, {"tactic": "simp only [mem_sInter, mem_inter_iff, mem_iInter, mem_compl_iff] at hx hy", "annotated_tactic": ["simp only [<a>mem_sInter</a>, <a>mem_inter_iff</a>, <a>mem_iInter</a>, <a>mem_compl_iff</a>] at hx hy", [{"full_name": "Set.mem_sInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [93, 9], "def_end_pos": [93, 19]}, {"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}, {"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}, {"full_name": "Set.mem_compl_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1658, 9], "def_end_pos": [1658, 22]}]], "state_before": "case intro.intro.intro.refine_2\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.5035\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\nf g : \u03b1 \u2192 \u03b2\np : Set \u03b1 \u2192 Prop\ns : Set \u03b1\nh : HasCountableSeparatingOn \u03b1 p s\nhs : s \u2208 l\nhl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nhSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s\nhS : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 (\u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)) \u2192 x = y\nx : \u03b1\nhx : x \u2208 s \u2229 \u22c2\u2080 (S \u2229 l.sets) \u2229 \u22c2 U \u2208 S, \u22c2 (_ : U\u1d9c \u2208 l), U\u1d9c\ny : \u03b1\nhy : y \u2208 s \u2229 \u22c2\u2080 (S \u2229 l.sets) \u2229 \u22c2 U \u2208 S, \u22c2 (_ : U\u1d9c \u2208 l), U\u1d9c\n\u22a2 x = y", "state_after": "case intro.intro.intro.refine_2\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.5035\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\nf g : \u03b1 \u2192 \u03b2\np : Set \u03b1 \u2192 Prop\ns : Set \u03b1\nh : HasCountableSeparatingOn \u03b1 p s\nhs : s \u2208 l\nhl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nhSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s\nhS : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 (\u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)) \u2192 x = y\nx y : \u03b1\nhx : (x \u2208 s \u2227 \u2200 (t : Set \u03b1), t \u2208 S \u2227 t \u2208 l.sets \u2192 x \u2208 t) \u2227 \u2200 (i : Set \u03b1), i \u2208 S \u2192 i\u1d9c \u2208 l \u2192 \u00acx \u2208 i\nhy : (y \u2208 s \u2227 \u2200 (t : Set \u03b1), t \u2208 S \u2227 t \u2208 l.sets \u2192 y \u2208 t) \u2227 \u2200 (i : Set \u03b1), i \u2208 S \u2192 i\u1d9c \u2208 l \u2192 \u00acy \u2208 i\n\u22a2 x = y"}, {"tactic": "refine hS x hx.1.1 y hy.1.1 (fun s hsS \u21a6 ?_)", "annotated_tactic": ["refine hS x hx.1.1 y hy.1.1 (fun s hsS \u21a6 ?_)", []], "state_before": "case intro.intro.intro.refine_2\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.5035\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\nf g : \u03b1 \u2192 \u03b2\np : Set \u03b1 \u2192 Prop\ns : Set \u03b1\nh : HasCountableSeparatingOn \u03b1 p s\nhs : s \u2208 l\nhl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nhSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s\nhS : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 (\u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)) \u2192 x = y\nx y : \u03b1\nhx : (x \u2208 s \u2227 \u2200 (t : Set \u03b1), t \u2208 S \u2227 t \u2208 l.sets \u2192 x \u2208 t) \u2227 \u2200 (i : Set \u03b1), i \u2208 S \u2192 i\u1d9c \u2208 l \u2192 \u00acx \u2208 i\nhy : (y \u2208 s \u2227 \u2200 (t : Set \u03b1), t \u2208 S \u2227 t \u2208 l.sets \u2192 y \u2208 t) \u2227 \u2200 (i : Set \u03b1), i \u2208 S \u2192 i\u1d9c \u2208 l \u2192 \u00acy \u2208 i\n\u22a2 x = y", "state_after": "case intro.intro.intro.refine_2\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.5035\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\nf g : \u03b1 \u2192 \u03b2\np : Set \u03b1 \u2192 Prop\ns\u271d : Set \u03b1\nh : HasCountableSeparatingOn \u03b1 p s\u271d\nhs : s\u271d \u2208 l\nhl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nhSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s\nhS : \u2200 (x : \u03b1), x \u2208 s\u271d \u2192 \u2200 (y : \u03b1), y \u2208 s\u271d \u2192 (\u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)) \u2192 x = y\nx y : \u03b1\nhx : (x \u2208 s\u271d \u2227 \u2200 (t : Set \u03b1), t \u2208 S \u2227 t \u2208 l.sets \u2192 x \u2208 t) \u2227 \u2200 (i : Set \u03b1), i \u2208 S \u2192 i\u1d9c \u2208 l \u2192 \u00acx \u2208 i\nhy : (y \u2208 s\u271d \u2227 \u2200 (t : Set \u03b1), t \u2208 S \u2227 t \u2208 l.sets \u2192 y \u2208 t) \u2227 \u2200 (i : Set \u03b1), i \u2208 S \u2192 i\u1d9c \u2208 l \u2192 \u00acy \u2208 i\ns : Set \u03b1\nhsS : s \u2208 S\n\u22a2 x \u2208 s \u2194 y \u2208 s"}, {"tactic": "cases hl s (hSp s hsS) with\n| inl hsl => simp only [hx.1.2 s \u27e8hsS, hsl\u27e9, hy.1.2 s \u27e8hsS, hsl\u27e9]\n| inr hsl => simp only [hx.2 s hsS hsl, hy.2 s hsS hsl]", "annotated_tactic": ["cases hl s (hSp s hsS) with\n    | <a>inl</a> hsl => simp only [hx.1.2 s \u27e8hsS, hsl\u27e9, hy.1.2 s \u27e8hsS, hsl\u27e9]\n    | <a>inr</a> hsl => simp only [hx.2 s hsS hsl, hy.2 s hsS hsl]", [{"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}]], "state_before": "case intro.intro.intro.refine_2\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.5035\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\nf g : \u03b1 \u2192 \u03b2\np : Set \u03b1 \u2192 Prop\ns\u271d : Set \u03b1\nh : HasCountableSeparatingOn \u03b1 p s\u271d\nhs : s\u271d \u2208 l\nhl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nhSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s\nhS : \u2200 (x : \u03b1), x \u2208 s\u271d \u2192 \u2200 (y : \u03b1), y \u2208 s\u271d \u2192 (\u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)) \u2192 x = y\nx y : \u03b1\nhx : (x \u2208 s\u271d \u2227 \u2200 (t : Set \u03b1), t \u2208 S \u2227 t \u2208 l.sets \u2192 x \u2208 t) \u2227 \u2200 (i : Set \u03b1), i \u2208 S \u2192 i\u1d9c \u2208 l \u2192 \u00acx \u2208 i\nhy : (y \u2208 s\u271d \u2227 \u2200 (t : Set \u03b1), t \u2208 S \u2227 t \u2208 l.sets \u2192 y \u2208 t) \u2227 \u2200 (i : Set \u03b1), i \u2208 S \u2192 i\u1d9c \u2208 l \u2192 \u00acy \u2208 i\ns : Set \u03b1\nhsS : s \u2208 S\n\u22a2 x \u2208 s \u2194 y \u2208 s", "state_after": "no goals"}, {"tactic": "simp only [hx.1.2 s \u27e8hsS, hsl\u27e9, hy.1.2 s \u27e8hsS, hsl\u27e9]", "annotated_tactic": ["simp only [hx.1.2 s \u27e8hsS, hsl\u27e9, hy.1.2 s \u27e8hsS, hsl\u27e9]", []], "state_before": "case intro.intro.intro.refine_2.inl\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.5035\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\nf g : \u03b1 \u2192 \u03b2\np : Set \u03b1 \u2192 Prop\ns\u271d : Set \u03b1\nh : HasCountableSeparatingOn \u03b1 p s\u271d\nhs : s\u271d \u2208 l\nhl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nhSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s\nhS : \u2200 (x : \u03b1), x \u2208 s\u271d \u2192 \u2200 (y : \u03b1), y \u2208 s\u271d \u2192 (\u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)) \u2192 x = y\nx y : \u03b1\nhx : (x \u2208 s\u271d \u2227 \u2200 (t : Set \u03b1), t \u2208 S \u2227 t \u2208 l.sets \u2192 x \u2208 t) \u2227 \u2200 (i : Set \u03b1), i \u2208 S \u2192 i\u1d9c \u2208 l \u2192 \u00acx \u2208 i\nhy : (y \u2208 s\u271d \u2227 \u2200 (t : Set \u03b1), t \u2208 S \u2227 t \u2208 l.sets \u2192 y \u2208 t) \u2227 \u2200 (i : Set \u03b1), i \u2208 S \u2192 i\u1d9c \u2208 l \u2192 \u00acy \u2208 i\ns : Set \u03b1\nhsS : s \u2208 S\nhsl : s \u2208 l\n\u22a2 x \u2208 s \u2194 y \u2208 s", "state_after": "no goals"}, {"tactic": "simp only [hx.2 s hsS hsl, hy.2 s hsS hsl]", "annotated_tactic": ["simp only [hx.2 s hsS hsl, hy.2 s hsS hsl]", []], "state_before": "case intro.intro.intro.refine_2.inr\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.5035\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\nf g : \u03b1 \u2192 \u03b2\np : Set \u03b1 \u2192 Prop\ns\u271d : Set \u03b1\nh : HasCountableSeparatingOn \u03b1 p s\u271d\nhs : s\u271d \u2208 l\nhl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nhSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s\nhS : \u2200 (x : \u03b1), x \u2208 s\u271d \u2192 \u2200 (y : \u03b1), y \u2208 s\u271d \u2192 (\u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)) \u2192 x = y\nx y : \u03b1\nhx : (x \u2208 s\u271d \u2227 \u2200 (t : Set \u03b1), t \u2208 S \u2227 t \u2208 l.sets \u2192 x \u2208 t) \u2227 \u2200 (i : Set \u03b1), i \u2208 S \u2192 i\u1d9c \u2208 l \u2192 \u00acx \u2208 i\nhy : (y \u2208 s\u271d \u2227 \u2200 (t : Set \u03b1), t \u2208 S \u2227 t \u2208 l.sets \u2192 y \u2208 t) \u2227 \u2200 (i : Set \u03b1), i \u2208 S \u2192 i\u1d9c \u2208 l \u2192 \u00acy \u2208 i\ns : Set \u03b1\nhsS : s \u2208 S\nhsl : s\u1d9c \u2208 l\n\u22a2 x \u2208 s \u2194 y \u2208 s", "state_after": "no goals"}, {"tactic": "exact inter_mem\n  (inter_mem hs ((countable_sInter_mem (hSc.mono (inter_subset_left _ _))).2 fun _ h \u21a6 h.2))\n  ((countable_bInter_mem hSc).2 fun U hU \u21a6 iInter_mem.2 id)", "annotated_tactic": ["exact <a>inter_mem</a>\n      (<a>inter_mem</a> hs ((<a>countable_sInter_mem</a> (hSc.mono (<a>inter_subset_left</a> _ _))).2 fun _ h \u21a6 h.2))\n      ((<a>countable_bInter_mem</a> hSc).2 fun U hU \u21a6 <a>iInter_mem</a>.2 <a>id</a>)", [{"full_name": "Filter.inter_mem", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [156, 9], "def_end_pos": [156, 18]}, {"full_name": "Filter.inter_mem", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [156, 9], "def_end_pos": [156, 18]}, {"full_name": "countable_sInter_mem", "def_path": "Mathlib/Order/Filter/CountableInter.lean", "def_pos": [45, 9], "def_end_pos": [45, 29]}, {"full_name": "Set.inter_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [965, 9], "def_end_pos": [965, 26]}, {"full_name": "countable_bInter_mem", "def_path": "Mathlib/Order/Filter/CountableInter.lean", "def_pos": [54, 9], "def_end_pos": [54, 29]}, {"full_name": "Filter.iInter_mem", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [206, 9], "def_end_pos": [206, 19]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "state_before": "case intro.intro.intro.refine_3\n\u03b1 : Type u_1\n\u03b2 : Sort ?u.5035\nl : Filter \u03b1\ninst\u271d : CountableInterFilter l\nf g : \u03b1 \u2192 \u03b2\np : Set \u03b1 \u2192 Prop\ns : Set \u03b1\nh : HasCountableSeparatingOn \u03b1 p s\nhs : s \u2208 l\nhl : \u2200 (U : Set \u03b1), p U \u2192 U \u2208 l \u2228 U\u1d9c \u2208 l\nS : Set (Set \u03b1)\nhSc : Set.Countable S\nhSp : \u2200 (s : Set \u03b1), s \u2208 S \u2192 p s\nhS : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 (\u2200 (s : Set \u03b1), s \u2208 S \u2192 (x \u2208 s \u2194 y \u2208 s)) \u2192 x = y\n\u22a2 s \u2229 \u22c2\u2080 (S \u2229 l.sets) \u2229 \u22c2 U \u2208 S, \u22c2 (_ : U\u1d9c \u2208 l), U\u1d9c \u2208 l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Complex/Circle.lean", "full_name": "normSq_eq_of_mem_circle", "start": [66, 1], "end": [66, 87], "traced_tactics": [{"tactic": "simp [normSq_eq_abs]", "annotated_tactic": ["simp [<a>normSq_eq_abs</a>]", [{"full_name": "Complex.normSq_eq_abs", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [1148, 9], "def_end_pos": [1148, 22]}]], "state_before": "z : { x // x \u2208 circle }\n\u22a2 \u2191normSq \u2191z = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.count_zero", "start": [2354, 1], "end": [2355, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/Eval.lean", "full_name": "Polynomial.eval\u2082_finset_prod", "start": [1140, 1], "end": [1142, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Basic.lean", "full_name": "lt_or_lt_iff_ne", "start": [502, 1], "end": [503, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/AddTorsor.lean", "full_name": "Equiv.coe_vaddConst_symm", "start": [377, 1], "end": [378, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "ZNum.dvd_to_int", "start": [1562, 1], "end": [1563, 86], "traced_tactics": [{"tactic": "rw [\u2190 of_to_int n, e]", "annotated_tactic": ["rw [\u2190 <a>of_to_int</a> n, e]", [{"full_name": "ZNum.of_to_int", "def_path": "Mathlib/Data/Num/Lemmas.lean", "def_pos": [1540, 9], "def_end_pos": [1540, 18]}]], "state_before": "\u03b1 : Type u_1\nm n : ZNum\nx\u271d : \u2191m \u2223 \u2191n\nk : \u2124\ne : \u2191n = \u2191m * k\n\u22a2 n = m * \u2191k", "state_after": "\u03b1 : Type u_1\nm n : ZNum\nx\u271d : \u2191m \u2223 \u2191n\nk : \u2124\ne : \u2191n = \u2191m * k\n\u22a2 \u2191(\u2191m * k) = m * \u2191k"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\nm n : ZNum\nx\u271d : \u2191m \u2223 \u2191n\nk : \u2124\ne : \u2191n = \u2191m * k\n\u22a2 \u2191(\u2191m * k) = m * \u2191k", "state_after": "no goals"}, {"tactic": "simp [e]", "annotated_tactic": ["simp [e]", []], "state_before": "\u03b1 : Type u_1\nm n : ZNum\nx\u271d : m \u2223 n\nk : ZNum\ne : n = m * k\n\u22a2 \u2191n = \u2191m * \u2191k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.get?_set_eq_of_lt", "start": [916, 1], "end": [917, 73], "traced_tactics": [{"tactic": "rw [get?_set_eq, get?_eq_get h]", "annotated_tactic": ["rw [<a>get?_set_eq</a>, <a>get?_eq_get</a> h]", [{"full_name": "List.get?_set_eq", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [913, 9], "def_end_pos": [913, 20]}, {"full_name": "List.get?_eq_get", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [581, 9], "def_end_pos": [581, 20]}]], "state_before": "\u03b1 : Type u_1\na : \u03b1\nn : Nat\nl : List \u03b1\nh : n < length l\n\u22a2 get? (set l n a) n = some a", "state_after": "\u03b1 : Type u_1\na : \u03b1\nn : Nat\nl : List \u03b1\nh : n < length l\n\u22a2 (fun x => a) <$> some (get l { val := n, isLt := h }) = some a"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\na : \u03b1\nn : Nat\nl : List \u03b1\nh : n < length l\n\u22a2 (fun x => a) <$> some (get l { val := n, isLt := h }) = some a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Lattice.lean", "full_name": "AEMeasurable.const_inf", "start": [183, 1], "end": [185, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Ordinal/FixedPoint.lean", "full_name": "Ordinal.nfp_eq_self", "start": [491, 1], "end": [492, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/Parity.lean", "full_name": "Nat.Odd.of_mul_right", "start": [165, 1], "end": [166, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Egorov.lean", "full_name": "MeasureTheory.Egorov.measure_inter_notConvergentSeq_eq_zero", "start": [59, 1], "end": [70, 25], "traced_tactics": [{"tactic": "simp_rw [Metric.tendsto_atTop, ae_iff] at hfg", "annotated_tactic": ["simp_rw [<a>Metric.tendsto_atTop</a>, <a>ae_iff</a>] at hfg", [{"full_name": "Metric.tendsto_atTop", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1107, 9], "def_end_pos": [1107, 22]}, {"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [388, 9], "def_end_pos": [388, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nhfg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (s \u2229 \u22c2 j, notConvergentSeq f g n j) = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\n\u22a2 \u2191\u2191\u03bc (s \u2229 \u22c2 j, notConvergentSeq f g n j) = 0"}, {"tactic": "rw [\u2190 nonpos_iff_eq_zero, \u2190 hfg]", "annotated_tactic": ["rw [\u2190 <a>nonpos_iff_eq_zero</a>, \u2190 hfg]", [{"full_name": "nonpos_iff_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [237, 3], "def_end_pos": [237, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\n\u22a2 \u2191\u2191\u03bc (s \u2229 \u22c2 j, notConvergentSeq f g n j) = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\n\u22a2 \u2191\u2191\u03bc (s \u2229 \u22c2 j, notConvergentSeq f g n j) \u2264\n    \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)}"}, {"tactic": "refine' measure_mono fun x => _", "annotated_tactic": ["refine' <a>measure_mono</a> fun x => _", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\n\u22a2 \u2191\u2191\u03bc (s \u2229 \u22c2 j, notConvergentSeq f g n j) \u2264\n    \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\nx : \u03b1\n\u22a2 x \u2208 s \u2229 \u22c2 j, notConvergentSeq f g n j \u2192\n    x \u2208 {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)}"}, {"tactic": "simp only [Set.mem_inter_iff, Set.mem_iInter, ge_iff_le, mem_notConvergentSeq_iff]", "annotated_tactic": ["simp only [<a>Set.mem_inter_iff</a>, <a>Set.mem_iInter</a>, <a>ge_iff_le</a>, <a>mem_notConvergentSeq_iff</a>]", [{"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}, {"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}, {"full_name": "ge_iff_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [359, 9], "def_end_pos": [359, 18]}, {"full_name": "MeasureTheory.Egorov.mem_notConvergentSeq_iff", "def_path": "Mathlib/MeasureTheory/Function/Egorov.lean", "def_pos": [49, 9], "def_end_pos": [49, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\nx : \u03b1\n\u22a2 x \u2208 s \u2229 \u22c2 j, notConvergentSeq f g n j \u2192\n    x \u2208 {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\nx : \u03b1\n\u22a2 (x \u2208 s \u2227 \u2200 (i : \u03b9), \u2203 k x_1, 1 / (\u2191n + 1) < dist (f k x) (g x)) \u2192\n    x \u2208 {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), N \u2264 n \u2192 dist (f n a) (g a) < \u03b5)}"}, {"tactic": "push_neg", "annotated_tactic": ["push_neg", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\nx : \u03b1\n\u22a2 (x \u2208 s \u2227 \u2200 (i : \u03b9), \u2203 k x_1, 1 / (\u2191n + 1) < dist (f k x) (g x)) \u2192\n    x \u2208 {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), N \u2264 n \u2192 dist (f n a) (g a) < \u03b5)}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\nx : \u03b1\n\u22a2 (x \u2208 s \u2227 \u2200 (i : \u03b9), \u2203 k x_1, 1 / (\u2191n + 1) < dist (f k x) (g x)) \u2192\n    x \u2208 {a | a \u2208 s \u2227 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (N : \u03b9), \u2203 n, N \u2264 n \u2227 \u03b5 \u2264 dist (f n a) (g a)}"}, {"tactic": "rintro \u27e8hmem, hx\u27e9", "annotated_tactic": ["rintro \u27e8hmem, hx\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\nx : \u03b1\n\u22a2 (x \u2208 s \u2227 \u2200 (i : \u03b9), \u2203 k x_1, 1 / (\u2191n + 1) < dist (f k x) (g x)) \u2192\n    x \u2208 {a | a \u2208 s \u2227 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (N : \u03b9), \u2203 n, N \u2264 n \u2227 \u03b5 \u2264 dist (f n a) (g a)}", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\nx : \u03b1\nhmem : x \u2208 s\nhx : \u2200 (i : \u03b9), \u2203 k x_1, 1 / (\u2191n + 1) < dist (f k x) (g x)\n\u22a2 x \u2208 {a | a \u2208 s \u2227 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (N : \u03b9), \u2203 n, N \u2264 n \u2227 \u03b5 \u2264 dist (f n a) (g a)}"}, {"tactic": "refine' \u27e8hmem, 1 / (n + 1 : \u211d), Nat.one_div_pos_of_nat, fun N => _\u27e9", "annotated_tactic": ["refine' \u27e8hmem, 1 / (n + 1 : \u211d), <a>Nat.one_div_pos_of_nat</a>, fun N => _\u27e9", [{"full_name": "Nat.one_div_pos_of_nat", "def_path": "Mathlib/Data/Nat/Cast/Field.lean", "def_pos": [65, 9], "def_end_pos": [65, 27]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\nx : \u03b1\nhmem : x \u2208 s\nhx : \u2200 (i : \u03b9), \u2203 k x_1, 1 / (\u2191n + 1) < dist (f k x) (g x)\n\u22a2 x \u2208 {a | a \u2208 s \u2227 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (N : \u03b9), \u2203 n, N \u2264 n \u2227 \u03b5 \u2264 dist (f n a) (g a)}", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\nx : \u03b1\nhmem : x \u2208 s\nhx : \u2200 (i : \u03b9), \u2203 k x_1, 1 / (\u2191n + 1) < dist (f k x) (g x)\nN : \u03b9\n\u22a2 \u2203 n_1, N \u2264 n_1 \u2227 1 / (\u2191n + 1) \u2264 dist (f n_1 x) (g x)"}, {"tactic": "obtain \u27e8n, hn\u2081, hn\u2082\u27e9 := hx N", "annotated_tactic": ["obtain \u27e8n, hn\u2081, hn\u2082\u27e9 := hx N", []], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\nx : \u03b1\nhmem : x \u2208 s\nhx : \u2200 (i : \u03b9), \u2203 k x_1, 1 / (\u2191n + 1) < dist (f k x) (g x)\nN : \u03b9\n\u22a2 \u2203 n_1, N \u2264 n_1 \u2227 1 / (\u2191n + 1) \u2264 dist (f n_1 x) (g x)", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d\u00b9 : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn\u271d : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\nx : \u03b1\nhmem : x \u2208 s\nhx : \u2200 (i : \u03b9), \u2203 k x_1, 1 / (\u2191n\u271d + 1) < dist (f k x) (g x)\nN n : \u03b9\nhn\u2081 : N \u2264 n\nhn\u2082 : 1 / (\u2191n\u271d + 1) < dist (f n x) (g x)\n\u22a2 \u2203 n, N \u2264 n \u2227 1 / (\u2191n\u271d + 1) \u2264 dist (f n x) (g x)"}, {"tactic": "exact \u27e8n, hn\u2081, hn\u2082.le\u27e9", "annotated_tactic": ["exact \u27e8n, hn\u2081, hn\u2082.le\u27e9", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MetricSpace \u03b2\n\u03bc : Measure \u03b1\nn\u271d\u00b9 : \u2115\ni j : \u03b9\ns : Set \u03b1\n\u03b5 : \u211d\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b9\ninst\u271d : Nonempty \u03b9\nn\u271d : \u2115\nhfg : \u2191\u2191\u03bc {a | \u00ac(a \u2208 s \u2192 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u03b9), n \u2265 N \u2192 dist (f n a) (g a) < \u03b5)} = 0\nx : \u03b1\nhmem : x \u2208 s\nhx : \u2200 (i : \u03b9), \u2203 k x_1, 1 / (\u2191n\u271d + 1) < dist (f k x) (g x)\nN n : \u03b9\nhn\u2081 : N \u2264 n\nhn\u2082 : 1 / (\u2191n\u271d + 1) < dist (f n x) (g x)\n\u22a2 \u2203 n, N \u2264 n \u2227 1 / (\u2191n\u271d + 1) \u2264 dist (f n x) (g x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/RootsOfUnity/Minpoly.lean", "full_name": "IsPrimitiveRoot.minpoly_dvd_x_pow_sub_one", "start": [51, 1], "end": [56, 41], "traced_tactics": [{"tactic": "rcases n.eq_zero_or_pos with (rfl | h0)", "annotated_tactic": ["rcases n.eq_zero_or_pos with (rfl | h0)", []], "state_before": "n : \u2115\nK : Type u_1\ninst\u271d\u00b2 : CommRing K\n\u03bc : K\nh : IsPrimitiveRoot \u03bc n\ninst\u271d\u00b9 : IsDomain K\ninst\u271d : CharZero K\n\u22a2 minpoly \u2124 \u03bc \u2223 X ^ n - 1", "state_after": "case inl\nK : Type u_1\ninst\u271d\u00b2 : CommRing K\n\u03bc : K\ninst\u271d\u00b9 : IsDomain K\ninst\u271d : CharZero K\nh : IsPrimitiveRoot \u03bc 0\n\u22a2 minpoly \u2124 \u03bc \u2223 X ^ 0 - 1\n\ncase inr\nn : \u2115\nK : Type u_1\ninst\u271d\u00b2 : CommRing K\n\u03bc : K\nh : IsPrimitiveRoot \u03bc n\ninst\u271d\u00b9 : IsDomain K\ninst\u271d : CharZero K\nh0 : n > 0\n\u22a2 minpoly \u2124 \u03bc \u2223 X ^ n - 1"}, {"tactic": "apply minpoly.isIntegrallyClosed_dvd (isIntegral h h0)", "annotated_tactic": ["apply <a>minpoly.isIntegrallyClosed_dvd</a> (<a>isIntegral</a> h h0)", [{"full_name": "minpoly.isIntegrallyClosed_dvd", "def_path": "Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean", "def_pos": [76, 9], "def_end_pos": [76, 31]}, {"full_name": "IsPrimitiveRoot.isIntegral", "def_path": "Mathlib/RingTheory/RootsOfUnity/Minpoly.lean", "def_pos": [38, 9], "def_end_pos": [38, 19]}]], "state_before": "case inr\nn : \u2115\nK : Type u_1\ninst\u271d\u00b2 : CommRing K\n\u03bc : K\nh : IsPrimitiveRoot \u03bc n\ninst\u271d\u00b9 : IsDomain K\ninst\u271d : CharZero K\nh0 : n > 0\n\u22a2 minpoly \u2124 \u03bc \u2223 X ^ n - 1", "state_after": "case inr\nn : \u2115\nK : Type u_1\ninst\u271d\u00b2 : CommRing K\n\u03bc : K\nh : IsPrimitiveRoot \u03bc n\ninst\u271d\u00b9 : IsDomain K\ninst\u271d : CharZero K\nh0 : n > 0\n\u22a2 \u2191(Polynomial.aeval \u03bc) (X ^ n - 1) = 0"}, {"tactic": "simp only [((IsPrimitiveRoot.iff_def \u03bc n).mp h).left, aeval_X_pow, eq_intCast, Int.cast_one,\n  aeval_one, AlgHom.map_sub, sub_self]", "annotated_tactic": ["simp only [((<a>IsPrimitiveRoot.iff_def</a> \u03bc n).<a>mp</a> h).<a>left</a>, <a>aeval_X_pow</a>, <a>eq_intCast</a>, <a>Int.cast_one</a>,\n    <a>aeval_one</a>, <a>AlgHom.map_sub</a>, <a>sub_self</a>]", [{"full_name": "IsPrimitiveRoot.iff_def", "def_path": "Mathlib/RingTheory/RootsOfUnity/Basic.lean", "def_pos": [289, 10], "def_end_pos": [289, 33]}, {"full_name": "Iff.mp", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [90, 3], "def_end_pos": [90, 5]}, {"full_name": "And.left", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [504, 3], "def_end_pos": [504, 7]}, {"full_name": "Polynomial.aeval_X_pow", "def_path": "Mathlib/Data/Polynomial/AlgebraMap.lean", "def_pos": [214, 9], "def_end_pos": [214, 20]}, {"full_name": "eq_intCast", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [306, 9], "def_end_pos": [306, 19]}, {"full_name": "Int.cast_one", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [77, 9], "def_end_pos": [77, 17]}, {"full_name": "Polynomial.aeval_one", "def_path": "Mathlib/Data/Polynomial/AlgebraMap.lean", "def_pos": [225, 9], "def_end_pos": [225, 18]}, {"full_name": "AlgHom.map_sub", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [487, 19], "def_end_pos": [487, 26]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}]], "state_before": "case inr\nn : \u2115\nK : Type u_1\ninst\u271d\u00b2 : CommRing K\n\u03bc : K\nh : IsPrimitiveRoot \u03bc n\ninst\u271d\u00b9 : IsDomain K\ninst\u271d : CharZero K\nh0 : n > 0\n\u22a2 \u2191(Polynomial.aeval \u03bc) (X ^ n - 1) = 0", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case inl\nK : Type u_1\ninst\u271d\u00b2 : CommRing K\n\u03bc : K\ninst\u271d\u00b9 : IsDomain K\ninst\u271d : CharZero K\nh : IsPrimitiveRoot \u03bc 0\n\u22a2 minpoly \u2124 \u03bc \u2223 X ^ 0 - 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.mapsTo_iInter_iInter", "start": [1602, 1], "end": [1604, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/UrysohnsLemma.lean", "full_name": "Urysohns.CU.left_U_subset_right_C", "start": [119, 1], "end": [120, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/WittVector/InitTail.lean", "full_name": "WittVector.coeff_add_of_disjoint", "start": [114, 1], "end": [135, 47], "traced_tactics": [{"tactic": "let P : \u2115 \u2192 Prop := fun n => y.coeff n = 0", "annotated_tactic": ["let P : \u2115 \u2192 Prop := fun n => y.coeff n = 0", []], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\n\u22a2 coeff (x + y) n = coeff x n + coeff y n", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\n\u22a2 coeff (x + y) n = coeff x n + coeff y n"}, {"tactic": "haveI : DecidablePred P := Classical.decPred P", "annotated_tactic": ["haveI : <a>DecidablePred</a> P := <a>Classical.decPred</a> P", [{"full_name": "DecidablePred", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [828, 8], "def_end_pos": [828, 21]}, {"full_name": "Classical.decPred", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [975, 19], "def_end_pos": [975, 26]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\n\u22a2 coeff (x + y) n = coeff x n + coeff y n", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\n\u22a2 coeff (x + y) n = coeff x n + coeff y n"}, {"tactic": "set z := mk p fun n => if P n then x.coeff n else y.coeff n", "annotated_tactic": ["set z := <a>mk</a> p fun n => if P n then x.coeff n else y.coeff n", [{"full_name": "WittVector.mk", "def_path": "Mathlib/RingTheory/WittVector/Defs.lean", "def_pos": [55, 5], "def_end_pos": [55, 18]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\n\u22a2 coeff (x + y) n = coeff x n + coeff y n", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\n\u22a2 coeff (x + y) n = coeff x n + coeff y n"}, {"tactic": "ext1 n", "annotated_tactic": ["ext1 n", []], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\n\u22a2 select P z = x", "state_after": "case h\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nn : \u2115\n\u22a2 coeff (select P z) n = coeff x n"}, {"tactic": "rw [select, coeff_mk, coeff_mk]", "annotated_tactic": ["rw [<a>select</a>, <a>coeff_mk</a>, <a>coeff_mk</a>]", [{"full_name": "WittVector.select", "def_path": "Mathlib/RingTheory/WittVector/InitTail.lean", "def_pos": [58, 5], "def_end_pos": [58, 11]}, {"full_name": "WittVector.coeff_mk", "def_path": "Mathlib/RingTheory/WittVector/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 17]}, {"full_name": "WittVector.coeff_mk", "def_path": "Mathlib/RingTheory/WittVector/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 17]}]], "state_before": "case h\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nn : \u2115\n\u22a2 coeff (select P z) n = coeff x n", "state_after": "case h\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nn : \u2115\n\u22a2 (if P n then if P n then coeff x n else coeff y n else 0) = coeff x n"}, {"tactic": "split_ifs with hn", "annotated_tactic": ["split_ifs with hn", []], "state_before": "case h\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nn : \u2115\n\u22a2 (if P n then if P n then coeff x n else coeff y n else 0) = coeff x n", "state_after": "case pos\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nn : \u2115\nhn : P n\n\u22a2 coeff x n = coeff x n\n\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nn : \u2115\nhn : \u00acP n\n\u22a2 0 = coeff x n"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case pos\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nn : \u2115\nhn : P n\n\u22a2 coeff x n = coeff x n", "state_after": "no goals"}, {"tactic": "rw [(h n).resolve_right hn]", "annotated_tactic": ["rw [(h n).<a>resolve_right</a> hn]", [{"full_name": "Or.resolve_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [272, 9], "def_end_pos": [272, 25]}]], "state_before": "case neg\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nn : \u2115\nhn : \u00acP n\n\u22a2 0 = coeff x n", "state_after": "no goals"}, {"tactic": "ext1 n", "annotated_tactic": ["ext1 n", []], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\n\u22a2 select (fun i => \u00acP i) z = y", "state_after": "case h\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nn : \u2115\n\u22a2 coeff (select (fun i => \u00acP i) z) n = coeff y n"}, {"tactic": "rw [select, coeff_mk, coeff_mk]", "annotated_tactic": ["rw [<a>select</a>, <a>coeff_mk</a>, <a>coeff_mk</a>]", [{"full_name": "WittVector.select", "def_path": "Mathlib/RingTheory/WittVector/InitTail.lean", "def_pos": [58, 5], "def_end_pos": [58, 11]}, {"full_name": "WittVector.coeff_mk", "def_path": "Mathlib/RingTheory/WittVector/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 17]}, {"full_name": "WittVector.coeff_mk", "def_path": "Mathlib/RingTheory/WittVector/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 17]}]], "state_before": "case h\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nn : \u2115\n\u22a2 coeff (select (fun i => \u00acP i) z) n = coeff y n", "state_after": "case h\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nn : \u2115\n\u22a2 (if \u00acP n then if P n then coeff x n else coeff y n else 0) = coeff y n"}, {"tactic": "split_ifs with hn", "annotated_tactic": ["split_ifs with hn", []], "state_before": "case h\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nn : \u2115\n\u22a2 (if \u00acP n then if P n then coeff x n else coeff y n else 0) = coeff y n", "state_after": "case pos\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nn : \u2115\nhn : P n\n\u22a2 0 = coeff y n\n\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nn : \u2115\nhn : \u00acP n\n\u22a2 coeff y n = coeff y n"}, {"tactic": "exact hn.symm", "annotated_tactic": ["exact hn.symm", []], "state_before": "case pos\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nn : \u2115\nhn : P n\n\u22a2 0 = coeff y n", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u271d : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nn : \u2115\nhn : \u00acP n\n\u22a2 coeff y n = coeff y n", "state_after": "no goals"}, {"tactic": "rw [\u2190 hx, \u2190 hy, select_add_select_not P z]", "annotated_tactic": ["rw [\u2190 hx, \u2190 hy, <a>select_add_select_not</a> P z]", [{"full_name": "WittVector.select_add_select_not", "def_path": "Mathlib/RingTheory/WittVector/InitTail.lean", "def_pos": [88, 9], "def_end_pos": [88, 30]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nhy : select (fun i => \u00acP i) z = y\n\u22a2 coeff (x + y) n = coeff z n", "state_after": "no goals"}, {"tactic": "simp only [mk._eq_1]", "annotated_tactic": ["simp only [mk._eq_1]", []], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nhy : select (fun i => \u00acP i) z = y\n\u22a2 coeff z n = coeff x n + coeff y n", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nhy : select (fun i => \u00acP i) z = y\n\u22a2 (if coeff y n = 0 then coeff x n else coeff y n) = coeff x n + coeff y n"}, {"tactic": "split_ifs with y0", "annotated_tactic": ["split_ifs with y0", []], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nhy : select (fun i => \u00acP i) z = y\n\u22a2 (if coeff y n = 0 then coeff x n else coeff y n) = coeff x n + coeff y n", "state_after": "case pos\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nhy : select (fun i => \u00acP i) z = y\ny0 : coeff y n = 0\n\u22a2 coeff x n = coeff x n + coeff y n\n\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nhy : select (fun i => \u00acP i) z = y\ny0 : \u00accoeff y n = 0\n\u22a2 coeff y n = coeff x n + coeff y n"}, {"tactic": "rw [y0, add_zero]", "annotated_tactic": ["rw [y0, <a>add_zero</a>]", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "case pos\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nhy : select (fun i => \u00acP i) z = y\ny0 : coeff y n = 0\n\u22a2 coeff x n = coeff x n + coeff y n", "state_after": "no goals"}, {"tactic": "rw [h n |>.resolve_right y0, zero_add]", "annotated_tactic": ["rw [h n |>.<a>resolve_right</a> y0, <a>zero_add</a>]", [{"full_name": "Or.resolve_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [272, 9], "def_end_pos": [272, 25]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "case neg\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nP\u271d : \u2115 \u2192 Prop\nx y : \ud835\udd4e R\nh : \u2200 (n : \u2115), coeff x n = 0 \u2228 coeff y n = 0\nP : \u2115 \u2192 Prop := fun n => coeff y n = 0\nthis : DecidablePred P\nz : \ud835\udd4e R := mk p fun n => if P n then coeff x n else coeff y n\nhx : select P z = x\nhy : select (fun i => \u00acP i) z = y\ny0 : \u00accoeff y n = 0\n\u22a2 coeff y n = coeff x n + coeff y n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Parity.lean", "full_name": "Int.even_or_odd'", "start": [67, 1], "end": [68, 65], "traced_tactics": [{"tactic": "simpa only [two_mul, exists_or, Odd, Even] using even_or_odd n", "annotated_tactic": ["simpa only [<a>two_mul</a>, <a>exists_or</a>, <a>Odd</a>, <a>Even</a>] using <a>even_or_odd</a> n", [{"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}, {"full_name": "exists_or", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [429, 9], "def_end_pos": [429, 18]}, {"full_name": "Odd", "def_path": "Mathlib/Algebra/Parity.lean", "def_pos": [302, 5], "def_end_pos": [302, 8]}, {"full_name": "Even", "def_path": "Mathlib/Algebra/Parity.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Int.even_or_odd", "def_path": "Mathlib/Data/Int/Parity.lean", "def_pos": [63, 9], "def_end_pos": [63, 20]}]], "state_before": "m n\u271d n : \u2124\n\u22a2 \u2203 k, n = 2 * k \u2228 n = 2 * k + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.subset_insert_iff", "start": [2024, 1], "end": [2026, 43], "traced_tactics": [{"tactic": "simp only [subset_iff, or_iff_not_imp_left, mem_erase, mem_insert, and_imp]", "annotated_tactic": ["simp only [<a>subset_iff</a>, <a>or_iff_not_imp_left</a>, <a>mem_erase</a>, <a>mem_insert</a>, <a>and_imp</a>]", [{"full_name": "Finset.subset_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [371, 9], "def_end_pos": [371, 19]}, {"full_name": "or_iff_not_imp_left", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [360, 9], "def_end_pos": [360, 28]}, {"full_name": "Finset.mem_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1887, 9], "def_end_pos": [1887, 18]}, {"full_name": "Finset.mem_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1087, 9], "def_end_pos": [1087, 19]}, {"full_name": "and_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [313, 17], "def_end_pos": [313, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u v : Finset \u03b1\na\u271d b a : \u03b1\ns t : Finset \u03b1\n\u22a2 s \u2286 insert a t \u2194 erase s a \u2286 t", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u v : Finset \u03b1\na\u271d b a : \u03b1\ns t : Finset \u03b1\n\u22a2 (\u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u00acx = a \u2192 x \u2208 t) \u2194 \u2200 \u2983x : \u03b1\u2984, x \u2260 a \u2192 x \u2208 s \u2192 x \u2208 t"}, {"tactic": "exact forall_congr' fun x => forall_swap", "annotated_tactic": ["exact <a>forall_congr'</a> fun x => <a>forall_swap</a>", [{"full_name": "forall_congr'", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [376, 9], "def_end_pos": [376, 22]}, {"full_name": "forall_swap", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [667, 9], "def_end_pos": [667, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d t\u271d u v : Finset \u03b1\na\u271d b a : \u03b1\ns t : Finset \u03b1\n\u22a2 (\u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u00acx = a \u2192 x \u2208 t) \u2194 \u2200 \u2983x : \u03b1\u2984, x \u2260 a \u2192 x \u2208 s \u2192 x \u2208 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/ModelTheory/Syntax.lean", "full_name": "FirstOrder.Language.BoundedFormula.IsAtomic.relabel", "start": [699, 1], "end": [701, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Abelianization.lean", "full_name": "abelianizationCongr_of", "start": [225, 1], "end": [227, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "full_name": "PMF.toOuterMeasure_caratheodory", "start": [166, 1], "end": [170, 98], "traced_tactics": [{"tactic": "refine' eq_top_iff.2 <| le_trans (le_sInf fun x hx => _) (le_sum_caratheodory _)", "annotated_tactic": ["refine' <a>eq_top_iff</a>.2 <| <a>le_trans</a> (<a>le_sInf</a> fun x hx => _) (<a>le_sum_caratheodory</a> _)", [{"full_name": "eq_top_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [165, 9], "def_end_pos": [165, 19]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_sInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [269, 9], "def_end_pos": [269, 16]}, {"full_name": "MeasureTheory.OuterMeasure.le_sum_caratheodory", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1107, 9], "def_end_pos": [1107, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\n\u22a2 OuterMeasure.caratheodory (toOuterMeasure p) = \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nx : MeasurableSpace \u03b1\nhx : x \u2208 Set.range fun i => OuterMeasure.caratheodory (\u2191p i \u2022 dirac i)\n\u22a2 \u22a4 \u2264 x"}, {"tactic": "have \u27e8y, hy\u27e9 := hx", "annotated_tactic": ["have \u27e8y, hy\u27e9 := hx", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nx : MeasurableSpace \u03b1\nhx : x \u2208 Set.range fun i => OuterMeasure.caratheodory (\u2191p i \u2022 dirac i)\n\u22a2 \u22a4 \u2264 x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nx : MeasurableSpace \u03b1\nhx : x \u2208 Set.range fun i => OuterMeasure.caratheodory (\u2191p i \u2022 dirac i)\ny : \u03b1\nhy : (fun i => OuterMeasure.caratheodory (\u2191p i \u2022 dirac i)) y = x\n\u22a2 \u22a4 \u2264 x"}, {"tactic": "exact\n  ((le_of_eq (dirac_caratheodory y).symm).trans (le_smul_caratheodory _ _)).trans (le_of_eq hy)", "annotated_tactic": ["exact\n    ((<a>le_of_eq</a> (<a>dirac_caratheodory</a> y).<a>symm</a>).<a>trans</a> (<a>le_smul_caratheodory</a> _ _)).<a>trans</a> (<a>le_of_eq</a> hy)", [{"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "MeasureTheory.OuterMeasure.dirac_caratheodory", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1120, 9], "def_end_pos": [1120, 27]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "MeasureTheory.OuterMeasure.le_smul_caratheodory", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1112, 9], "def_end_pos": [1112, 29]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np : PMF \u03b1\ns t : Set \u03b1\nx : MeasurableSpace \u03b1\nhx : x \u2208 Set.range fun i => OuterMeasure.caratheodory (\u2191p i \u2022 dirac i)\ny : \u03b1\nhy : (fun i => OuterMeasure.caratheodory (\u2191p i \u2022 dirac i)) y = x\n\u22a2 \u22a4 \u2264 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "full_name": "MonoidHom.ker_restrict", "start": [2868, 1], "end": [2869, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.measure_toMeasurable_union", "start": [386, 1], "end": [389, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/ContinuousOn.lean", "full_name": "ContinuousWithinAt.mem_closure", "start": [773, 1], "end": [775, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/LocalExtr.lean", "full_name": "IsLocalMaxOn.sup", "start": [461, 8], "end": [463, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/FreeMonoid/Basic.lean", "full_name": "FreeMonoid.hom_eq", "start": [200, 1], "end": [202, 62], "traced_tactics": [{"tactic": "simp only [h, hxs, MonoidHom.map_mul]", "annotated_tactic": ["simp only [h, hxs, <a>MonoidHom.map_mul</a>]", [{"full_name": "MonoidHom.map_mul", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [938, 19], "def_end_pos": [938, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nM : Type u_4\ninst\u271d\u00b9 : Monoid M\nN : Type u_5\ninst\u271d : Monoid N\nf g : FreeMonoid \u03b1 \u2192* M\nh : \u2200 (x : \u03b1), \u2191f (of x) = \u2191g (of x)\nl : FreeMonoid \u03b1\nx : \u03b1\nxs : FreeMonoid \u03b1\nhxs : \u2191f xs = \u2191g xs\n\u22a2 \u2191f (of x * xs) = \u2191g (of x * xs)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RepresentationTheory/Rep.lean", "full_name": "Rep.MonoidalCategory.braiding_inv_apply", "start": [156, 1], "end": [158, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.unrev_ok", "start": [1384, 1], "end": [1387, 45], "traced_tactics": [{"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "q : \u039b'\ns : Option \u0393'\nS : K' \u2192 List \u0393'\n\u22a2 rev \u2260 main", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/BigOperators/Basic.lean", "full_name": "Finset.Equiv.prod_comp_finset", "start": [583, 1], "end": [592, 27], "traced_tactics": [{"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "\u03b9 : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\n\u03b9' : Type u_2\ninst\u271d : DecidableEq \u03b9\ne : \u03b9 \u2243 \u03b9'\nf : \u03b9' \u2192 \u03b2\ns' : Finset \u03b9'\ns : Finset \u03b9\nh : s = image (\u2191e.symm) s'\n\u22a2 \u220f i' in s', f i' = \u220f i in s, f (\u2191e i)", "state_after": "\u03b9 : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\n\u03b9' : Type u_2\ninst\u271d : DecidableEq \u03b9\ne : \u03b9 \u2243 \u03b9'\nf : \u03b9' \u2192 \u03b2\ns' : Finset \u03b9'\ns : Finset \u03b9\nh : s = image (\u2191e.symm) s'\n\u22a2 \u220f i' in s', f i' = \u220f i in image (\u2191e.symm) s', f (\u2191e i)"}, {"tactic": "refine'\n  Finset.prod_bij' (fun i' _hi' => e.symm i') (fun a ha => Finset.mem_image_of_mem _ ha)\n    (fun a _ha => by simp_rw [e.apply_symm_apply]) (fun i _hi => e i) (fun a ha => _)\n    (fun a _ha => e.apply_symm_apply a) fun a _ha => e.symm_apply_apply a", "annotated_tactic": ["refine'\n    <a>Finset.prod_bij'</a> (fun i' _hi' => e.symm i') (fun a ha => <a>Finset.mem_image_of_mem</a> _ ha)\n      (fun a _ha => by simp_rw [e.apply_symm_apply]) (fun i _hi => e i) (fun a ha => _)\n      (fun a _ha => e.apply_symm_apply a) fun a _ha => e.symm_apply_apply a", [{"full_name": "Finset.prod_bij'", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [564, 9], "def_end_pos": [564, 18]}, {"full_name": "Finset.mem_image_of_mem", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [334, 9], "def_end_pos": [334, 25]}]], "state_before": "\u03b9 : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\n\u03b9' : Type u_2\ninst\u271d : DecidableEq \u03b9\ne : \u03b9 \u2243 \u03b9'\nf : \u03b9' \u2192 \u03b2\ns' : Finset \u03b9'\ns : Finset \u03b9\nh : s = image (\u2191e.symm) s'\n\u22a2 \u220f i' in s', f i' = \u220f i in image (\u2191e.symm) s', f (\u2191e i)", "state_after": "\u03b9 : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\n\u03b9' : Type u_2\ninst\u271d : DecidableEq \u03b9\ne : \u03b9 \u2243 \u03b9'\nf : \u03b9' \u2192 \u03b2\ns' : Finset \u03b9'\ns : Finset \u03b9\nh : s = image (\u2191e.symm) s'\na : \u03b9\nha : a \u2208 image (\u2191e.symm) s'\n\u22a2 (fun i _hi => \u2191e i) a ha \u2208 s'"}, {"tactic": "rcases Finset.mem_image.mp ha with \u27e8i', hi', rfl\u27e9", "annotated_tactic": ["rcases Finset.mem_image.mp ha with \u27e8i', hi', rfl\u27e9", []], "state_before": "\u03b9 : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\n\u03b9' : Type u_2\ninst\u271d : DecidableEq \u03b9\ne : \u03b9 \u2243 \u03b9'\nf : \u03b9' \u2192 \u03b2\ns' : Finset \u03b9'\ns : Finset \u03b9\nh : s = image (\u2191e.symm) s'\na : \u03b9\nha : a \u2208 image (\u2191e.symm) s'\n\u22a2 (fun i _hi => \u2191e i) a ha \u2208 s'", "state_after": "case intro.intro\n\u03b9 : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\n\u03b9' : Type u_2\ninst\u271d : DecidableEq \u03b9\ne : \u03b9 \u2243 \u03b9'\nf : \u03b9' \u2192 \u03b2\ns' : Finset \u03b9'\ns : Finset \u03b9\nh : s = image (\u2191e.symm) s'\ni' : \u03b9'\nhi' : i' \u2208 s'\nha : \u2191e.symm i' \u2208 image (\u2191e.symm) s'\n\u22a2 (fun i _hi => \u2191e i) (\u2191e.symm i') ha \u2208 s'"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "state_before": "case intro.intro\n\u03b9 : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\n\u03b9' : Type u_2\ninst\u271d : DecidableEq \u03b9\ne : \u03b9 \u2243 \u03b9'\nf : \u03b9' \u2192 \u03b2\ns' : Finset \u03b9'\ns : Finset \u03b9\nh : s = image (\u2191e.symm) s'\ni' : \u03b9'\nhi' : i' \u2208 s'\nha : \u2191e.symm i' \u2208 image (\u2191e.symm) s'\n\u22a2 (fun i _hi => \u2191e i) (\u2191e.symm i') ha \u2208 s'", "state_after": "case intro.intro\n\u03b9 : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\n\u03b9' : Type u_2\ninst\u271d : DecidableEq \u03b9\ne : \u03b9 \u2243 \u03b9'\nf : \u03b9' \u2192 \u03b2\ns' : Finset \u03b9'\ns : Finset \u03b9\nh : s = image (\u2191e.symm) s'\ni' : \u03b9'\nhi' : i' \u2208 s'\nha : \u2191e.symm i' \u2208 image (\u2191e.symm) s'\n\u22a2 \u2191e (\u2191e.symm i') \u2208 s'"}, {"tactic": "rwa [e.apply_symm_apply]", "annotated_tactic": ["rwa [e.apply_symm_apply]", []], "state_before": "case intro.intro\n\u03b9 : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\n\u03b9' : Type u_2\ninst\u271d : DecidableEq \u03b9\ne : \u03b9 \u2243 \u03b9'\nf : \u03b9' \u2192 \u03b2\ns' : Finset \u03b9'\ns : Finset \u03b9\nh : s = image (\u2191e.symm) s'\ni' : \u03b9'\nhi' : i' \u2208 s'\nha : \u2191e.symm i' \u2208 image (\u2191e.symm) s'\n\u22a2 \u2191e (\u2191e.symm i') \u2208 s'", "state_after": "no goals"}, {"tactic": "simp_rw [e.apply_symm_apply]", "annotated_tactic": ["simp_rw [e.apply_symm_apply]", []], "state_before": "\u03b9 : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na\u271d : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\n\u03b9' : Type u_2\ninst\u271d : DecidableEq \u03b9\ne : \u03b9 \u2243 \u03b9'\nf : \u03b9' \u2192 \u03b2\ns' : Finset \u03b9'\ns : Finset \u03b9\nh : s = image (\u2191e.symm) s'\na : \u03b9'\n_ha : a \u2208 s'\n\u22a2 f a = f (\u2191e ((fun i' _hi' => \u2191e.symm i') a _ha))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Multilinear/Basic.lean", "full_name": "MultilinearMap.map_update_zero", "start": [179, 1], "end": [180, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Commutator.lean", "full_name": "commutatorElement_self", "start": [54, 1], "end": [55, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/NormedSpace/Units.lean", "full_name": "NormedRing.inverse_add_nth_order", "start": [148, 1], "end": [156, 10], "traced_tactics": [{"tactic": "filter_upwards [inverse_add x, hzero.eventually (inverse_one_sub_nth_order n)] with t ht ht'", "annotated_tactic": ["filter_upwards [<a>inverse_add</a> x, hzero.eventually (<a>inverse_one_sub_nth_order</a> n)] with t ht ht'", [{"full_name": "NormedRing.inverse_add", "def_path": "Mathlib/Analysis/NormedSpace/Units.lean", "def_pos": [119, 9], "def_end_pos": [119, 20]}, {"full_name": "NormedRing.inverse_one_sub_nth_order", "def_path": "Mathlib/Analysis/NormedSpace/Units.lean", "def_pos": [137, 9], "def_end_pos": [137, 34]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\u02e3\nn : \u2115\nhzero : Tendsto (fun x_1 => -\u2191x\u207b\u00b9 * x_1) (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u22a2 \u2200\u1da0 (t : R) in \ud835\udcdd 0, inverse (\u2191x + t) = (\u2211 i in range n, (-\u2191x\u207b\u00b9 * t) ^ i) * \u2191x\u207b\u00b9 + (-\u2191x\u207b\u00b9 * t) ^ n * inverse (\u2191x + t)", "state_after": "case h\nR : Type u_1\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\u02e3\nn : \u2115\nhzero : Tendsto (fun x_1 => -\u2191x\u207b\u00b9 * x_1) (\ud835\udcdd 0) (\ud835\udcdd 0)\nt : R\nht : inverse (\u2191x + t) = inverse (1 + \u2191x\u207b\u00b9 * t) * \u2191x\u207b\u00b9\nht' : inverse (1 - -\u2191x\u207b\u00b9 * t) = \u2211 i in range n, (-\u2191x\u207b\u00b9 * t) ^ i + (-\u2191x\u207b\u00b9 * t) ^ n * inverse (1 - -\u2191x\u207b\u00b9 * t)\n\u22a2 inverse (\u2191x + t) = (\u2211 i in range n, (-\u2191x\u207b\u00b9 * t) ^ i) * \u2191x\u207b\u00b9 + (-\u2191x\u207b\u00b9 * t) ^ n * inverse (\u2191x + t)"}, {"tactic": "rw [neg_mul, sub_neg_eq_add] at ht'", "annotated_tactic": ["rw [<a>neg_mul</a>, <a>sub_neg_eq_add</a>] at ht'", [{"full_name": "neg_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 16]}, {"full_name": "sub_neg_eq_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [453, 3], "def_end_pos": [453, 14]}]], "state_before": "case h\nR : Type u_1\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\u02e3\nn : \u2115\nhzero : Tendsto (fun x_1 => -\u2191x\u207b\u00b9 * x_1) (\ud835\udcdd 0) (\ud835\udcdd 0)\nt : R\nht : inverse (\u2191x + t) = inverse (1 + \u2191x\u207b\u00b9 * t) * \u2191x\u207b\u00b9\nht' : inverse (1 - -\u2191x\u207b\u00b9 * t) = \u2211 i in range n, (-\u2191x\u207b\u00b9 * t) ^ i + (-\u2191x\u207b\u00b9 * t) ^ n * inverse (1 - -\u2191x\u207b\u00b9 * t)\n\u22a2 inverse (\u2191x + t) = (\u2211 i in range n, (-\u2191x\u207b\u00b9 * t) ^ i) * \u2191x\u207b\u00b9 + (-\u2191x\u207b\u00b9 * t) ^ n * inverse (\u2191x + t)", "state_after": "case h\nR : Type u_1\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\u02e3\nn : \u2115\nhzero : Tendsto (fun x_1 => -\u2191x\u207b\u00b9 * x_1) (\ud835\udcdd 0) (\ud835\udcdd 0)\nt : R\nht : inverse (\u2191x + t) = inverse (1 + \u2191x\u207b\u00b9 * t) * \u2191x\u207b\u00b9\nht' : inverse (1 + \u2191x\u207b\u00b9 * t) = \u2211 i in range n, (-(\u2191x\u207b\u00b9 * t)) ^ i + (-(\u2191x\u207b\u00b9 * t)) ^ n * inverse (1 + \u2191x\u207b\u00b9 * t)\n\u22a2 inverse (\u2191x + t) = (\u2211 i in range n, (-\u2191x\u207b\u00b9 * t) ^ i) * \u2191x\u207b\u00b9 + (-\u2191x\u207b\u00b9 * t) ^ n * inverse (\u2191x + t)"}, {"tactic": "conv_lhs => rw [ht, ht', add_mul, \u2190 neg_mul, mul_assoc]", "annotated_tactic": ["conv_lhs => rw [ht, ht', <a>add_mul</a>, \u2190 <a>neg_mul</a>, <a>mul_assoc</a>]", [{"full_name": "add_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [91, 7], "def_end_pos": [91, 14]}, {"full_name": "neg_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 16]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "case h\nR : Type u_1\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\u02e3\nn : \u2115\nhzero : Tendsto (fun x_1 => -\u2191x\u207b\u00b9 * x_1) (\ud835\udcdd 0) (\ud835\udcdd 0)\nt : R\nht : inverse (\u2191x + t) = inverse (1 + \u2191x\u207b\u00b9 * t) * \u2191x\u207b\u00b9\nht' : inverse (1 + \u2191x\u207b\u00b9 * t) = \u2211 i in range n, (-(\u2191x\u207b\u00b9 * t)) ^ i + (-(\u2191x\u207b\u00b9 * t)) ^ n * inverse (1 + \u2191x\u207b\u00b9 * t)\n\u22a2 inverse (\u2191x + t) = (\u2211 i in range n, (-\u2191x\u207b\u00b9 * t) ^ i) * \u2191x\u207b\u00b9 + (-\u2191x\u207b\u00b9 * t) ^ n * inverse (\u2191x + t)", "state_after": "case h\nR : Type u_1\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\u02e3\nn : \u2115\nhzero : Tendsto (fun x_1 => -\u2191x\u207b\u00b9 * x_1) (\ud835\udcdd 0) (\ud835\udcdd 0)\nt : R\nht : inverse (\u2191x + t) = inverse (1 + \u2191x\u207b\u00b9 * t) * \u2191x\u207b\u00b9\nht' : inverse (1 + \u2191x\u207b\u00b9 * t) = \u2211 i in range n, (-(\u2191x\u207b\u00b9 * t)) ^ i + (-(\u2191x\u207b\u00b9 * t)) ^ n * inverse (1 + \u2191x\u207b\u00b9 * t)\n\u22a2 (\u2211 i in range n, (-\u2191x\u207b\u00b9 * t) ^ i) * \u2191x\u207b\u00b9 + (-\u2191x\u207b\u00b9 * t) ^ n * (inverse (1 + \u2191x\u207b\u00b9 * t) * \u2191x\u207b\u00b9) =\n    (\u2211 i in range n, (-\u2191x\u207b\u00b9 * t) ^ i) * \u2191x\u207b\u00b9 + (-\u2191x\u207b\u00b9 * t) ^ n * inverse (\u2191x + t)"}, {"tactic": "rw [ht]", "annotated_tactic": ["rw [ht]", []], "state_before": "case h\nR : Type u_1\ninst\u271d\u00b9 : NormedRing R\ninst\u271d : CompleteSpace R\nx : R\u02e3\nn : \u2115\nhzero : Tendsto (fun x_1 => -\u2191x\u207b\u00b9 * x_1) (\ud835\udcdd 0) (\ud835\udcdd 0)\nt : R\nht : inverse (\u2191x + t) = inverse (1 + \u2191x\u207b\u00b9 * t) * \u2191x\u207b\u00b9\nht' : inverse (1 + \u2191x\u207b\u00b9 * t) = \u2211 i in range n, (-(\u2191x\u207b\u00b9 * t)) ^ i + (-(\u2191x\u207b\u00b9 * t)) ^ n * inverse (1 + \u2191x\u207b\u00b9 * t)\n\u22a2 (\u2211 i in range n, (-\u2191x\u207b\u00b9 * t) ^ i) * \u2191x\u207b\u00b9 + (-\u2191x\u207b\u00b9 * t) ^ n * (inverse (1 + \u2191x\u207b\u00b9 * t) * \u2191x\u207b\u00b9) =\n    (\u2211 i in range n, (-\u2191x\u207b\u00b9 * t) ^ i) * \u2191x\u207b\u00b9 + (-\u2191x\u207b\u00b9 * t) ^ n * inverse (\u2191x + t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Prod.lean", "full_name": "LinearMap.fst_surjective", "start": [84, 1], "end": [84, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/Basic.lean", "full_name": "Nat.mul_div_le_mul_div_assoc", "start": [673, 1], "end": [677, 81], "traced_tactics": [{"tactic": "simp [hc0]", "annotated_tactic": ["simp [hc0]", []], "state_before": "m n k a b c : \u2115\nhc0 : c = 0\n\u22a2 a * (b / c) \u2264 a * b / c", "state_after": "no goals"}, {"tactic": "rw [mul_assoc]", "annotated_tactic": ["rw [<a>mul_assoc</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "m n k a b c : \u2115\nhc0 : \u00acc = 0\n\u22a2 a * (b / c) * c \u2264 a * b", "state_after": "m n k a b c : \u2115\nhc0 : \u00acc = 0\n\u22a2 a * (b / c * c) \u2264 a * b"}, {"tactic": "exact Nat.mul_le_mul_left _ (Nat.div_mul_le_self _ _)", "annotated_tactic": ["exact <a>Nat.mul_le_mul_left</a> _ (<a>Nat.div_mul_le_self</a> _ _)", [{"full_name": "Nat.mul_le_mul_left", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [437, 9], "def_end_pos": [437, 24]}, {"full_name": "Nat.div_mul_le_self", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [613, 9], "def_end_pos": [613, 24]}]], "state_before": "m n k a b c : \u2115\nhc0 : \u00acc = 0\n\u22a2 a * (b / c * c) \u2264 a * b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/SesquilinearForm.lean", "full_name": "LinearMap.IsRefl.flip_isRefl_iff", "start": [189, 1], "end": [190, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Matrix/BilinearForm.lean", "full_name": "isAdjointPair_toBilin", "start": [455, 1], "end": [467, 6], "traced_tactics": [{"tactic": "rw [BilinForm.isAdjointPair_iff_compLeft_eq_compRight]", "annotated_tactic": ["rw [<a>BilinForm.isAdjointPair_iff_compLeft_eq_compRight</a>]", [{"full_name": "BilinForm.isAdjointPair_iff_compLeft_eq_compRight", "def_path": "Mathlib/LinearAlgebra/BilinearForm.lean", "def_pos": [1048, 9], "def_end_pos": [1048, 48]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\n\u22a2 BilinForm.IsAdjointPair (\u2191(toBilin b) J) (\u2191(toBilin b) J\u2083) (\u2191(toLin b b) A) (\u2191(toLin b b) A') \u2194\n    IsAdjointPair J J\u2083 A A'", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\n\u22a2 BilinForm.compLeft (\u2191(toBilin b) J\u2083) (\u2191(toLin b b) A) = BilinForm.compRight (\u2191(toBilin b) J) (\u2191(toLin b b) A') \u2194\n    IsAdjointPair J J\u2083 A A'"}, {"tactic": "rw [h, BilinForm.toMatrix_compLeft, BilinForm.toMatrix_compRight, LinearMap.toMatrix_toLin,\n  LinearMap.toMatrix_toLin, BilinForm.toMatrix_toBilin, BilinForm.toMatrix_toBilin]", "annotated_tactic": ["rw [h, <a>BilinForm.toMatrix_compLeft</a>, <a>BilinForm.toMatrix_compRight</a>, <a>LinearMap.toMatrix_toLin</a>,\n    <a>LinearMap.toMatrix_toLin</a>, <a>BilinForm.toMatrix_toBilin</a>, <a>BilinForm.toMatrix_toBilin</a>]", [{"full_name": "BilinForm.toMatrix_compLeft", "def_path": "Mathlib/LinearAlgebra/Matrix/BilinearForm.lean", "def_pos": [384, 9], "def_end_pos": [384, 36]}, {"full_name": "BilinForm.toMatrix_compRight", "def_path": "Mathlib/LinearAlgebra/Matrix/BilinearForm.lean", "def_pos": [389, 9], "def_end_pos": [389, 37]}, {"full_name": "LinearMap.toMatrix_toLin", "def_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "def_pos": [567, 9], "def_end_pos": [567, 33]}, {"full_name": "LinearMap.toMatrix_toLin", "def_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "def_pos": [567, 9], "def_end_pos": [567, 33]}, {"full_name": "BilinForm.toMatrix_toBilin", "def_path": "Mathlib/LinearAlgebra/Matrix/BilinearForm.lean", "def_pos": [350, 9], "def_end_pos": [350, 35]}, {"full_name": "BilinForm.toMatrix_toBilin", "def_path": "Mathlib/LinearAlgebra/Matrix/BilinearForm.lean", "def_pos": [350, 9], "def_end_pos": [350, 35]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nh : \u2200 (B B' : BilinForm R\u2083 M\u2083), B = B' \u2194 \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B'\n\u22a2 BilinForm.compLeft (\u2191(toBilin b) J\u2083) (\u2191(toLin b b) A) = BilinForm.compRight (\u2191(toBilin b) J) (\u2191(toLin b b) A') \u2194\n    IsAdjointPair J J\u2083 A A'", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nh : \u2200 (B B' : BilinForm R\u2083 M\u2083), B = B' \u2194 \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B'\n\u22a2 A\u1d40 * J\u2083 = J * A' \u2194 IsAdjointPair J J\u2083 A A'"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nh : \u2200 (B B' : BilinForm R\u2083 M\u2083), B = B' \u2194 \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B'\n\u22a2 A\u1d40 * J\u2083 = J * A' \u2194 IsAdjointPair J J\u2083 A A'", "state_after": "no goals"}, {"tactic": "intro B B'", "annotated_tactic": ["intro B B'", []], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\n\u22a2 \u2200 (B B' : BilinForm R\u2083 M\u2083), B = B' \u2194 \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B'", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nB B' : BilinForm R\u2083 M\u2083\n\u22a2 B = B' \u2194 \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B'"}, {"tactic": "constructor <;> intro h", "annotated_tactic": ["constructor <;> intro h", []], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nB B' : BilinForm R\u2083 M\u2083\n\u22a2 B = B' \u2194 \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B'", "state_after": "case mp\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nB B' : BilinForm R\u2083 M\u2083\nh : B = B'\n\u22a2 \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B'\n\ncase mpr\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nB B' : BilinForm R\u2083 M\u2083\nh : \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B'\n\u22a2 B = B'"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "case mp\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nB B' : BilinForm R\u2083 M\u2083\nh : B = B'\n\u22a2 \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B'", "state_after": "no goals"}, {"tactic": "exact (BilinForm.toMatrix b).injective h", "annotated_tactic": ["exact (<a>BilinForm.toMatrix</a> b).<a>injective</a> h", [{"full_name": "BilinForm.toMatrix", "def_path": "Mathlib/LinearAlgebra/Matrix/BilinearForm.lean", "def_pos": [291, 19], "def_end_pos": [291, 37]}, {"full_name": "LinearEquiv.injective", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [555, 19], "def_end_pos": [555, 28]}]], "state_before": "case mpr\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nB B' : BilinForm R\u2083 M\u2083\nh : \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B'\n\u22a2 B = B'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Rat/Defs.lean", "full_name": "Rat.mul_add", "start": [283, 11], "end": [284, 65], "traced_tactics": [{"tactic": "rw [Rat.mul_comm, Rat.add_mul, Rat.mul_comm, Rat.mul_comm c a]", "annotated_tactic": ["rw [<a>Rat.mul_comm</a>, <a>Rat.add_mul</a>, 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"https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "full_name": "affineSpan_le", "start": [710, 1], "end": [712, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/WellFoundedSet.lean", "full_name": "Set.PartiallyWellOrderedOn.image_of_monotone_on", "start": [302, 1], "end": [308, 48], "traced_tactics": [{"tactic": "intro g' hg'", "annotated_tactic": ["intro g' hg'", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nr' : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\na : \u03b1\nhs : PartiallyWellOrderedOn s r\nhf : \u2200 (a\u2081 : \u03b1), a\u2081 \u2208 s \u2192 \u2200 (a\u2082 : \u03b1), a\u2082 \u2208 s \u2192 r a\u2081 a\u2082 \u2192 r' (f a\u2081) (f a\u2082)\n\u22a2 PartiallyWellOrderedOn (f '' s) r'", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nr' : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\na : \u03b1\nhs : PartiallyWellOrderedOn s r\nhf : \u2200 (a\u2081 : \u03b1), a\u2081 \u2208 s \u2192 \u2200 (a\u2082 : \u03b1), a\u2082 \u2208 s \u2192 r a\u2081 a\u2082 \u2192 r' (f a\u2081) (f a\u2082)\ng' : \u2115 \u2192 \u03b2\nhg' : \u2200 (n : \u2115), g' n \u2208 f '' s\n\u22a2 \u2203 m n, m < n \u2227 r' (g' m) (g' n)"}, {"tactic": "choose g hgs heq using hg'", "annotated_tactic": ["choose g hgs heq using hg'", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nr' : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\na : \u03b1\nhs : PartiallyWellOrderedOn s r\nhf : \u2200 (a\u2081 : \u03b1), a\u2081 \u2208 s \u2192 \u2200 (a\u2082 : \u03b1), a\u2082 \u2208 s \u2192 r a\u2081 a\u2082 \u2192 r' (f a\u2081) (f a\u2082)\ng' : \u2115 \u2192 \u03b2\nhg' : \u2200 (n : \u2115), g' n \u2208 f '' s\n\u22a2 \u2203 m n, m < n \u2227 r' (g' m) (g' n)", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nr' : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\na : \u03b1\nhs : PartiallyWellOrderedOn s r\nhf : \u2200 (a\u2081 : \u03b1), a\u2081 \u2208 s \u2192 \u2200 (a\u2082 : \u03b1), a\u2082 \u2208 s \u2192 r a\u2081 a\u2082 \u2192 r' (f a\u2081) (f a\u2082)\ng' : \u2115 \u2192 \u03b2\ng : \u2115 \u2192 \u03b1\nhgs : \u2200 (n : \u2115), g n \u2208 s\nheq : \u2200 (n : \u2115), f (g n) = g' n\n\u22a2 \u2203 m n, m < n \u2227 r' (g' m) (g' n)"}, {"tactic": "obtain rfl : f \u2218 g = g'", "annotated_tactic": ["obtain rfl : f \u2218 g = g'", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nr' : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\na : \u03b1\nhs : PartiallyWellOrderedOn s r\nhf : \u2200 (a\u2081 : \u03b1), a\u2081 \u2208 s \u2192 \u2200 (a\u2082 : \u03b1), a\u2082 \u2208 s \u2192 r a\u2081 a\u2082 \u2192 r' (f a\u2081) (f a\u2082)\ng' : \u2115 \u2192 \u03b2\ng : \u2115 \u2192 \u03b1\nhgs : \u2200 (n : \u2115), g n \u2208 s\nheq : \u2200 (n : \u2115), f (g n) = g' n\n\u22a2 \u2203 m n, m < n \u2227 r' (g' m) (g' n)", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nr' : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\na : \u03b1\nhs : PartiallyWellOrderedOn s r\nhf : \u2200 (a\u2081 : \u03b1), a\u2081 \u2208 s \u2192 \u2200 (a\u2082 : \u03b1), a\u2082 \u2208 s \u2192 r a\u2081 a\u2082 \u2192 r' (f a\u2081) (f a\u2082)\ng' : \u2115 \u2192 \u03b2\ng : \u2115 \u2192 \u03b1\nhgs : \u2200 (n : \u2115), g n \u2208 s\nheq : \u2200 (n : \u2115), f (g n) = g' n\n\u22a2 f \u2218 g = g'\n\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nr' : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\na : \u03b1\nhs : PartiallyWellOrderedOn s r\nhf : \u2200 (a\u2081 : \u03b1), a\u2081 \u2208 s \u2192 \u2200 (a\u2082 : \u03b1), a\u2082 \u2208 s \u2192 r a\u2081 a\u2082 \u2192 r' (f a\u2081) (f a\u2082)\ng : \u2115 \u2192 \u03b1\nhgs : \u2200 (n : \u2115), g n \u2208 s\nheq : \u2200 (n : \u2115), f (g n) = (f \u2218 g) n\n\u22a2 \u2203 m n, m < n \u2227 r' ((f \u2218 g) m) ((f \u2218 g) n)"}, {"tactic": "exact funext heq", "annotated_tactic": ["exact <a>funext</a> heq", [{"full_name": "funext", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1555, 9], "def_end_pos": [1555, 15]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nr' : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\na : \u03b1\nhs : PartiallyWellOrderedOn s r\nhf : \u2200 (a\u2081 : \u03b1), a\u2081 \u2208 s \u2192 \u2200 (a\u2082 : \u03b1), a\u2082 \u2208 s \u2192 r a\u2081 a\u2082 \u2192 r' (f a\u2081) (f a\u2082)\ng' : \u2115 \u2192 \u03b2\ng : \u2115 \u2192 \u03b1\nhgs : \u2200 (n : \u2115), g n \u2208 s\nheq : \u2200 (n : \u2115), f (g n) = g' n\n\u22a2 f \u2218 g = g'\n\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nr' : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\na : \u03b1\nhs : PartiallyWellOrderedOn s r\nhf : \u2200 (a\u2081 : \u03b1), a\u2081 \u2208 s \u2192 \u2200 (a\u2082 : \u03b1), a\u2082 \u2208 s \u2192 r a\u2081 a\u2082 \u2192 r' (f a\u2081) (f a\u2082)\ng : \u2115 \u2192 \u03b1\nhgs : \u2200 (n : \u2115), g n \u2208 s\nheq : \u2200 (n : \u2115), f (g n) = (f \u2218 g) n\n\u22a2 \u2203 m n, m < n \u2227 r' ((f \u2218 g) m) ((f \u2218 g) n)", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nr' : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\na : \u03b1\nhs : PartiallyWellOrderedOn s r\nhf : \u2200 (a\u2081 : \u03b1), a\u2081 \u2208 s \u2192 \u2200 (a\u2082 : \u03b1), a\u2082 \u2208 s \u2192 r a\u2081 a\u2082 \u2192 r' (f a\u2081) (f a\u2082)\ng : \u2115 \u2192 \u03b1\nhgs : \u2200 (n : \u2115), g n \u2208 s\nheq : \u2200 (n : \u2115), f (g n) = (f \u2218 g) n\n\u22a2 \u2203 m n, m < n \u2227 r' ((f \u2218 g) m) ((f \u2218 g) n)"}, {"tactic": "obtain \u27e8m, n, hlt, hmn\u27e9 := hs g hgs", "annotated_tactic": ["obtain \u27e8m, n, hlt, hmn\u27e9 := hs g hgs", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nr' : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\na : \u03b1\nhs : PartiallyWellOrderedOn s r\nhf : \u2200 (a\u2081 : \u03b1), a\u2081 \u2208 s \u2192 \u2200 (a\u2082 : \u03b1), a\u2082 \u2208 s \u2192 r a\u2081 a\u2082 \u2192 r' (f a\u2081) (f a\u2082)\ng : \u2115 \u2192 \u03b1\nhgs : \u2200 (n : \u2115), g n \u2208 s\nheq : \u2200 (n : \u2115), f (g n) = (f \u2218 g) n\n\u22a2 \u2203 m n, m < n \u2227 r' ((f \u2218 g) m) ((f \u2218 g) n)", "state_after": "case intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nr' : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\na : \u03b1\nhs : PartiallyWellOrderedOn s r\nhf : \u2200 (a\u2081 : \u03b1), a\u2081 \u2208 s \u2192 \u2200 (a\u2082 : \u03b1), a\u2082 \u2208 s \u2192 r a\u2081 a\u2082 \u2192 r' (f a\u2081) (f a\u2082)\ng : \u2115 \u2192 \u03b1\nhgs : \u2200 (n : \u2115), g n \u2208 s\nheq : \u2200 (n : \u2115), f (g n) = (f \u2218 g) n\nm n : \u2115\nhlt : m < n\nhmn : r (g m) (g n)\n\u22a2 \u2203 m n, m < n \u2227 r' ((f \u2218 g) m) ((f \u2218 g) n)"}, {"tactic": "exact \u27e8m, n, hlt, hf _ (hgs m) _ (hgs n) hmn\u27e9", "annotated_tactic": ["exact \u27e8m, n, hlt, hf _ (hgs m) _ (hgs n) hmn\u27e9", []], "state_before": "case intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nr' : \u03b2 \u2192 \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\na : \u03b1\nhs : PartiallyWellOrderedOn s r\nhf : \u2200 (a\u2081 : \u03b1), a\u2081 \u2208 s \u2192 \u2200 (a\u2082 : \u03b1), a\u2082 \u2208 s \u2192 r a\u2081 a\u2082 \u2192 r' (f a\u2081) (f a\u2082)\ng : \u2115 \u2192 \u03b1\nhgs : \u2200 (n : \u2115), g n \u2208 s\nheq : \u2200 (n : \u2115), f (g n) = (f \u2218 g) n\nm n : \u2115\nhlt : m < n\nhmn : r (g m) (g n)\n\u22a2 \u2203 m n, m < n \u2227 r' ((f \u2218 g) m) ((f \u2218 g) n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "MeasureTheory.integral_iUnion_ae", "start": [257, 1], "end": [260, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finsupp/Defs.lean", "full_name": "Finsupp.single_eq_zero", "start": [438, 1], "end": [439, 50], "traced_tactics": [{"tactic": "simp [FunLike.ext_iff, single_eq_set_indicator]", "annotated_tactic": ["simp [<a>FunLike.ext_iff</a>, <a>single_eq_set_indicator</a>]", [{"full_name": "FunLike.ext_iff", "def_path": "Mathlib/Data/FunLike/Basic.lean", "def_pos": [181, 9], "def_end_pos": [181, 16]}, {"full_name": "Finsupp.single_eq_set_indicator", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [317, 9], "def_end_pos": [317, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na a' : \u03b1\nb : M\n\u22a2 single a b = 0 \u2194 b = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/MonoidLocalization.lean", "full_name": "Submonoid.LocalizationMap.lift_eq", "start": [1009, 1], "end": [1010, 88], "traced_tactics": [{"tactic": "rw [lift_spec, \u2190 g.map_mul]", "annotated_tactic": ["rw [<a>lift_spec</a>, \u2190 g.map_mul]", [{"full_name": "Submonoid.LocalizationMap.lift_spec", "def_path": "Mathlib/GroupTheory/MonoidLocalization.lean", "def_pos": [959, 9], "def_end_pos": [959, 18]}]], "state_before": "M : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx : M\n\u22a2 \u2191(lift f hg) (\u2191(toMap f) x) = \u2191g x", "state_after": "M : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx : M\n\u22a2 \u2191g (sec f (\u2191(toMap f) x)).1 = \u2191g (\u2191(sec f (\u2191(toMap f) x)).2 * x)"}, {"tactic": "exact f.eq_of_eq hg (by rw [sec_spec', f.toMap.map_mul])", "annotated_tactic": ["exact f.eq_of_eq hg (by rw [<a>sec_spec'</a>, f.toMap.map_mul])", [{"full_name": "Submonoid.LocalizationMap.sec_spec'", "def_path": "Mathlib/GroupTheory/MonoidLocalization.lean", "def_pos": [608, 9], "def_end_pos": [608, 18]}]], "state_before": "M : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx : M\n\u22a2 \u2191g (sec f (\u2191(toMap f) x)).1 = \u2191g (\u2191(sec f (\u2191(toMap f) x)).2 * x)", "state_after": "no goals"}, {"tactic": "rw [sec_spec', f.toMap.map_mul]", "annotated_tactic": ["rw [<a>sec_spec'</a>, f.toMap.map_mul]", [{"full_name": "Submonoid.LocalizationMap.sec_spec'", "def_path": "Mathlib/GroupTheory/MonoidLocalization.lean", "def_pos": [608, 9], "def_end_pos": [608, 18]}]], "state_before": "M : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : LocalizationMap S N\ng : M \u2192* P\nhg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y)\nx : M\n\u22a2 \u2191(toMap f) (sec f (\u2191(toMap f) x)).1 = \u2191(toMap f) (\u2191(sec f (\u2191(toMap f) x)).2 * x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "full_name": "MeasureTheory.AEEqFun.inf_le_right", "start": [575, 11], "end": [579, 21], "traced_tactics": [{"tactic": "rw [\u2190 coeFn_le]", "annotated_tactic": ["rw [\u2190 <a>coeFn_le</a>]", [{"full_name": "MeasureTheory.AEEqFun.coeFn_le", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [513, 9], "def_end_pos": [513, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\ninst\u271d\u00b9 : SemilatticeInf \u03b2\ninst\u271d : ContinuousInf \u03b2\nf g : \u03b1 \u2192\u2098[\u03bc] \u03b2\n\u22a2 f \u2293 g \u2264 g", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\ninst\u271d\u00b9 : SemilatticeInf \u03b2\ninst\u271d : ContinuousInf \u03b2\nf g : \u03b1 \u2192\u2098[\u03bc] \u03b2\n\u22a2 \u2191(f \u2293 g) \u2264\u1d50[\u03bc] \u2191g"}, {"tactic": "filter_upwards [coeFn_inf f g] with _ ha", "annotated_tactic": ["filter_upwards [<a>coeFn_inf</a> f g] with _ ha", [{"full_name": "MeasureTheory.AEEqFun.coeFn_inf", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [564, 9], "def_end_pos": [564, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\ninst\u271d\u00b9 : SemilatticeInf \u03b2\ninst\u271d : ContinuousInf \u03b2\nf g : \u03b1 \u2192\u2098[\u03bc] \u03b2\n\u22a2 \u2191(f \u2293 g) \u2264\u1d50[\u03bc] \u2191g", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\ninst\u271d\u00b9 : SemilatticeInf \u03b2\ninst\u271d : ContinuousInf \u03b2\nf g : \u03b1 \u2192\u2098[\u03bc] \u03b2\na\u271d : \u03b1\nha : \u2191(f \u2293 g) a\u271d = \u2191f a\u271d \u2293 \u2191g a\u271d\n\u22a2 \u2191(f \u2293 g) a\u271d \u2264 \u2191g a\u271d"}, {"tactic": "rw [ha]", "annotated_tactic": ["rw [ha]", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\ninst\u271d\u00b9 : SemilatticeInf \u03b2\ninst\u271d : ContinuousInf \u03b2\nf g : \u03b1 \u2192\u2098[\u03bc] \u03b2\na\u271d : \u03b1\nha : \u2191(f \u2293 g) a\u271d = \u2191f a\u271d \u2293 \u2191g a\u271d\n\u22a2 \u2191(f \u2293 g) a\u271d \u2264 \u2191g a\u271d", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\ninst\u271d\u00b9 : SemilatticeInf \u03b2\ninst\u271d : ContinuousInf \u03b2\nf g : \u03b1 \u2192\u2098[\u03bc] \u03b2\na\u271d : \u03b1\nha : \u2191(f \u2293 g) a\u271d = \u2191f a\u271d \u2293 \u2191g a\u271d\n\u22a2 \u2191f a\u271d \u2293 \u2191g a\u271d \u2264 \u2191g a\u271d"}, {"tactic": "exact inf_le_right", "annotated_tactic": ["exact <a>inf_le_right</a>", [{"full_name": "inf_le_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [399, 9], "def_end_pos": [399, 21]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\ninst\u271d\u00b9 : SemilatticeInf \u03b2\ninst\u271d : ContinuousInf \u03b2\nf g : \u03b1 \u2192\u2098[\u03bc] \u03b2\na\u271d : \u03b1\nha : \u2191(f \u2293 g) a\u271d = \u2191f a\u271d \u2293 \u2191g a\u271d\n\u22a2 \u2191f a\u271d \u2293 \u2191g a\u271d \u2264 \u2191g a\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/ContinuousOn.lean", "full_name": "ContinuousWithinAt.comp'", "start": [918, 1], "end": [921, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Halting.lean", "full_name": "RePred.of_eq", "start": [147, 1], "end": [149, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Init/Data/Bool/Lemmas.lean", "full_name": "Bool.decide_false_iff", "start": [145, 1], "end": [146, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Heyting/Basic.lean", "full_name": "sdiff_triangle", "start": [715, 1], "end": [717, 42], "traced_tactics": [{"tactic": "rw [sdiff_le_iff, sup_left_comm, \u2190 sdiff_le_iff]", "annotated_tactic": ["rw [<a>sdiff_le_iff</a>, <a>sup_left_comm</a>, \u2190 <a>sdiff_le_iff</a>]", [{"full_name": "sdiff_le_iff", "def_path": "Mathlib/Order/Heyting/Basic.lean", "def_pos": [500, 9], "def_end_pos": [500, 21]}, {"full_name": "sup_left_comm", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [275, 9], "def_end_pos": [275, 22]}, {"full_name": "sdiff_le_iff", "def_path": "Mathlib/Order/Heyting/Basic.lean", "def_pos": [500, 9], "def_end_pos": [500, 21]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : GeneralizedCoheytingAlgebra \u03b1\na\u271d b\u271d c\u271d d a b c : \u03b1\n\u22a2 a \\ c \u2264 a \\ b \u2294 b \\ c", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : GeneralizedCoheytingAlgebra \u03b1\na\u271d b\u271d c\u271d d a b c : \u03b1\n\u22a2 a \\ (a \\ b) \u2264 c \u2294 b \\ c"}, {"tactic": "exact sdiff_sdiff_le.trans le_sup_sdiff", "annotated_tactic": ["exact sdiff_sdiff_le.trans <a>le_sup_sdiff</a>", [{"full_name": "le_sup_sdiff", "def_path": "Mathlib/Order/Heyting/Basic.lean", "def_pos": [530, 9], "def_end_pos": [530, 21]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : GeneralizedCoheytingAlgebra \u03b1\na\u271d b\u271d c\u271d d a b c : \u03b1\n\u22a2 a \\ (a \\ b) \u2264 c \u2294 b \\ c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "full_name": "Metric.self_subset_cthickening", "start": [1183, 1], "end": [1184, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.aeval_def", "start": [1465, 1], "end": [1466, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Basic.lean", "full_name": "lt_of_eq_of_lt", "start": [191, 1], "end": [192, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "Monotone.map_iSup_of_continuousAt", "start": [2811, 1], "end": [2814, 74], "traced_tactics": [{"tactic": "rw [iSup, Mf.map_sSup_of_continuousAt Cf fbot, \u2190 range_comp, iSup]", "annotated_tactic": ["rw [<a>iSup</a>, Mf.map_sSup_of_continuousAt Cf fbot, \u2190 <a>range_comp</a>, <a>iSup</a>]", [{"full_name": "iSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [78, 5], "def_end_pos": [78, 9]}, {"full_name": "Set.range_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [744, 9], "def_end_pos": [744, 19]}, {"full_name": "iSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [78, 5], "def_end_pos": [78, 9]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2076 : CompleteLinearOrder \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : OrderTopology \u03b1\ninst\u271d\u00b3 : CompleteLinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : Nonempty \u03b3\n\u03b9 : Sort u_1\nf : \u03b1 \u2192 \u03b2\ng : \u03b9 \u2192 \u03b1\nCf : ContinuousAt f (iSup g)\nMf : Monotone f\nfbot : f \u22a5 = \u22a5\n\u22a2 f (\u2a06 i, g i) = \u2a06 i, f (g i)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2076 : CompleteLinearOrder \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : OrderTopology \u03b1\ninst\u271d\u00b3 : CompleteLinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : Nonempty \u03b3\n\u03b9 : Sort u_1\nf : \u03b1 \u2192 \u03b2\ng : \u03b9 \u2192 \u03b1\nCf : ContinuousAt f (iSup g)\nMf : Monotone f\nfbot : f \u22a5 = \u22a5\n\u22a2 sSup (range (f \u2218 g)) = sSup (range fun i => f (g i))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2076 : CompleteLinearOrder \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : OrderTopology \u03b1\ninst\u271d\u00b3 : CompleteLinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : Nonempty \u03b3\n\u03b9 : Sort u_1\nf : \u03b1 \u2192 \u03b2\ng : \u03b9 \u2192 \u03b1\nCf : ContinuousAt f (iSup g)\nMf : Monotone f\nfbot : f \u22a5 = \u22a5\n\u22a2 sSup (range (f \u2218 g)) = sSup (range fun i => f (g i))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "full_name": "isComplete_iff_ultrafilter", "start": [353, 1], "end": [358, 72], "traced_tactics": [{"tactic": "refine' \u27e8fun h l => h l, fun H => isComplete_iff_clusterPt.2 fun l hl hls => _\u27e9", "annotated_tactic": ["refine' \u27e8fun h l => h l, fun H => <a>isComplete_iff_clusterPt</a>.2 fun l hl hls => _\u27e9", [{"full_name": "isComplete_iff_clusterPt", "def_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "def_pos": [347, 9], "def_end_pos": [347, 33]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\ns : Set \u03b1\n\u22a2 IsComplete s \u2194 \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2191l \u2264 \ud835\udcdf s \u2192 \u2203 x, x \u2208 s \u2227 \u2191l \u2264 \ud835\udcdd x", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\ns : Set \u03b1\nH : \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2191l \u2264 \ud835\udcdf s \u2192 \u2203 x, x \u2208 s \u2227 \u2191l \u2264 \ud835\udcdd x\nl : Filter \u03b1\nhl : Cauchy l\nhls : l \u2264 \ud835\udcdf s\n\u22a2 \u2203 x, x \u2208 s \u2227 ClusterPt x l"}, {"tactic": "haveI := hl.1", "annotated_tactic": ["haveI := hl.1", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\ns : Set \u03b1\nH : \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2191l \u2264 \ud835\udcdf s \u2192 \u2203 x, x \u2208 s \u2227 \u2191l \u2264 \ud835\udcdd x\nl : Filter \u03b1\nhl : Cauchy l\nhls : l \u2264 \ud835\udcdf s\n\u22a2 \u2203 x, x \u2208 s \u2227 ClusterPt x l", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\ns : Set \u03b1\nH : \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2191l \u2264 \ud835\udcdf s \u2192 \u2203 x, x \u2208 s \u2227 \u2191l \u2264 \ud835\udcdd x\nl : Filter \u03b1\nhl : Cauchy l\nhls : l \u2264 \ud835\udcdf s\nthis : NeBot l\n\u22a2 \u2203 x, x \u2208 s \u2227 ClusterPt x l"}, {"tactic": "rcases H (Ultrafilter.of l) hl.ultrafilter_of ((Ultrafilter.of_le l).trans hls) with \u27e8x, hxs, hxl\u27e9", "annotated_tactic": ["rcases H (<a>Ultrafilter.of</a> l) hl.ultrafilter_of ((<a>Ultrafilter.of_le</a> l).<a>trans</a> hls) with \u27e8x, hxs, hxl\u27e9", [{"full_name": "Ultrafilter.of", "def_path": "Mathlib/Order/Filter/Ultrafilter.lean", "def_pos": [380, 19], "def_end_pos": [380, 21]}, {"full_name": "Ultrafilter.of_le", "def_path": "Mathlib/Order/Filter/Ultrafilter.lean", "def_pos": [384, 9], "def_end_pos": [384, 14]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\ns : Set \u03b1\nH : \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2191l \u2264 \ud835\udcdf s \u2192 \u2203 x, x \u2208 s \u2227 \u2191l \u2264 \ud835\udcdd x\nl : Filter \u03b1\nhl : Cauchy l\nhls : l \u2264 \ud835\udcdf s\nthis : NeBot l\n\u22a2 \u2203 x, x \u2208 s \u2227 ClusterPt x l", "state_after": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\ns : Set \u03b1\nH : \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2191l \u2264 \ud835\udcdf s \u2192 \u2203 x, x \u2208 s \u2227 \u2191l \u2264 \ud835\udcdd x\nl : Filter \u03b1\nhl : Cauchy l\nhls : l \u2264 \ud835\udcdf s\nthis : NeBot l\nx : \u03b1\nhxs : x \u2208 s\nhxl : \u2191(Ultrafilter.of l) \u2264 \ud835\udcdd x\n\u22a2 \u2203 x, x \u2208 s \u2227 ClusterPt x l"}, {"tactic": "exact \u27e8x, hxs, (ClusterPt.of_le_nhds hxl).mono (Ultrafilter.of_le l)\u27e9", "annotated_tactic": ["exact \u27e8x, hxs, (<a>ClusterPt.of_le_nhds</a> hxl).<a>mono</a> (<a>Ultrafilter.of_le</a> l)\u27e9", [{"full_name": "ClusterPt.of_le_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1131, 9], "def_end_pos": [1131, 29]}, {"full_name": "ClusterPt.mono", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1144, 9], "def_end_pos": [1144, 23]}, {"full_name": "Ultrafilter.of_le", "def_path": "Mathlib/Order/Filter/Ultrafilter.lean", "def_pos": [384, 9], "def_end_pos": [384, 14]}]], "state_before": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\ns : Set \u03b1\nH : \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2191l \u2264 \ud835\udcdf s \u2192 \u2203 x, x \u2208 s \u2227 \u2191l \u2264 \ud835\udcdd x\nl : Filter \u03b1\nhl : Cauchy l\nhls : l \u2264 \ud835\udcdf s\nthis : NeBot l\nx : \u03b1\nhxs : x \u2208 s\nhxl : \u2191(Ultrafilter.of l) \u2264 \ud835\udcdd x\n\u22a2 \u2203 x, x \u2208 s \u2227 ClusterPt x l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Group/Basic.lean", "full_name": "eq_div_iff_mul_eq''", "start": [939, 1], "end": [939, 92], "traced_tactics": [{"tactic": "rw [eq_div_iff_mul_eq', mul_comm]", "annotated_tactic": ["rw [<a>eq_div_iff_mul_eq'</a>, <a>mul_comm</a>]", [{"full_name": "eq_div_iff_mul_eq'", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [825, 9], "def_end_pos": [825, 27]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nG : Type u_3\ninst\u271d : CommGroup G\na b c d : G\n\u22a2 a = b / c \u2194 c * a = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean", "full_name": "HasStrictDerivAt.const_smul", "start": [122, 8], "end": [124, 49], "traced_tactics": [{"tactic": "simpa using (hf.rst.imnst_smul c).hasStrictDerivAt", "annotated_tactic": ["simpa using (hf.rst.imnst_smul c).<a>hasStrictDerivAt</a>", [{"full_name": "HasStrictFDerivAt.hasStrictDerivAt", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [208, 19], "def_end_pos": [208, 53]}]], "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nR : Type u_1\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : SMulCommClass \ud835\udd5c R F\ninst\u271d : ContinuousConstSMul R F\nc : R\nhf : HasStrictDerivAt f f' x\n\u22a2 HasStrictDerivAt (fun y => c \u2022 f y) (c \u2022 f') x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "full_name": "WithBot.add_coe_eq_bot_iff", "start": [633, 1], "end": [634, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Pi.lean", "full_name": "Filter.compl_mem_coprod\u1d62", "start": [205, 1], "end": [207, 59], "traced_tactics": [{"tactic": "simp only [Filter.coprod\u1d62, mem_iSup, compl_mem_comap]", "annotated_tactic": ["simp only [<a>Filter.coprod\u1d62</a>, <a>mem_iSup</a>, <a>compl_mem_comap</a>]", [{"full_name": "Filter.coprod\u1d62", "def_path": "Mathlib/Order/Filter/Pi.lean", "def_pos": [197, 15], "def_end_pos": [197, 22]}, {"full_name": "Filter.mem_iSup", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [582, 9], "def_end_pos": [582, 17]}, {"full_name": "Filter.compl_mem_comap", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1963, 9], "def_end_pos": [1963, 24]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\ns : Set ((i : \u03b9) \u2192 \u03b1 i)\n\u22a2 s\u1d9c \u2208 Filter.coprod\u1d62 f \u2194 \u2200 (i : \u03b9), (eval i '' s)\u1d9c \u2208 f i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Nat.Primrec.add", "start": [130, 1], "end": [132, 54], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "p : \u2115\n\u22a2 unpaired (fun z n => Nat.rec (id z) (fun y IH => succ (unpair (unpair (Nat.pair z (Nat.pair y IH))).2).2) n) p =\n    unpaired (fun x x_1 => x + x_1) p", "state_after": "p : \u2115\n\u22a2 Nat.rec (unpair p).1 (fun y IH => succ IH) (unpair p).2 = (unpair p).1 + (unpair p).2"}, {"tactic": "induction p.unpair.2 <;> simp [*, add_succ]", "annotated_tactic": ["induction p.unpair.2 <;> simp [*, <a>add_succ</a>]", [{"full_name": "Nat.add_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 17]}]], "state_before": "p : \u2115\n\u22a2 Nat.rec (unpair p).1 (fun y IH => succ IH) (unpair p).2 = (unpair p).1 + (unpair p).2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/BumpFunction/Basic.lean", "full_name": "ContDiffBump.one_lt_rOut_div_rIn", "start": [116, 1], "end": [118, 22], "traced_tactics": [{"tactic": "rw [one_lt_div f.rIn_pos]", "annotated_tactic": ["rw [<a>one_lt_div</a> f.rIn_pos]", [{"full_name": "one_lt_div", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [428, 9], "def_end_pos": [428, 19]}]], "state_before": "E : Type u_1\nX : Type u_2\nc : E\nf : ContDiffBump c\n\u22a2 1 < f.rOut / f.rIn", "state_after": "E : Type u_1\nX : Type u_2\nc : E\nf : ContDiffBump c\n\u22a2 f.rIn < f.rOut"}, {"tactic": "exact f.rIn_lt_rOut", "annotated_tactic": ["exact f.rIn_lt_rOut", []], "state_before": "E : Type u_1\nX : Type u_2\nc : E\nf : ContDiffBump c\n\u22a2 f.rIn < f.rOut", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.iUnion_or", "start": [879, 1], "end": [881, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Instances/ENNReal.lean", "full_name": "ENNReal.exists_frequently_lt_of_liminf_ne_top", "start": [708, 1], "end": [714, 68], "traced_tactics": [{"tactic": "by_contra h", "annotated_tactic": ["by_contra h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Type u_4\nl : Filter \u03b9\nx : \u03b9 \u2192 \u211d\nhx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4\n\u22a2 \u2203 R, \u2203\u1da0 (n : \u03b9) in l, x n < R", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Type u_4\nl : Filter \u03b9\nx : \u03b9 \u2192 \u211d\nhx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4\nh : \u00ac\u2203 R, \u2203\u1da0 (n : \u03b9) in l, x n < R\n\u22a2 False"}, {"tactic": "simp_rw [not_exists, not_frequently, not_lt] at h", "annotated_tactic": ["simp_rw [<a>not_exists</a>, <a>not_frequently</a>, <a>not_lt</a>] at h", [{"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"full_name": "Filter.not_frequently", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1331, 9], "def_end_pos": [1331, 23]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Type u_4\nl : Filter \u03b9\nx : \u03b9 \u2192 \u211d\nhx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4\nh : \u00ac\u2203 R, \u2203\u1da0 (n : \u03b9) in l, x n < R\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Type u_4\nl : Filter \u03b9\nx : \u03b9 \u2192 \u211d\nhx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4\nh : \u2200 (x_1 : \u211d), \u2200\u1da0 (x_2 : \u03b9) in l, x_1 \u2264 x x_2\n\u22a2 False"}, {"tactic": "refine hx (ENNReal.eq_top_of_forall_nnreal_le fun r => le_limsInf_of_le (by isBoundedDefault) ?_)", "annotated_tactic": ["refine hx (<a>ENNReal.eq_top_of_forall_nnreal_le</a> fun r => <a>le_limsInf_of_le</a> (by isBoundedDefault) ?_)", [{"full_name": "ENNReal.eq_top_of_forall_nnreal_le", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1734, 9], "def_end_pos": [1734, 35]}, {"full_name": "Filter.le_limsInf_of_le", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [498, 9], "def_end_pos": [498, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Type u_4\nl : Filter \u03b9\nx : \u03b9 \u2192 \u211d\nhx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4\nh : \u2200 (x_1 : \u211d), \u2200\u1da0 (x_2 : \u03b9) in l, x_1 \u2264 x x_2\n\u22a2 False", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr\u271d p q : \u211d\u22650\nx\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Type u_4\nl : Filter \u03b9\nx : \u03b9 \u2192 \u211d\nhx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4\nh : \u2200 (x_1 : \u211d), \u2200\u1da0 (x_2 : \u03b9) in l, x_1 \u2264 x x_2\nr : \u211d\u22650\n\u22a2 \u2200\u1da0 (n : \u211d\u22650\u221e) in map (fun n => \u2191(\u2191Real.nnabs (x n))) l, \u2191r \u2264 n"}, {"tactic": "simp only [eventually_map, ENNReal.coe_le_coe]", "annotated_tactic": ["simp only [<a>eventually_map</a>, <a>ENNReal.coe_le_coe</a>]", [{"full_name": "Filter.eventually_map", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 23]}, {"full_name": "ENNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [349, 28], "def_end_pos": [349, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr\u271d p q : \u211d\u22650\nx\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Type u_4\nl : Filter \u03b9\nx : \u03b9 \u2192 \u211d\nhx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4\nh : \u2200 (x_1 : \u211d), \u2200\u1da0 (x_2 : \u03b9) in l, x_1 \u2264 x x_2\nr : \u211d\u22650\n\u22a2 \u2200\u1da0 (n : \u211d\u22650\u221e) in map (fun n => \u2191(\u2191Real.nnabs (x n))) l, \u2191r \u2264 n", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr\u271d p q : \u211d\u22650\nx\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Type u_4\nl : Filter \u03b9\nx : \u03b9 \u2192 \u211d\nhx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4\nh : \u2200 (x_1 : \u211d), \u2200\u1da0 (x_2 : \u03b9) in l, x_1 \u2264 x x_2\nr : \u211d\u22650\n\u22a2 \u2200\u1da0 (a : \u03b9) in l, r \u2264 \u2191Real.nnabs (x a)"}, {"tactic": "filter_upwards [h r] with i hi using hi.trans (le_abs_self (x i))", "annotated_tactic": ["filter_upwards [h r] with i hi using hi.trans (<a>le_abs_self</a> (x i))", [{"full_name": "le_abs_self", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [57, 9], "def_end_pos": [57, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr\u271d p q : \u211d\u22650\nx\u271d y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u03b9 : Type u_4\nl : Filter \u03b9\nx : \u03b9 \u2192 \u211d\nhx : liminf (fun n => \u2191(\u2191Real.nnabs (x n))) l \u2260 \u22a4\nh : \u2200 (x_1 : \u211d), \u2200\u1da0 (x_2 : \u03b9) in l, x_1 \u2264 x x_2\nr : \u211d\u22650\n\u22a2 \u2200\u1da0 (a : \u03b9) in l, r \u2264 \u2191Real.nnabs (x a)", "state_after": "no goals"}, {"tactic": "isBoundedDefault", "annotated_tactic": ["isBoundedDefault", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type 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"commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.succ_to_nat", "start": [296, 1], "end": [297, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/UpperLower/Basic.lean", "full_name": "UpperSet.coe_subset_coe", "start": [548, 1], "end": [549, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Sites/Surjective.lean", "full_name": "CategoryTheory.isLocallySurjective_iff_imagePresheaf_sheafify_eq_top'", "start": [81, 1], "end": [85, 53], "traced_tactics": [{"tactic": "simp only [Subpresheaf.ext_iff, Function.funext_iff, Set.ext_iff, top_subpresheaf_obj,\n  Set.top_eq_univ, Set.mem_univ, iff_true_iff]", "annotated_tactic": ["simp only [<a>Subpresheaf.ext_iff</a>, <a>Function.funext_iff</a>, <a>Set.ext_iff</a>, <a>top_subpresheaf_obj</a>,\n    <a>Set.top_eq_univ</a>, <a>Set.mem_univ</a>, <a>iff_true_iff</a>]", [{"full_name": "CategoryTheory.GrothendieckTopology.Subpresheaf.ext_iff", "def_path": "Mathlib/CategoryTheory/Sites/Subsheaf.lean", "def_pos": [46, 3], "def_end_pos": [46, 6]}, {"full_name": "Function.funext_iff", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [82, 9], "def_end_pos": [82, 19]}, {"full_name": "Set.ext_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [241, 9], "def_end_pos": [241, 16]}, {"full_name": "CategoryTheory.GrothendieckTopology.top_subpresheaf_obj", "def_path": "Mathlib/CategoryTheory/Sites/Subsheaf.lean", "def_pos": [350, 9], "def_end_pos": [350, 28]}, {"full_name": "Set.top_eq_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [101, 9], "def_end_pos": [101, 20]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}, {"full_name": "iff_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [196, 9], "def_end_pos": [196, 21]}]], "state_before": "C : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 Type w\nf : F \u27f6 G\n\u22a2 IsLocallySurjective J f \u2194 Subpresheaf.sheafify J (imagePresheaf f) = \u22a4", "state_after": "C : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 Type w\nf : F \u27f6 G\n\u22a2 IsLocallySurjective J f \u2194 \u2200 (a : C\u1d52\u1d56) (x : G.obj a), x \u2208 Subpresheaf.obj (Subpresheaf.sheafify J (imagePresheaf f)) a"}, {"tactic": "exact \u27e8fun H U => H (unop U), fun H U => H (op U)\u27e9", "annotated_tactic": ["exact \u27e8fun H U => H (<a>unop</a> U), fun H U => H (<a>op</a> U)\u27e9", [{"full_name": "Opposite.unop", "def_path": "Mathlib/Data/Opposite.lean", "def_pos": [37, 3], "def_end_pos": [37, 7]}, {"full_name": "Opposite.op", "def_path": "Mathlib/Data/Opposite.lean", "def_pos": [53, 5], "def_end_pos": [53, 7]}]], "state_before": "C : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 Type w\nf : F \u27f6 G\n\u22a2 IsLocallySurjective J f \u2194 \u2200 (a : C\u1d52\u1d56) (x : G.obj a), x \u2208 Subpresheaf.obj (Subpresheaf.sheafify J (imagePresheaf f)) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean", "full_name": "strictConvexOn_pow", "start": [40, 1], "end": [45, 50], "traced_tactics": [{"tactic": "apply StrictMonoOn.strictConvexOn_of_deriv (convex_Ici _) (continuousOn_pow _)", "annotated_tactic": ["apply <a>StrictMonoOn.strictConvexOn_of_deriv</a> (<a>convex_Ici</a> _) (<a>continuousOn_pow</a> _)", [{"full_name": "StrictMonoOn.strictConvexOn_of_deriv", "def_path": "Mathlib/Analysis/Calculus/MeanValue.lean", "def_pos": [1119, 9], "def_end_pos": [1119, 45]}, {"full_name": "convex_Ici", "def_path": "Mathlib/Analysis/Convex/Basic.lean", "def_pos": [286, 9], "def_end_pos": [286, 19]}, {"full_name": "continuousOn_pow", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [583, 9], "def_end_pos": [583, 25]}]], "state_before": "n : \u2115\nhn : 2 \u2264 n\n\u22a2 StrictConvexOn \u211d (Ici 0) fun x => x ^ n", "state_after": "n : \u2115\nhn : 2 \u2264 n\n\u22a2 StrictMonoOn (deriv fun x => x ^ n) (interior (Ici 0))"}, {"tactic": "rw [deriv_pow', interior_Ici]", "annotated_tactic": ["rw [<a>deriv_pow'</a>, <a>interior_Ici</a>]", [{"full_name": "deriv_pow'", "def_path": "Mathlib/Analysis/Calculus/Deriv/Pow.lean", "def_pos": [92, 9], "def_end_pos": [92, 19]}, {"full_name": "interior_Ici", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [2357, 9], "def_end_pos": [2357, 21]}]], "state_before": "n : \u2115\nhn : 2 \u2264 n\n\u22a2 StrictMonoOn (deriv fun x => x ^ n) (interior (Ici 0))", "state_after": "n : \u2115\nhn : 2 \u2264 n\n\u22a2 StrictMonoOn (fun x => \u2191n * x ^ (n - 1)) (Ioi 0)"}, {"tactic": "exact fun x (hx : 0 < x) y hy hxy =>\n  mul_lt_mul_of_pos_left (pow_lt_pow_of_lt_left hxy hx.le <| Nat.sub_pos_of_lt hn)\n    (Nat.cast_pos.2 <| zero_lt_two.trans_le hn)", "annotated_tactic": ["exact fun x (hx : 0 < x) y hy hxy =>\n    <a>mul_lt_mul_of_pos_left</a> (<a>pow_lt_pow_of_lt_left</a> hxy hx.le <| <a>Nat.sub_pos_of_lt</a> hn)\n      (<a>Nat.cast_pos</a>.2 <| zero_lt_two.trans_le hn)", [{"full_name": "mul_lt_mul_of_pos_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [160, 9], "def_end_pos": [160, 31]}, {"full_name": "pow_lt_pow_of_lt_left", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [471, 9], "def_end_pos": [471, 30]}, {"full_name": "Nat.sub_pos_of_lt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [404, 19], "def_end_pos": [404, 32]}, {"full_name": "Nat.cast_pos", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "n : \u2115\nhn : 2 \u2264 n\n\u22a2 StrictMonoOn (fun x => \u2191n * x ^ (n - 1)) (Ioi 0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Alternating/Basic.lean", "full_name": "AlternatingMap.curryLeft_compLinearMap", "start": [1062, 1], "end": [1068, 11], "traced_tactics": [{"tactic": "refine' Fin.cases _ _", "annotated_tactic": ["refine' <a>Fin.cases</a> _ _", [{"full_name": "Fin.cases", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [613, 21], "def_end_pos": [613, 26]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9\u2079 : Semiring R\nM : Type u_2\ninst\u271d\u00b9\u2078 : AddCommMonoid M\ninst\u271d\u00b9\u2077 : Module R M\nN : Type u_3\ninst\u271d\u00b9\u2076 : AddCommMonoid N\ninst\u271d\u00b9\u2075 : Module R N\nP : Type u_4\ninst\u271d\u00b9\u2074 : AddCommMonoid P\ninst\u271d\u00b9\u00b3 : Module R P\nM' : Type u_5\ninst\u271d\u00b9\u00b2 : AddCommGroup M'\ninst\u271d\u00b9\u00b9 : Module R M'\nN' : Type u_6\ninst\u271d\u00b9\u2070 : AddCommGroup N'\ninst\u271d\u2079 : Module R N'\n\u03b9 : Type u_7\n\u03b9' : Type u_8\n\u03b9'' : Type u_9\nR' : Type u_10\nM'' : Type u_11\nM\u2082'' : Type u_12\nN'' : Type u_13\nN\u2082'' : Type u_14\ninst\u271d\u2078 : CommSemiring R'\ninst\u271d\u2077 : AddCommMonoid M''\ninst\u271d\u2076 : AddCommMonoid M\u2082''\ninst\u271d\u2075 : AddCommMonoid N''\ninst\u271d\u2074 : AddCommMonoid N\u2082''\ninst\u271d\u00b3 : Module R' M''\ninst\u271d\u00b2 : Module R' M\u2082''\ninst\u271d\u00b9 : Module R' N''\ninst\u271d : Module R' N\u2082''\nn : \u2115\ng : M\u2082'' \u2192\u2097[R'] M''\nf : AlternatingMap R' M'' N'' (Fin (Nat.succ n))\nm : M\u2082''\nv : Fin n \u2192 M\u2082''\n\u22a2 \u2200 (x : Fin (Nat.succ n)),\n    \u2191((fun x => g) x) (Matrix.vecCons m v x) = Matrix.vecCons (\u2191g m) (fun i => \u2191((fun x => g) i) (v i)) x", "state_after": "case refine'_1\nR : Type u_1\ninst\u271d\u00b9\u2079 : Semiring R\nM : Type u_2\ninst\u271d\u00b9\u2078 : AddCommMonoid M\ninst\u271d\u00b9\u2077 : Module R M\nN : Type u_3\ninst\u271d\u00b9\u2076 : AddCommMonoid N\ninst\u271d\u00b9\u2075 : Module R N\nP : Type u_4\ninst\u271d\u00b9\u2074 : AddCommMonoid P\ninst\u271d\u00b9\u00b3 : Module R P\nM' : Type u_5\ninst\u271d\u00b9\u00b2 : AddCommGroup M'\ninst\u271d\u00b9\u00b9 : Module R M'\nN' : Type u_6\ninst\u271d\u00b9\u2070 : AddCommGroup N'\ninst\u271d\u2079 : Module R N'\n\u03b9 : Type u_7\n\u03b9' : Type u_8\n\u03b9'' : Type u_9\nR' : Type u_10\nM'' : Type u_11\nM\u2082'' : Type u_12\nN'' : Type u_13\nN\u2082'' : Type u_14\ninst\u271d\u2078 : CommSemiring R'\ninst\u271d\u2077 : AddCommMonoid M''\ninst\u271d\u2076 : AddCommMonoid M\u2082''\ninst\u271d\u2075 : AddCommMonoid N''\ninst\u271d\u2074 : AddCommMonoid N\u2082''\ninst\u271d\u00b3 : Module R' M''\ninst\u271d\u00b2 : Module R' M\u2082''\ninst\u271d\u00b9 : Module R' N''\ninst\u271d : Module R' N\u2082''\nn : \u2115\ng : M\u2082'' \u2192\u2097[R'] M''\nf : AlternatingMap R' M'' N'' (Fin (Nat.succ n))\nm : M\u2082''\nv : Fin n \u2192 M\u2082''\n\u22a2 \u2191((fun x => g) 0) (Matrix.vecCons m v 0) = Matrix.vecCons (\u2191g m) (fun i => \u2191((fun x => g) i) (v i)) 0\n\ncase refine'_2\nR : Type u_1\ninst\u271d\u00b9\u2079 : Semiring R\nM : Type u_2\ninst\u271d\u00b9\u2078 : AddCommMonoid M\ninst\u271d\u00b9\u2077 : Module R M\nN : Type u_3\ninst\u271d\u00b9\u2076 : AddCommMonoid N\ninst\u271d\u00b9\u2075 : Module R N\nP : Type u_4\ninst\u271d\u00b9\u2074 : AddCommMonoid P\ninst\u271d\u00b9\u00b3 : Module R P\nM' : Type u_5\ninst\u271d\u00b9\u00b2 : AddCommGroup M'\ninst\u271d\u00b9\u00b9 : Module R M'\nN' : Type u_6\ninst\u271d\u00b9\u2070 : AddCommGroup N'\ninst\u271d\u2079 : Module R N'\n\u03b9 : Type u_7\n\u03b9' : Type u_8\n\u03b9'' : Type u_9\nR' : Type u_10\nM'' : Type u_11\nM\u2082'' : Type u_12\nN'' : Type u_13\nN\u2082'' : Type u_14\ninst\u271d\u2078 : CommSemiring R'\ninst\u271d\u2077 : AddCommMonoid M''\ninst\u271d\u2076 : AddCommMonoid M\u2082''\ninst\u271d\u2075 : AddCommMonoid N''\ninst\u271d\u2074 : AddCommMonoid N\u2082''\ninst\u271d\u00b3 : Module R' M''\ninst\u271d\u00b2 : Module R' M\u2082''\ninst\u271d\u00b9 : Module R' N''\ninst\u271d : Module R' N\u2082''\nn : \u2115\ng : M\u2082'' \u2192\u2097[R'] M''\nf : AlternatingMap R' M'' N'' (Fin (Nat.succ n))\nm : M\u2082''\nv : Fin n \u2192 M\u2082''\n\u22a2 \u2200 (i : Fin n),\n    \u2191((fun x => g) (Fin.succ i)) (Matrix.vecCons m v (Fin.succ i)) =\n      Matrix.vecCons (\u2191g m) (fun i => \u2191((fun x => g) i) (v i)) (Fin.succ i)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case refine'_1\nR : Type u_1\ninst\u271d\u00b9\u2079 : Semiring R\nM : Type u_2\ninst\u271d\u00b9\u2078 : AddCommMonoid M\ninst\u271d\u00b9\u2077 : Module R M\nN : Type u_3\ninst\u271d\u00b9\u2076 : AddCommMonoid N\ninst\u271d\u00b9\u2075 : Module R N\nP : Type u_4\ninst\u271d\u00b9\u2074 : AddCommMonoid P\ninst\u271d\u00b9\u00b3 : Module R P\nM' : Type u_5\ninst\u271d\u00b9\u00b2 : AddCommGroup M'\ninst\u271d\u00b9\u00b9 : Module R M'\nN' : Type u_6\ninst\u271d\u00b9\u2070 : AddCommGroup N'\ninst\u271d\u2079 : Module R N'\n\u03b9 : Type u_7\n\u03b9' : Type u_8\n\u03b9'' : Type u_9\nR' : Type u_10\nM'' : Type u_11\nM\u2082'' : Type u_12\nN'' : Type u_13\nN\u2082'' : Type u_14\ninst\u271d\u2078 : CommSemiring R'\ninst\u271d\u2077 : AddCommMonoid M''\ninst\u271d\u2076 : AddCommMonoid M\u2082''\ninst\u271d\u2075 : AddCommMonoid N''\ninst\u271d\u2074 : AddCommMonoid N\u2082''\ninst\u271d\u00b3 : Module R' M''\ninst\u271d\u00b2 : Module R' M\u2082''\ninst\u271d\u00b9 : Module R' N''\ninst\u271d : Module R' N\u2082''\nn : \u2115\ng : M\u2082'' \u2192\u2097[R'] M''\nf : AlternatingMap R' M'' N'' (Fin (Nat.succ n))\nm : M\u2082''\nv : Fin n \u2192 M\u2082''\n\u22a2 \u2191((fun x => g) 0) (Matrix.vecCons m v 0) = Matrix.vecCons (\u2191g m) (fun i => \u2191((fun x => g) i) (v i)) 0", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case refine'_2\nR : Type u_1\ninst\u271d\u00b9\u2079 : Semiring R\nM : Type u_2\ninst\u271d\u00b9\u2078 : AddCommMonoid M\ninst\u271d\u00b9\u2077 : Module R M\nN : Type u_3\ninst\u271d\u00b9\u2076 : AddCommMonoid N\ninst\u271d\u00b9\u2075 : Module R N\nP : Type u_4\ninst\u271d\u00b9\u2074 : AddCommMonoid P\ninst\u271d\u00b9\u00b3 : Module R P\nM' : Type u_5\ninst\u271d\u00b9\u00b2 : AddCommGroup M'\ninst\u271d\u00b9\u00b9 : Module R M'\nN' : Type u_6\ninst\u271d\u00b9\u2070 : AddCommGroup N'\ninst\u271d\u2079 : Module R N'\n\u03b9 : Type u_7\n\u03b9' : Type u_8\n\u03b9'' : Type u_9\nR' : Type u_10\nM'' : Type u_11\nM\u2082'' : Type u_12\nN'' : Type u_13\nN\u2082'' : Type u_14\ninst\u271d\u2078 : CommSemiring R'\ninst\u271d\u2077 : AddCommMonoid M''\ninst\u271d\u2076 : AddCommMonoid M\u2082''\ninst\u271d\u2075 : AddCommMonoid N''\ninst\u271d\u2074 : AddCommMonoid N\u2082''\ninst\u271d\u00b3 : Module R' M''\ninst\u271d\u00b2 : Module R' M\u2082''\ninst\u271d\u00b9 : Module R' N''\ninst\u271d : Module R' N\u2082''\nn : \u2115\ng : M\u2082'' \u2192\u2097[R'] M''\nf : AlternatingMap R' M'' N'' (Fin (Nat.succ n))\nm : M\u2082''\nv : Fin n \u2192 M\u2082''\n\u22a2 \u2200 (i : Fin n),\n    \u2191((fun x => g) (Fin.succ i)) (Matrix.vecCons m v (Fin.succ i)) =\n      Matrix.vecCons (\u2191g m) (fun i => \u2191((fun x => g) i) (v i)) (Fin.succ i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Module.lean", "full_name": "smul_lt_smul_of_neg", "start": [53, 1], "end": [55, 50], "traced_tactics": [{"tactic": "rw [\u2190 neg_neg c, neg_smul, neg_smul (-c), neg_lt_neg_iff]", "annotated_tactic": ["rw [\u2190 <a>neg_neg</a> c, <a>neg_smul</a>, <a>neg_smul</a> (-c), <a>neg_lt_neg_iff</a>]", [{"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [799, 3], "def_end_pos": [799, 14]}, {"full_name": "neg_smul", "def_path": "Mathlib/Algebra/Module/Basic.lean", "def_pos": [278, 9], "def_end_pos": [278, 17]}, {"full_name": "neg_smul", "def_path": "Mathlib/Algebra/Module/Basic.lean", "def_pos": [278, 9], "def_end_pos": [278, 17]}, {"full_name": "neg_lt_neg_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [381, 3], "def_end_pos": [381, 14]}]], "state_before": "k : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nh : a < b\nhc : c < 0\n\u22a2 c \u2022 b < c \u2022 a", "state_after": "k : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nh : a < b\nhc : c < 0\n\u22a2 -c \u2022 a < -c \u2022 b"}, {"tactic": "exact smul_lt_smul_of_pos h (neg_pos_of_neg hc)", "annotated_tactic": ["exact <a>smul_lt_smul_of_pos</a> h (<a>neg_pos_of_neg</a> hc)", [{"full_name": "smul_lt_smul_of_pos", "def_path": "Mathlib/Algebra/Order/SMul.lean", "def_pos": [90, 19], "def_end_pos": [90, 38]}, {"full_name": "neg_pos_of_neg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [613, 24], "def_end_pos": [613, 38]}]], "state_before": "k : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nh : a < b\nhc : c < 0\n\u22a2 -c \u2022 a < -c \u2022 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/MeanValue.lean", "full_name": "convexOn_of_deriv2_nonneg", "start": [1183, 1], "end": [1188, 14], "traced_tactics": [{"tactic": "rwa [interior_interior]", "annotated_tactic": ["rwa [<a>interior_interior</a>]", [{"full_name": "interior_interior", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [355, 9], "def_end_pos": [355, 26]}]], "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nhf'' : DifferentiableOn \u211d (deriv f) (interior D)\nhf''_nonneg : \u2200 (x : \u211d), x \u2208 interior D \u2192 0 \u2264 deriv^[2] f x\n\u22a2 DifferentiableOn \u211d (deriv f) (interior (interior D))", "state_after": "no goals"}, {"tactic": "rwa [interior_interior]", "annotated_tactic": ["rwa [<a>interior_interior</a>]", [{"full_name": "interior_interior", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [355, 9], "def_end_pos": [355, 26]}]], "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nhf'' : DifferentiableOn \u211d (deriv f) (interior D)\nhf''_nonneg : \u2200 (x : \u211d), x \u2208 interior D \u2192 0 \u2264 deriv^[2] f x\n\u22a2 \u2200 (x : \u211d), x \u2208 interior (interior D) \u2192 0 \u2264 deriv (deriv f) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Order.lean", "full_name": "nhds_induced", "start": [863, 1], "end": [866, 35], "traced_tactics": [{"tactic": "ext s", "annotated_tactic": ["ext s", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf\u271d : \u03b1 \u2192 \u03b2\n\u03b9 : Sort u_2\nT : TopologicalSpace \u03b1\nf : \u03b2 \u2192 \u03b1\na : \u03b2\n\u22a2 \ud835\udcdd a = comap f (\ud835\udcdd (f a))", "state_after": "case a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf\u271d : \u03b1 \u2192 \u03b2\n\u03b9 : Sort u_2\nT : TopologicalSpace \u03b1\nf : \u03b2 \u2192 \u03b1\na : \u03b2\ns : Set \u03b2\n\u22a2 s \u2208 \ud835\udcdd a \u2194 s \u2208 comap f (\ud835\udcdd (f a))"}, {"tactic": "rw [mem_nhds_induced, mem_comap]", "annotated_tactic": ["rw [<a>mem_nhds_induced</a>, <a>mem_comap</a>]", [{"full_name": "mem_nhds_induced", "def_path": "Mathlib/Topology/Order.lean", "def_pos": [852, 9], "def_end_pos": [852, 25]}, {"full_name": "Filter.mem_comap", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2128, 17], "def_end_pos": [2128, 26]}]], "state_before": "case a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf\u271d : \u03b1 \u2192 \u03b2\n\u03b9 : Sort u_2\nT : TopologicalSpace \u03b1\nf : \u03b2 \u2192 \u03b1\na : \u03b2\ns : Set \u03b2\n\u22a2 s \u2208 \ud835\udcdd a \u2194 s \u2208 comap f (\ud835\udcdd (f a))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/SuccPred/LinearLocallyFinite.lean", "full_name": "LinearLocallyFiniteOrder.le_succFn", "start": [73, 1], "end": [75, 31], "traced_tactics": [{"tactic": "rw [le_isGLB_iff (succFn_spec i), mem_lowerBounds]", "annotated_tactic": ["rw [<a>le_isGLB_iff</a> (<a>succFn_spec</a> i), <a>mem_lowerBounds</a>]", [{"full_name": "le_isGLB_iff", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [314, 9], "def_end_pos": [314, 21]}, {"full_name": "LinearLocallyFiniteOrder.succFn_spec", "def_path": "Mathlib/Order/SuccPred/LinearLocallyFinite.lean", "def_pos": [69, 9], "def_end_pos": [69, 20]}, {"full_name": "mem_lowerBounds", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 24]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : LinearOrder \u03b9\ni : \u03b9\n\u22a2 i \u2264 succFn i", "state_after": "\u03b9 : Type u_1\ninst\u271d : LinearOrder \u03b9\ni : \u03b9\n\u22a2 \u2200 (x : \u03b9), x \u2208 Set.Ioi i \u2192 i \u2264 x"}, {"tactic": "exact fun x hx \u21a6 le_of_lt hx", "annotated_tactic": ["exact fun x hx \u21a6 <a>le_of_lt</a> hx", [{"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : LinearOrder \u03b9\ni : \u03b9\n\u22a2 \u2200 (x : \u03b9), x \u2208 Set.Ioi i \u2192 i \u2264 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Disjoint.lean", "full_name": "Complementeds.coe_bot", "start": [734, 1], "end": [734, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "full_name": "AEMeasurable.div_const", "start": [320, 1], "end": [322, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Dynamics/Ergodic/Ergodic.lean", "full_name": "MeasureTheory.MeasurePreserving.preErgodic_of_preErgodic_conjugate", "start": [89, 1], "end": [96, 90], "traced_tactics": [{"tactic": "intro s hs\u2080 hs\u2081", "annotated_tactic": ["intro s hs\u2080 hs\u2081", []], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g\nhf : PreErgodic f\nf' : \u03b2 \u2192 \u03b2\nh_comm : g \u2218 f = f' \u2218 g\n\u22a2 \u2200 \u2983s : Set \u03b2\u2984, MeasurableSet s \u2192 f' \u207b\u00b9' s = s \u2192 s =\u1da0[ae \u03bc'] \u2205 \u2228 s =\u1da0[ae \u03bc'] univ", "state_after": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g\nhf : PreErgodic f\nf' : \u03b2 \u2192 \u03b2\nh_comm : g \u2218 f = f' \u2218 g\ns : Set \u03b2\nhs\u2080 : MeasurableSet s\nhs\u2081 : f' \u207b\u00b9' s = s\n\u22a2 s =\u1da0[ae \u03bc'] \u2205 \u2228 s =\u1da0[ae \u03bc'] univ"}, {"tactic": "replace hs\u2081 : f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s", "annotated_tactic": ["replace hs\u2081 : f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s", []], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g\nhf : PreErgodic f\nf' : \u03b2 \u2192 \u03b2\nh_comm : g \u2218 f = f' \u2218 g\ns : Set \u03b2\nhs\u2080 : MeasurableSet s\nhs\u2081 : f' \u207b\u00b9' s = s\n\u22a2 s =\u1da0[ae \u03bc'] \u2205 \u2228 s =\u1da0[ae \u03bc'] univ", "state_after": "case hs\u2081\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g\nhf : PreErgodic f\nf' : \u03b2 \u2192 \u03b2\nh_comm : g \u2218 f = f' \u2218 g\ns : Set \u03b2\nhs\u2080 : MeasurableSet s\nhs\u2081 : f' \u207b\u00b9' s = s\n\u22a2 f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s\n\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g\nhf : PreErgodic f\nf' : \u03b2 \u2192 \u03b2\nh_comm : g \u2218 f = f' \u2218 g\ns : Set \u03b2\nhs\u2080 : MeasurableSet s\nhs\u2081 : f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s\n\u22a2 s =\u1da0[ae \u03bc'] \u2205 \u2228 s =\u1da0[ae \u03bc'] univ"}, {"tactic": "cases' hf.ae_empty_or_univ (hg.measurable hs\u2080) hs\u2081 with hs\u2082 hs\u2082 <;> [left; right]", "annotated_tactic": ["cases' hf.ae_empty_or_univ (hg.measurable hs\u2080) hs\u2081 with hs\u2082 hs\u2082 <;> [left; right]", []], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g\nhf : PreErgodic f\nf' : \u03b2 \u2192 \u03b2\nh_comm : g \u2218 f = f' \u2218 g\ns : Set \u03b2\nhs\u2080 : MeasurableSet s\nhs\u2081 : f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s\n\u22a2 s =\u1da0[ae \u03bc'] \u2205 \u2228 s =\u1da0[ae \u03bc'] univ", "state_after": "case inl.h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g\nhf : PreErgodic f\nf' : \u03b2 \u2192 \u03b2\nh_comm : g \u2218 f = f' \u2218 g\ns : Set \u03b2\nhs\u2080 : MeasurableSet s\nhs\u2081 : f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s\nhs\u2082 : g \u207b\u00b9' s =\u1da0[ae \u03bc] \u2205\n\u22a2 s =\u1da0[ae \u03bc'] \u2205\n\ncase inr.h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g\nhf : PreErgodic f\nf' : \u03b2 \u2192 \u03b2\nh_comm : g \u2218 f = f' \u2218 g\ns : Set \u03b2\nhs\u2080 : MeasurableSet s\nhs\u2081 : f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s\nhs\u2082 : g \u207b\u00b9' s =\u1da0[ae \u03bc] univ\n\u22a2 s =\u1da0[ae \u03bc'] univ"}, {"tactic": "rw [\u2190 preimage_comp, h_comm, preimage_comp, hs\u2081]", "annotated_tactic": ["rw [\u2190 <a>preimage_comp</a>, h_comm, <a>preimage_comp</a>, hs\u2081]", [{"full_name": "Set.preimage_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [163, 9], "def_end_pos": [163, 22]}, {"full_name": "Set.preimage_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [163, 9], "def_end_pos": [163, 22]}]], "state_before": "case hs\u2081\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g\nhf : PreErgodic f\nf' : \u03b2 \u2192 \u03b2\nh_comm : g \u2218 f = f' \u2218 g\ns : Set \u03b2\nhs\u2080 : MeasurableSet s\nhs\u2081 : f' \u207b\u00b9' s = s\n\u22a2 f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s", "state_after": "no goals"}, {"tactic": "simpa only [ae_eq_empty, hg.measure_preimage hs\u2080] using hs\u2082", "annotated_tactic": ["simpa only [<a>ae_eq_empty</a>, hg.measure_preimage hs\u2080] using hs\u2082", [{"full_name": "MeasureTheory.ae_eq_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [456, 9], "def_end_pos": [456, 20]}]], "state_before": "case inl.h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g\nhf : PreErgodic f\nf' : \u03b2 \u2192 \u03b2\nh_comm : g \u2218 f = f' \u2218 g\ns : Set \u03b2\nhs\u2080 : MeasurableSet s\nhs\u2081 : f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s\nhs\u2082 : g \u207b\u00b9' s =\u1da0[ae \u03bc] \u2205\n\u22a2 s =\u1da0[ae \u03bc'] \u2205", "state_after": "no goals"}, {"tactic": "simpa only [ae_eq_univ, \u2190 preimage_compl, hg.measure_preimage hs\u2080.compl] using hs\u2082", "annotated_tactic": ["simpa only [<a>ae_eq_univ</a>, \u2190 <a>preimage_compl</a>, hg.measure_preimage hs\u2080.compl] using hs\u2082", [{"full_name": "MeasureTheory.ae_eq_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [462, 9], "def_end_pos": [462, 19]}, {"full_name": "Set.preimage_compl", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [97, 9], "def_end_pos": [97, 23]}]], "state_before": "case inr.h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns\u271d : Set \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\nm' : MeasurableSpace \u03b2\n\u03bc' : Measure \u03b2\ns' : Set \u03b2\ng : \u03b1 \u2192 \u03b2\nhg : MeasurePreserving g\nhf : PreErgodic f\nf' : \u03b2 \u2192 \u03b2\nh_comm : g \u2218 f = f' \u2218 g\ns : Set \u03b2\nhs\u2080 : MeasurableSet s\nhs\u2081 : f \u207b\u00b9' (g \u207b\u00b9' s) = g \u207b\u00b9' s\nhs\u2082 : g \u207b\u00b9' s =\u1da0[ae \u03bc] univ\n\u22a2 s =\u1da0[ae \u03bc'] univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/ModelTheory/Basic.lean", "full_name": "FirstOrder.Language.Equiv.coe_toEmbedding", "start": [826, 1], "end": [827, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean", "full_name": "Matrix.isUnit_of_left_inverse", "start": [164, 1], "end": [165, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Localization/Away/Basic.lean", "full_name": "selfZpow_of_nonneg", "start": [213, 1], "end": [214, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/AList.lean", "full_name": "AList.keys_erase", "start": [236, 1], "end": [237, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Functor/Const.lean", "full_name": "CategoryTheory.Functor.const.unop_functor_op_obj_map", "start": [82, 1], "end": [84, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean", "full_name": "UniformFun.hasBasis_nhds", "start": [334, 11], "end": [336, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "full_name": "UniqueFactorizationMonoid.normalizedFactors_mul", "start": [659, 1], "end": [678, 58], "traced_tactics": [{"tactic": "have h : (normalize : \u03b1 \u2192 \u03b1) = Associates.out \u2218 Associates.mk := by\n  ext\n  rw [Function.comp_apply, Associates.out_mk]", "annotated_tactic": ["have h : (<a>normalize</a> : \u03b1 \u2192 \u03b1) = <a>Associates.out</a> \u2218 <a>Associates.mk</a> := by\n    ext\n    rw [<a>Function.comp_apply</a>, <a>Associates.out_mk</a>]", [{"full_name": "normalize", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [100, 5], "def_end_pos": [100, 14]}, {"full_name": "Associates.out", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [214, 15], "def_end_pos": [214, 18]}, {"full_name": "Associates.mk", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [762, 18], "def_end_pos": [762, 20]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "Associates.out_mk", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [220, 9], "def_end_pos": [220, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 normalizedFactors (x * y) = normalizedFactors x + normalizedFactors y", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 normalizedFactors (x * y) = normalizedFactors x + normalizedFactors y"}, {"tactic": "rw [\u2190 Multiset.map_id' (normalizedFactors (x * y)), \u2190 Multiset.map_id' (normalizedFactors x), \u2190\n  Multiset.map_id' (normalizedFactors y), \u2190 Multiset.map_congr rfl normalize_normalized_factor, \u2190\n  Multiset.map_congr rfl normalize_normalized_factor, \u2190\n  Multiset.map_congr rfl normalize_normalized_factor, \u2190 Multiset.map_add, h, \u2190\n  Multiset.map_map Associates.out, eq_comm, \u2190 Multiset.map_map Associates.out]", "annotated_tactic": ["rw [\u2190 <a>Multiset.map_id'</a> (<a>normalizedFactors</a> (x * y)), \u2190 <a>Multiset.map_id'</a> (<a>normalizedFactors</a> x), \u2190\n    <a>Multiset.map_id'</a> (<a>normalizedFactors</a> y), \u2190 <a>Multiset.map_congr</a> <a>rfl</a> <a>normalize_normalized_factor</a>, \u2190\n    <a>Multiset.map_congr</a> <a>rfl</a> <a>normalize_normalized_factor</a>, \u2190\n    <a>Multiset.map_congr</a> <a>rfl</a> <a>normalize_normalized_factor</a>, \u2190 <a>Multiset.map_add</a>, h, \u2190\n    <a>Multiset.map_map</a> <a>Associates.out</a>, <a>eq_comm</a>, \u2190 <a>Multiset.map_map</a> <a>Associates.out</a>]", [{"full_name": "Multiset.map_id'", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1292, 9], "def_end_pos": [1292, 16]}, {"full_name": "UniqueFactorizationMonoid.normalizedFactors", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [546, 19], "def_end_pos": [546, 36]}, {"full_name": "Multiset.map_id'", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1292, 9], "def_end_pos": [1292, 16]}, {"full_name": "UniqueFactorizationMonoid.normalizedFactors", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [546, 19], "def_end_pos": [546, 36]}, {"full_name": "Multiset.map_id'", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1292, 9], "def_end_pos": [1292, 16]}, {"full_name": "UniqueFactorizationMonoid.normalizedFactors", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [546, 19], "def_end_pos": [546, 36]}, {"full_name": "Multiset.map_congr", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1147, 9], "def_end_pos": [1147, 18]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "UniqueFactorizationMonoid.normalize_normalized_factor", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [585, 9], "def_end_pos": [585, 36]}, {"full_name": "Multiset.map_congr", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1147, 9], "def_end_pos": [1147, 18]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "UniqueFactorizationMonoid.normalize_normalized_factor", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [585, 9], "def_end_pos": [585, 36]}, {"full_name": "Multiset.map_congr", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1147, 9], "def_end_pos": [1147, 18]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "UniqueFactorizationMonoid.normalize_normalized_factor", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [585, 9], "def_end_pos": [585, 36]}, {"full_name": "Multiset.map_add", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1196, 9], "def_end_pos": [1196, 16]}, {"full_name": "Multiset.map_map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1283, 9], "def_end_pos": [1283, 16]}, {"full_name": "Associates.out", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [214, 15], "def_end_pos": [214, 18]}, {"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}, {"full_name": "Multiset.map_map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1283, 9], "def_end_pos": [1283, 16]}, {"full_name": "Associates.out", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [214, 15], "def_end_pos": [214, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 normalizedFactors (x * y) = normalizedFactors x + normalizedFactors y", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 Multiset.map Associates.out (Multiset.map Associates.mk (normalizedFactors x + normalizedFactors y)) =\n    Multiset.map Associates.out (Multiset.map Associates.mk (normalizedFactors (x * y)))"}, {"tactic": "refine' congr rfl _", "annotated_tactic": ["refine' <a>congr</a> <a>rfl</a> _", [{"full_name": "congr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [359, 9], "def_end_pos": [359, 14]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 Multiset.map Associates.out (Multiset.map Associates.mk (normalizedFactors x + normalizedFactors y)) =\n    Multiset.map Associates.out (Multiset.map Associates.mk (normalizedFactors (x * y)))", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 Multiset.map Associates.mk (normalizedFactors x + normalizedFactors y) =\n    Multiset.map Associates.mk (normalizedFactors (x * y))"}, {"tactic": "apply Multiset.map_mk_eq_map_mk_of_rel", "annotated_tactic": ["apply <a>Multiset.map_mk_eq_map_mk_of_rel</a>", [{"full_name": "Multiset.map_mk_eq_map_mk_of_rel", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2870, 9], "def_end_pos": [2870, 32]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 Multiset.map Associates.mk (normalizedFactors x + normalizedFactors y) =\n    Multiset.map Associates.mk (normalizedFactors (x * y))", "state_after": "case hst\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 Multiset.Rel Setoid.r (normalizedFactors x + normalizedFactors y) (normalizedFactors (x * y))"}, {"tactic": "apply factors_unique", "annotated_tactic": ["apply <a>factors_unique</a>", [{"full_name": "UniqueFactorizationMonoid.factors_unique", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [240, 9], "def_end_pos": [240, 23]}]], "state_before": "case hst\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 Multiset.Rel Setoid.r (normalizedFactors x + normalizedFactors y) (normalizedFactors (x * y))", "state_after": "case hst.hf\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 \u2200 (x_1 : \u03b1), x_1 \u2208 normalizedFactors x + normalizedFactors y \u2192 Irreducible x_1\n\ncase hst.hg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 \u2200 (x_1 : \u03b1), x_1 \u2208 normalizedFactors (x * y) \u2192 Irreducible x_1\n\ncase hst.h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 Multiset.prod (normalizedFactors x + normalizedFactors y) ~\u1d64 Multiset.prod (normalizedFactors (x * y))"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 \u2191normalize = Associates.out \u2218 Associates.mk", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nx\u271d : \u03b1\n\u22a2 \u2191normalize x\u271d = (Associates.out \u2218 Associates.mk) x\u271d"}, {"tactic": "rw [Function.comp_apply, Associates.out_mk]", "annotated_tactic": ["rw [<a>Function.comp_apply</a>, <a>Associates.out_mk</a>]", [{"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "Associates.out_mk", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [220, 9], "def_end_pos": [220, 15]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nx\u271d : \u03b1\n\u22a2 \u2191normalize x\u271d = (Associates.out \u2218 Associates.mk) x\u271d", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case hst.hf\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 \u2200 (x_1 : \u03b1), x_1 \u2208 normalizedFactors x + normalizedFactors y \u2192 Irreducible x_1", "state_after": "case hst.hf\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx\u271d y : \u03b1\nhx\u271d : x\u271d \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\nx : \u03b1\nhx : x \u2208 normalizedFactors x\u271d + normalizedFactors y\n\u22a2 Irreducible x"}, {"tactic": "rcases Multiset.mem_add.1 hx with (hx | hx) <;> exact irreducible_of_normalized_factor x hx", "annotated_tactic": ["rcases <a>Multiset.mem_add</a>.1 hx with (hx | hx) <;> exact <a>irreducible_of_normalized_factor</a> x hx", [{"full_name": "Multiset.mem_add", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [683, 9], "def_end_pos": [683, 16]}, {"full_name": "UniqueFactorizationMonoid.irreducible_of_normalized_factor", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [580, 9], "def_end_pos": [580, 41]}]], "state_before": "case hst.hf\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx\u271d y : \u03b1\nhx\u271d : x\u271d \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\nx : \u03b1\nhx : x \u2208 normalizedFactors x\u271d + normalizedFactors y\n\u22a2 Irreducible x", "state_after": "no goals"}, {"tactic": "exact irreducible_of_normalized_factor", "annotated_tactic": ["exact <a>irreducible_of_normalized_factor</a>", [{"full_name": "UniqueFactorizationMonoid.irreducible_of_normalized_factor", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [580, 9], "def_end_pos": [580, 41]}]], "state_before": "case hst.hg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 \u2200 (x_1 : \u03b1), x_1 \u2208 normalizedFactors (x * y) \u2192 Irreducible x_1", "state_after": "no goals"}, {"tactic": "rw [Multiset.prod_add]", "annotated_tactic": ["rw [<a>Multiset.prod_add</a>]", [{"full_name": "Multiset.prod_add", "def_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "def_pos": [115, 9], "def_end_pos": [115, 17]}]], "state_before": "case hst.h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 Multiset.prod (normalizedFactors x + normalizedFactors y) ~\u1d64 Multiset.prod (normalizedFactors (x * y))", "state_after": "case hst.h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 Multiset.prod (normalizedFactors x) * Multiset.prod (normalizedFactors y) ~\u1d64 Multiset.prod (normalizedFactors (x * y))"}, {"tactic": "exact\n  ((normalizedFactors_prod hx).mul_mul (normalizedFactors_prod hy)).trans\n    (normalizedFactors_prod (mul_ne_zero hx hy)).symm", "annotated_tactic": ["exact\n      ((<a>normalizedFactors_prod</a> hx).<a>mul_mul</a> (<a>normalizedFactors_prod</a> hy)).<a>trans</a>\n        (<a>normalizedFactors_prod</a> (<a>mul_ne_zero</a> hx hy)).<a>symm</a>", [{"full_name": "UniqueFactorizationMonoid.normalizedFactors_prod", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [561, 9], "def_end_pos": [561, 31]}, {"full_name": "Associated.mul_mul", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [523, 9], "def_end_pos": [523, 27]}, {"full_name": "UniqueFactorizationMonoid.normalizedFactors_prod", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [561, 9], "def_end_pos": [561, 31]}, {"full_name": "Associated.trans", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [410, 19], "def_end_pos": [410, 24]}, {"full_name": "UniqueFactorizationMonoid.normalizedFactors_prod", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [561, 9], "def_end_pos": [561, 31]}, {"full_name": "mul_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [88, 9], "def_end_pos": [88, 20]}, {"full_name": "Associated.symm", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [398, 19], "def_end_pos": [398, 23]}]], "state_before": "case hst.h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : NormalizationMonoid \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\nx y : \u03b1\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2191normalize = Associates.out \u2218 Associates.mk\n\u22a2 Multiset.prod (normalizedFactors x) * Multiset.prod (normalizedFactors y) ~\u1d64 Multiset.prod (normalizedFactors (x * y))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Homotopy/Equiv.lean", "full_name": "ContinuousMap.HomotopyEquiv.continuous", "start": [68, 1], "end": [69, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Ordinal/Notation.lean", "full_name": "ONote.opow_def", "start": [705, 1], "end": [706, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Algebra/Equiv.lean", "full_name": "AlgEquiv.coe_algHom", "start": [267, 1], "end": [268, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "full_name": "MeasureTheory.aemeasurable_withDensity_ennreal_iff", "start": [238, 1], "end": [261, 83], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 AEMeasurable g \u2194 AEMeasurable fun x => \u2191(f x) * g x", "state_after": "case mp\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 AEMeasurable g \u2192 AEMeasurable fun x => \u2191(f x) * g x\n\ncase mpr\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 (AEMeasurable fun x => \u2191(f x) * g x) \u2192 AEMeasurable g"}, {"tactic": "rintro \u27e8g', g'meas, hg'\u27e9", "annotated_tactic": ["rintro \u27e8g', g'meas, hg'\u27e9", []], "state_before": "case mp\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 AEMeasurable g \u2192 AEMeasurable fun x => \u2191(f x) * g x", "state_after": "case mp.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\n\u22a2 AEMeasurable fun x => \u2191(f x) * g x"}, {"tactic": "have A : MeasurableSet { x : \u03b1 | f x \u2260 0 } := (hf (measurableSet_singleton 0)).compl", "annotated_tactic": ["have A : <a>MeasurableSet</a> { x : \u03b1 | f x \u2260 0 } := (hf (<a>measurableSet_singleton</a> 0)).<a>compl</a>", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasurableSingletonClass.measurableSet_singleton", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [269, 3], "def_end_pos": [269, 26]}, {"full_name": "MeasurableSet.compl", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [87, 19], "def_end_pos": [87, 38]}]], "state_before": "case mp.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\n\u22a2 AEMeasurable fun x => \u2191(f x) * g x", "state_after": "case mp.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 AEMeasurable fun x => \u2191(f x) * g x"}, {"tactic": "refine' \u27e8fun x => f x * g' x, hf.rst.ime_nnreal_ennreal.smul g'meas, _\u27e9", "annotated_tactic": ["refine' \u27e8fun x => f x * g' x, hf.rst.ime_nnreal_ennreal.smul g'meas, _\u27e9", []], "state_before": "case mp.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 AEMeasurable fun x => \u2191(f x) * g x", "state_after": "case mp.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] fun x => \u2191(f x) * g' x"}, {"tactic": "apply ae_of_ae_restrict_of_ae_restrict_compl { x | f x \u2260 0 }", "annotated_tactic": ["apply <a>ae_of_ae_restrict_of_ae_restrict_compl</a> { x | f x \u2260 0 }", [{"full_name": "MeasureTheory.ae_of_ae_restrict_of_ae_restrict_compl", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2611, 9], "def_end_pos": [2611, 47]}]], "state_before": "case mp.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] fun x => \u2191(f x) * g' x", "state_after": "case mp.intro.intro.ht\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202restrict \u03bc {x | f x \u2260 0}, (fun x => \u2191(f x) * g x) x = (fun x => \u2191(f x) * g' x) x\n\ncase mp.intro.intro.htc\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202restrict \u03bc {x | f x \u2260 0}\u1d9c, (fun x => \u2191(f x) * g x) x = (fun x => \u2191(f x) * g' x) x"}, {"tactic": "rw [EventuallyEq, ae_withDensity_iff hf.rst.ime_nnreal_ennreal] at hg'", "annotated_tactic": ["rw [<a>EventuallyEq</a>, <a>ae_withDensity_iff</a> hf.rst.ime_nnreal_ennreal] at hg'", [{"full_name": "Filter.EventuallyEq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1438, 5], "def_end_pos": [1438, 17]}, {"full_name": "MeasureTheory.ae_withDensity_iff", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [223, 9], "def_end_pos": [223, 27]}]], "state_before": "case mp.intro.intro.ht\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202restrict \u03bc {x | f x \u2260 0}, (fun x => \u2191(f x) * g x) x = (fun x => \u2191(f x) * g' x) x", "state_after": "case mp.intro.intro.ht\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202restrict \u03bc {x | f x \u2260 0}, (fun x => \u2191(f x) * g x) x = (fun x => \u2191(f x) * g' x) x"}, {"tactic": "rw [ae_restrict_iff' A]", "annotated_tactic": ["rw [<a>ae_restrict_iff'</a> A]", [{"full_name": "MeasureTheory.ae_restrict_iff'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2572, 9], "def_end_pos": [2572, 25]}]], "state_before": "case mp.intro.intro.ht\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202restrict \u03bc {x | f x \u2260 0}, (fun x => \u2191(f x) * g x) x = (fun x => \u2191(f x) * g' x) x", "state_after": "case mp.intro.intro.ht\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 {x | f x \u2260 0} \u2192 (fun x => \u2191(f x) * g x) x = (fun x => \u2191(f x) * g' x) x"}, {"tactic": "filter_upwards [hg']", "annotated_tactic": ["filter_upwards [hg']", []], "state_before": "case mp.intro.intro.ht\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 {x | f x \u2260 0} \u2192 (fun x => \u2191(f x) * g x) x = (fun x => \u2191(f x) * g' x) x", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200 (a : \u03b1), (\u2191(f a) \u2260 0 \u2192 g a = g' a) \u2192 f a \u2260 0 \u2192 \u2191(f a) * g a = \u2191(f a) * g' a"}, {"tactic": "intro a ha h'a", "annotated_tactic": ["intro a ha h'a", []], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200 (a : \u03b1), (\u2191(f a) \u2260 0 \u2192 g a = g' a) \u2192 f a \u2260 0 \u2192 \u2191(f a) * g a = \u2191(f a) * g' a", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\na : \u03b1\nha : \u2191(f a) \u2260 0 \u2192 g a = g' a\nh'a : f a \u2260 0\n\u22a2 \u2191(f a) * g a = \u2191(f a) * g' a"}, {"tactic": "have : (f a : \u211d\u22650\u221e) \u2260 0 := by simpa only [Ne.def, coe_eq_zero] using h'a", "annotated_tactic": ["have : (f a : \u211d\u22650\u221e) \u2260 0 := by simpa only [<a>Ne.def</a>, <a>coe_eq_zero</a>] using h'a", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "ENNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [368, 28], "def_end_pos": [368, 39]}]], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\na : \u03b1\nha : \u2191(f a) \u2260 0 \u2192 g a = g' a\nh'a : f a \u2260 0\n\u22a2 \u2191(f a) * g a = \u2191(f a) * g' a", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\na : \u03b1\nha : \u2191(f a) \u2260 0 \u2192 g a = g' a\nh'a : f a \u2260 0\nthis : \u2191(f a) \u2260 0\n\u22a2 \u2191(f a) * g a = \u2191(f a) * g' a"}, {"tactic": "rw [ha this]", "annotated_tactic": ["rw [ha this]", []], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\na : \u03b1\nha : \u2191(f a) \u2260 0 \u2192 g a = g' a\nh'a : f a \u2260 0\nthis : \u2191(f a) \u2260 0\n\u22a2 \u2191(f a) * g a = \u2191(f a) * g' a", "state_after": "no goals"}, {"tactic": "simpa only [Ne.def, coe_eq_zero] using h'a", "annotated_tactic": ["simpa only [<a>Ne.def</a>, <a>coe_eq_zero</a>] using h'a", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "ENNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [368, 28], "def_end_pos": [368, 39]}]], "state_before": "\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = g' x\nA : MeasurableSet {x | f x \u2260 0}\na : \u03b1\nha : \u2191(f a) \u2260 0 \u2192 g a = g' a\nh'a : f a \u2260 0\n\u22a2 \u2191(f a) \u2260 0", "state_after": "no goals"}, {"tactic": "filter_upwards [ae_restrict_mem A.compl]", "annotated_tactic": ["filter_upwards [<a>ae_restrict_mem</a> A.compl]", [{"full_name": "MeasureTheory.ae_restrict_mem", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2586, 9], "def_end_pos": [2586, 24]}]], "state_before": "case mp.intro.intro.htc\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202restrict \u03bc {x | f x \u2260 0}\u1d9c, (fun x => \u2191(f x) * g x) x = (fun x => \u2191(f x) * g' x) x", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200 (a : \u03b1), a \u2208 {x | f x \u2260 0}\u1d9c \u2192 \u2191(f a) * g a = \u2191(f a) * g' a"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\n\u22a2 \u2200 (a : \u03b1), a \u2208 {x | f x \u2260 0}\u1d9c \u2192 \u2191(f a) * g a = \u2191(f a) * g' a", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\nx : \u03b1\nhx : x \u2208 {x | f x \u2260 0}\u1d9c\n\u22a2 \u2191(f x) * g x = \u2191(f x) * g' x"}, {"tactic": "simp only [Classical.not_not, mem_setOf_eq, mem_compl_iff] at hx", "annotated_tactic": ["simp only [<a>Classical.not_not</a>, <a>mem_setOf_eq</a>, <a>mem_compl_iff</a>] at hx", [{"full_name": "Classical.not_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [683, 24], "def_end_pos": [683, 31]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "Set.mem_compl_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1658, 9], "def_end_pos": [1658, 22]}]], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\nx : \u03b1\nhx : x \u2208 {x | f x \u2260 0}\u1d9c\n\u22a2 \u2191(f x) * g x = \u2191(f x) * g' x", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\nx : \u03b1\nhx : f x = 0\n\u22a2 \u2191(f x) * g x = \u2191(f x) * g' x"}, {"tactic": "simp [hx]", "annotated_tactic": ["simp [hx]", []], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] g'\nA : MeasurableSet {x | f x \u2260 0}\nx : \u03b1\nhx : f x = 0\n\u22a2 \u2191(f x) * g x = \u2191(f x) * g' x", "state_after": "no goals"}, {"tactic": "rintro \u27e8g', g'meas, hg'\u27e9", "annotated_tactic": ["rintro \u27e8g', g'meas, hg'\u27e9", []], "state_before": "case mpr\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 (AEMeasurable fun x => \u2191(f x) * g x) \u2192 AEMeasurable g", "state_after": "case mpr.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] g'\n\u22a2 AEMeasurable g"}, {"tactic": "refine' \u27e8fun x => ((f x)\u207b\u00b9 : \u211d\u22650\u221e) * g' x, hf.rst.ime_nnreal_ennreal.inv.smul g'meas, _\u27e9", "annotated_tactic": ["refine' \u27e8fun x => ((f x)\u207b\u00b9 : \u211d\u22650\u221e) * g' x, hf.rst.ime_nnreal_ennreal.inv.smul g'meas, _\u27e9", []], "state_before": "case mpr.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] g'\n\u22a2 AEMeasurable g", "state_after": "case mpr.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] g'\n\u22a2 g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] fun x => (\u2191(f x))\u207b\u00b9 * g' x"}, {"tactic": "rw [EventuallyEq, ae_withDensity_iff hf.rst.ime_nnreal_ennreal]", "annotated_tactic": ["rw [<a>EventuallyEq</a>, <a>ae_withDensity_iff</a> hf.rst.ime_nnreal_ennreal]", [{"full_name": "Filter.EventuallyEq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1438, 5], "def_end_pos": [1438, 17]}, {"full_name": "MeasureTheory.ae_withDensity_iff", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [223, 9], "def_end_pos": [223, 27]}]], "state_before": "case mpr.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] g'\n\u22a2 g =\u1da0[ae (withDensity \u03bc fun x => \u2191(f x))] fun x => (\u2191(f x))\u207b\u00b9 * g' x", "state_after": "case mpr.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] g'\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = (\u2191(f x))\u207b\u00b9 * g' x"}, {"tactic": "filter_upwards [hg']", "annotated_tactic": ["filter_upwards [hg']", []], "state_before": "case mpr.intro.intro\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] g'\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191(f x) \u2260 0 \u2192 g x = (\u2191(f x))\u207b\u00b9 * g' x", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] g'\n\u22a2 \u2200 (a : \u03b1), \u2191(f a) * g a = g' a \u2192 \u2191(f a) \u2260 0 \u2192 g a = (\u2191(f a))\u207b\u00b9 * g' a"}, {"tactic": "intro x hx h'x", "annotated_tactic": ["intro x hx h'x", []], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] g'\n\u22a2 \u2200 (a : \u03b1), \u2191(f a) * g a = g' a \u2192 \u2191(f a) \u2260 0 \u2192 g a = (\u2191(f a))\u207b\u00b9 * g' a", "state_after": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] g'\nx : \u03b1\nhx : \u2191(f x) * g x = g' x\nh'x : \u2191(f x) \u2260 0\n\u22a2 g x = (\u2191(f x))\u207b\u00b9 * g' x"}, {"tactic": "rw [\u2190 hx, \u2190 mul_assoc, ENNReal.inv_mul_cancel h'x ENNReal.coe_ne_top, one_mul]", "annotated_tactic": ["rw [\u2190 hx, \u2190 <a>mul_assoc</a>, <a>ENNReal.inv_mul_cancel</a> h'x <a>ENNReal.coe_ne_top</a>, <a>one_mul</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "ENNReal.inv_mul_cancel", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1424, 19], "def_end_pos": [1424, 33]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "case h\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\nhf : Measurable f\ng g' : \u03b1 \u2192 \u211d\u22650\u221e\ng'meas : Measurable g'\nhg' : (fun x => \u2191(f x) * g x) =\u1da0[ae \u03bc] g'\nx : \u03b1\nhx : \u2191(f x) * g x = g' x\nh'x : \u2191(f x) \u2260 0\n\u22a2 g x = (\u2191(f x))\u207b\u00b9 * g' x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean", "full_name": "strictConcaveOn_cos_Icc", "start": [175, 1], "end": [178, 31], "traced_tactics": [{"tactic": "apply strictConcaveOn_of_deriv2_neg (convex_Icc _ _) continuousOn_cos fun x hx => ?_", "annotated_tactic": ["apply <a>strictConcaveOn_of_deriv2_neg</a> (<a>convex_Icc</a> _ _) <a>continuousOn_cos</a> fun x hx => ?_", [{"full_name": "strictConcaveOn_of_deriv2_neg", "def_path": "Mathlib/Analysis/Calculus/MeanValue.lean", "def_pos": [1219, 9], "def_end_pos": [1219, 38]}, {"full_name": "convex_Icc", "def_path": "Mathlib/Analysis/Convex/Basic.lean", "def_pos": [290, 9], "def_end_pos": [290, 19]}, {"full_name": "Real.continuousOn_cos", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 25]}]], "state_before": "\u22a2 StrictConcaveOn \u211d (Icc (-(\u03c0 / 2)) (\u03c0 / 2)) cos", "state_after": "x : \u211d\nhx : x \u2208 interior (Icc (-(\u03c0 / 2)) (\u03c0 / 2))\n\u22a2 deriv^[2] cos x < 0"}, {"tactic": "rw [interior_Icc] at hx", "annotated_tactic": ["rw [<a>interior_Icc</a>] at hx", [{"full_name": "interior_Icc", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [2371, 9], "def_end_pos": [2371, 21]}]], "state_before": "x : \u211d\nhx : x \u2208 interior (Icc (-(\u03c0 / 2)) (\u03c0 / 2))\n\u22a2 deriv^[2] cos x < 0", "state_after": "x : \u211d\nhx : x \u2208 Ioo (-(\u03c0 / 2)) (\u03c0 / 2)\n\u22a2 deriv^[2] cos x < 0"}, {"tactic": "simp [cos_pos_of_mem_Ioo hx]", "annotated_tactic": ["simp [<a>cos_pos_of_mem_Ioo</a> hx]", [{"full_name": "Real.cos_pos_of_mem_Ioo", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "def_pos": [473, 9], "def_end_pos": [473, 27]}]], "state_before": "x : \u211d\nhx : x \u2208 Ioo (-(\u03c0 / 2)) (\u03c0 / 2)\n\u22a2 deriv^[2] cos x < 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Kaehler.lean", "full_name": "KaehlerDifferential.derivationQuotKerTotal_lift_comp_total", "start": [537, 1], "end": [544, 58], "traced_tactics": [{"tactic": "apply Finsupp.lhom_ext", "annotated_tactic": ["apply <a>Finsupp.lhom_ext</a>", [{"full_name": "Finsupp.lhom_ext", "def_path": "Mathlib/LinearAlgebra/Finsupp.lean", "def_pos": [64, 9], "def_end_pos": [64, 17]}]], "state_before": "R : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\n\u22a2 LinearMap.comp (Derivation.liftKaehlerDifferential (derivationQuotKerTotal R S))\n      (Finsupp.total S (\u03a9[S\u2044R]) S \u2191(D R S)) =\n    Submodule.mkQ (kerTotal R S)", "state_after": "case h\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\n\u22a2 \u2200 (a b : S),\n    (\u2191(LinearMap.comp (Derivation.liftKaehlerDifferential (derivationQuotKerTotal R S))\n            (Finsupp.total S (\u03a9[S\u2044R]) S \u2191(D R S)))\n        fun\u2080 | a => b) =\n      b\ud835\udda3a"}, {"tactic": "intro a b", "annotated_tactic": ["intro a b", []], "state_before": "case h\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\n\u22a2 \u2200 (a b : S),\n    (\u2191(LinearMap.comp (Derivation.liftKaehlerDifferential (derivationQuotKerTotal R S))\n            (Finsupp.total S (\u03a9[S\u2044R]) S \u2191(D R S)))\n        fun\u2080 | a => b) =\n      b\ud835\udda3a", "state_after": "case h\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\na b : S\n\u22a2 (\u2191(LinearMap.comp (Derivation.liftKaehlerDifferential (derivationQuotKerTotal R S))\n          (Finsupp.total S (\u03a9[S\u2044R]) S \u2191(D R S)))\n      fun\u2080 | a => b) =\n    b\ud835\udda3a"}, {"tactic": "conv_rhs => rw [\u2190 Finsupp.smul_single_one a b, LinearMap.map_smul]", "annotated_tactic": ["conv_rhs => rw [\u2190 <a>Finsupp.smul_single_one</a> a b, <a>LinearMap.map_smul</a>]", [{"full_name": "Finsupp.smul_single_one", "def_path": "Mathlib/Data/Finsupp/Basic.lean", "def_pos": [1598, 9], "def_end_pos": [1598, 24]}, {"full_name": "LinearMap.map_smul", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [350, 19], "def_end_pos": [350, 27]}]], "state_before": "case h\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\na b : S\n\u22a2 (\u2191(LinearMap.comp (Derivation.liftKaehlerDifferential (derivationQuotKerTotal R S))\n          (Finsupp.total S (\u03a9[S\u2044R]) S \u2191(D R S)))\n      fun\u2080 | a => b) =\n    b\ud835\udda3a", "state_after": "case h\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\na b : S\n\u22a2 (\u2191(LinearMap.comp (Derivation.liftKaehlerDifferential (derivationQuotKerTotal R S))\n          (Finsupp.total S (\u03a9[S\u2044R]) S \u2191(D R S)))\n      fun\u2080 | a => b) =\n    b \u2022 1\ud835\udda3a"}, {"tactic": "simp [KaehlerDifferential.derivationQuotKerTotal_apply]", "annotated_tactic": ["simp [<a>KaehlerDifferential.derivationQuotKerTotal_apply</a>]", [{"full_name": "KaehlerDifferential.derivationQuotKerTotal_apply", "def_path": "Mathlib/RingTheory/Kaehler.lean", "def_pos": [532, 9], "def_end_pos": [532, 57]}]], "state_before": "case h\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\na b : S\n\u22a2 (\u2191(LinearMap.comp (Derivation.liftKaehlerDifferential (derivationQuotKerTotal R S))\n          (Finsupp.total S (\u03a9[S\u2044R]) S \u2191(D R S)))\n      fun\u2080 | a => b) =\n    b \u2022 1\ud835\udda3a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/SimpleGraph/Basic.lean", "full_name": "SimpleGraph.singleton_disjoint_neighborFinset", "start": [1435, 1], "end": [1436, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "full_name": "edist_ofMul", "start": [1199, 1], "end": [1200, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBNode.insert_toList_zoom_nil", "start": [542, 1], "end": [544, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Tactic/ComputeDegree.lean", "full_name": "Mathlib.Tactic.ComputeDegree.natDegree_C_le", "start": [86, 1], "end": [86, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/CubicDiscriminant.lean", "full_name": "Cubic.natDegree_of_c_eq_zero", "start": [429, 1], "end": [431, 42], "traced_tactics": [{"tactic": "rw [of_c_eq_zero ha hb hc, natDegree_C]", "annotated_tactic": ["rw [<a>of_c_eq_zero</a> ha hb hc, <a>natDegree_C</a>]", [{"full_name": "Cubic.of_c_eq_zero", "def_path": "Mathlib/Algebra/CubicDiscriminant.lean", "def_pos": [153, 9], "def_end_pos": [153, 21]}, {"full_name": "Polynomial.natDegree_C", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [265, 9], "def_end_pos": [265, 20]}]], "state_before": "R : Type u_1\nS : Type u_2\nF : Type u_3\nK : Type u_4\nP Q : Cubic R\na b c d a' b' c' d' : R\ninst\u271d : Semiring R\nha : P.a = 0\nhb : P.b = 0\nhc : P.c = 0\n\u22a2 natDegree (toPoly P) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/WittVector/Truncated.lean", "full_name": "WittVector.mem_ker_truncate", "start": [343, 1], "end": [347, 23], "traced_tactics": [{"tactic": "simp only [RingHom.mem_ker, truncate, truncateFun, RingHom.coe_mk, TruncatedWittVector.ext_iff,\n  TruncatedWittVector.coeff_mk, coeff_zero]", "annotated_tactic": ["simp only [<a>RingHom.mem_ker</a>, <a>truncate</a>, <a>truncateFun</a>, <a>RingHom.coe_mk</a>, <a>TruncatedWittVector.ext_iff</a>,\n    <a>TruncatedWittVector.coeff_mk</a>, <a>coeff_zero</a>]", [{"full_name": "RingHom.mem_ker", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [2074, 9], "def_end_pos": [2074, 16]}, {"full_name": "WittVector.truncate", "def_path": "Mathlib/RingTheory/WittVector/Truncated.lean", "def_pos": [320, 19], "def_end_pos": [320, 27]}, {"full_name": "WittVector.truncateFun", "def_path": "Mathlib/RingTheory/WittVector/Truncated.lean", "def_pos": [136, 5], "def_end_pos": [136, 16]}, {"full_name": "RingHom.coe_mk", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [449, 9], "def_end_pos": [449, 15]}, {"full_name": "TruncatedWittVector.ext_iff", "def_path": "Mathlib/RingTheory/WittVector/Truncated.lean", "def_pos": [90, 9], "def_end_pos": [90, 16]}, {"full_name": "TruncatedWittVector.coeff_mk", "def_path": "Mathlib/RingTheory/WittVector/Truncated.lean", "def_pos": [95, 9], "def_end_pos": [95, 17]}, {"full_name": "TruncatedWittVector.coeff_zero", "def_path": "Mathlib/RingTheory/WittVector/Truncated.lean", "def_pos": [211, 9], "def_end_pos": [211, 19]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\n\u22a2 x \u2208 RingHom.ker (truncate n) \u2194 \u2200 (i : \u2115), i < n \u2192 coeff x i = 0", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\n\u22a2 (\u2200 (i : Fin n),\n      TruncatedWittVector.coeff i\n          (\u2191{ toOneHom := { toFun := truncateFun n, map_one' := (_ : truncateFun n 1 = 1) },\n                map_mul' := (_ : \u2200 (x y : \ud835\udd4e R), truncateFun n (x * y) = truncateFun n x * truncateFun n y) }\n            x) =\n        0) \u2194\n    \u2200 (i : \u2115), i < n \u2192 coeff x i = 0"}, {"tactic": "exact Fin.forall_iff", "annotated_tactic": ["exact <a>Fin.forall_iff</a>", [{"full_name": "Fin.forall_iff", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [45, 9], "def_end_pos": [45, 19]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\n\u22a2 (\u2200 (i : Fin n),\n      TruncatedWittVector.coeff i\n          (\u2191{ toOneHom := { toFun := truncateFun n, map_one' := (_ : truncateFun n 1 = 1) },\n                map_mul' := (_ : \u2200 (x y : \ud835\udd4e R), truncateFun n (x * y) = truncateFun n x * truncateFun n y) }\n            x) =\n        0) \u2194\n    \u2200 (i : \u2115), i < n \u2192 coeff x i = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/IndicatorFunction.lean", "full_name": "mulIndicator_eventuallyLE_mulIndicator", "start": [56, 1], "end": [58, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.countP_filter", "start": [2275, 1], "end": [2276, 93], "traced_tactics": [{"tactic": "simp [countP_eq_card_filter]", "annotated_tactic": ["simp [<a>countP_eq_card_filter</a>]", [{"full_name": "Multiset.countP_eq_card_filter", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2229, 9], "def_end_pos": [2229, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\nq : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred q\ns : Multiset \u03b1\n\u22a2 countP p (filter q s) = countP (fun a => p a \u2227 q a) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Log.lean", "full_name": "Int.clog_natCast", "start": [231, 1], "end": [234, 83], "traced_tactics": [{"tactic": "cases' n with n", "annotated_tactic": ["cases' n with n", []], "state_before": "R : Type u_1\ninst\u271d\u00b9 : LinearOrderedSemifield R\ninst\u271d : FloorSemiring R\nb n : \u2115\n\u22a2 clog b \u2191n = \u2191(Nat.clog b n)", "state_after": "case zero\nR : Type u_1\ninst\u271d\u00b9 : LinearOrderedSemifield R\ninst\u271d : FloorSemiring R\nb : \u2115\n\u22a2 clog b \u2191Nat.zero = \u2191(Nat.clog b Nat.zero)\n\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : LinearOrderedSemifield R\ninst\u271d : FloorSemiring R\nb n : \u2115\n\u22a2 clog b \u2191(Nat.succ n) = \u2191(Nat.clog b (Nat.succ n))"}, {"tactic": "simp [clog_of_right_le_one]", "annotated_tactic": ["simp [<a>clog_of_right_le_one</a>]", [{"full_name": "Int.clog_of_right_le_one", "def_path": "Mathlib/Data/Int/Log.lean", "def_pos": [193, 9], "def_end_pos": [193, 29]}]], "state_before": "case zero\nR : Type u_1\ninst\u271d\u00b9 : LinearOrderedSemifield R\ninst\u271d : FloorSemiring R\nb : \u2115\n\u22a2 clog b \u2191Nat.zero = \u2191(Nat.clog b Nat.zero)", "state_after": "no goals"}, {"tactic": "rw [clog_of_one_le_right, (Nat.ceil_eq_iff (Nat.succ_ne_zero n)).mpr] <;> simp", "annotated_tactic": ["rw [<a>clog_of_one_le_right</a>, (<a>Nat.ceil_eq_iff</a> (<a>Nat.succ_ne_zero</a> n)).<a>mpr</a>] <;> simp", [{"full_name": "Int.clog_of_one_le_right", "def_path": "Mathlib/Data/Int/Log.lean", "def_pos": [189, 9], "def_end_pos": [189, 29]}, {"full_name": "Nat.ceil_eq_iff", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [358, 9], "def_end_pos": [358, 20]}, {"full_name": "Nat.succ_ne_zero", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [432, 9], "def_end_pos": [432, 21]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "case succ\nR : Type u_1\ninst\u271d\u00b9 : LinearOrderedSemifield R\ninst\u271d : FloorSemiring R\nb n : \u2115\n\u22a2 clog b \u2191(Nat.succ n) = \u2191(Nat.clog b (Nat.succ n))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Field/Basic.lean", "full_name": "lt_inv_of_neg", "start": [777, 1], "end": [778, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/PGroup.lean", "full_name": "IsPGroup.to_sup_of_normal_right", "start": [312, 1], "end": [316, 30], "traced_tactics": [{"tactic": "rw [\u2190 QuotientGroup.ker_mk' K, \u2190 Subgroup.comap_map_eq]", "annotated_tactic": ["rw [\u2190 <a>QuotientGroup.ker_mk'</a> K, \u2190 <a>Subgroup.comap_map_eq</a>]", [{"full_name": "QuotientGroup.ker_mk'", "def_path": "Mathlib/GroupTheory/QuotientGroup.lean", "def_pos": [133, 9], "def_end_pos": [133, 16]}, {"full_name": "Subgroup.comap_map_eq", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [3094, 9], "def_end_pos": [3094, 21]}]], "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K : Subgroup G\nhH : IsPGroup p { x // x \u2208 H }\nhK : IsPGroup p { x // x \u2208 K }\ninst\u271d : Subgroup.Normal K\n\u22a2 IsPGroup p { x // x \u2208 H \u2294 K }", "state_after": "p : \u2115\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K : Subgroup G\nhH : IsPGroup p { x // x \u2208 H }\nhK : IsPGroup p { x // x \u2208 K }\ninst\u271d : Subgroup.Normal K\n\u22a2 IsPGroup p { x // x \u2208 Subgroup.comap (QuotientGroup.mk' K) (Subgroup.map (QuotientGroup.mk' K) H) }"}, {"tactic": "apply (hH.map (QuotientGroup.mk' K)).comap_of_ker_isPGroup", "annotated_tactic": ["apply (hH.map (<a>QuotientGroup.mk'</a> K)).<a>comap_of_ker_isPGroup</a>", [{"full_name": "QuotientGroup.mk'", "def_path": "Mathlib/GroupTheory/QuotientGroup.lean", "def_pos": [76, 5], "def_end_pos": [76, 8]}, {"full_name": "IsPGroup.comap_of_ker_isPGroup", "def_path": "Mathlib/GroupTheory/PGroup.lean", "def_pos": [287, 9], "def_end_pos": [287, 30]}]], "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K : Subgroup G\nhH : IsPGroup p { x // x \u2208 H }\nhK : IsPGroup p { x // x \u2208 K }\ninst\u271d : Subgroup.Normal K\n\u22a2 IsPGroup p { x // x \u2208 Subgroup.comap (QuotientGroup.mk' K) (Subgroup.map (QuotientGroup.mk' K) H) }", "state_after": "case h\u03d5\np : \u2115\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K : Subgroup G\nhH : IsPGroup p { x // x \u2208 H }\nhK : IsPGroup p { x // x \u2208 K }\ninst\u271d : Subgroup.Normal K\n\u22a2 IsPGroup p { x // x \u2208 MonoidHom.ker (QuotientGroup.mk' K) }"}, {"tactic": "rwa [QuotientGroup.ker_mk']", "annotated_tactic": ["rwa [<a>QuotientGroup.ker_mk'</a>]", [{"full_name": "QuotientGroup.ker_mk'", "def_path": "Mathlib/GroupTheory/QuotientGroup.lean", "def_pos": [133, 9], "def_end_pos": [133, 16]}]], "state_before": "case h\u03d5\np : \u2115\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K : Subgroup G\nhH : IsPGroup p { x // x \u2208 H }\nhK : IsPGroup p { x // x \u2208 K }\ninst\u271d : Subgroup.Normal K\n\u22a2 IsPGroup p { x // x \u2208 MonoidHom.ker (QuotientGroup.mk' K) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Variables.lean", "full_name": "MvPolynomial.degreeOf_X", "start": [519, 1], "end": [524, 46], "traced_tactics": [{"tactic": "by_cases c : i = j", "annotated_tactic": ["by_cases c : i = j", []], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b2 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : DecidableEq \u03c3\ni j : \u03c3\ninst\u271d : Nontrivial R\n\u22a2 degreeOf i (X j) = if i = j then 1 else 0", "state_after": "case pos\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b2 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : DecidableEq \u03c3\ni j : \u03c3\ninst\u271d : Nontrivial R\nc : i = j\n\u22a2 degreeOf i (X j) = if i = j then 1 else 0\n\ncase neg\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b2 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : DecidableEq \u03c3\ni j : \u03c3\ninst\u271d : Nontrivial R\nc : \u00aci = j\n\u22a2 degreeOf i (X j) = if i = j then 1 else 0"}, {"tactic": "simp [c, if_false, degreeOf_def, degrees_X]", "annotated_tactic": ["simp [c, <a>if_false</a>, <a>degreeOf_def</a>, <a>degrees_X</a>]", [{"full_name": "if_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [729, 17], "def_end_pos": [729, 25]}, {"full_name": "MvPolynomial.degreeOf_def", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [490, 9], "def_end_pos": [490, 21]}, {"full_name": "MvPolynomial.degrees_X", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [122, 9], "def_end_pos": [122, 18]}]], "state_before": "case neg\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b2 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : DecidableEq \u03c3\ni j : \u03c3\ninst\u271d : Nontrivial R\nc : \u00aci = j\n\u22a2 degreeOf i (X j) = if i = j then 1 else 0", "state_after": "no goals"}, {"tactic": "simp only [c, if_true, eq_self_iff_true, degreeOf_def, degrees_X, Multiset.count_singleton]", "annotated_tactic": ["simp only [c, <a>if_true</a>, <a>eq_self_iff_true</a>, <a>degreeOf_def</a>, <a>degrees_X</a>, <a>Multiset.count_singleton</a>]", [{"full_name": "if_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [727, 17], "def_end_pos": [727, 24]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "MvPolynomial.degreeOf_def", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [490, 9], "def_end_pos": [490, 21]}, {"full_name": "MvPolynomial.degrees_X", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [122, 9], "def_end_pos": [122, 18]}, {"full_name": "Multiset.count_singleton", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2389, 9], "def_end_pos": [2389, 24]}]], "state_before": "case pos\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b2 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : DecidableEq \u03c3\ni j : \u03c3\ninst\u271d : Nontrivial R\nc : i = j\n\u22a2 degreeOf i (X j) = if i = j then 1 else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/Degree/Lemmas.lean", "full_name": "Polynomial.coeff_sub_eq_neg_right_of_lt", "start": [344, 1], "end": [345, 60], "traced_tactics": [{"tactic": "rwa [sub_eq_add_neg, coeff_add_eq_right_of_lt, coeff_neg]", "annotated_tactic": ["rwa [<a>sub_eq_add_neg</a>, <a>coeff_add_eq_right_of_lt</a>, <a>coeff_neg</a>]", [{"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "Polynomial.coeff_add_eq_right_of_lt", "def_path": "Mathlib/Data/Polynomial/Degree/Lemmas.lean", "def_pos": [197, 9], "def_end_pos": [197, 33]}, {"full_name": "Polynomial.coeff_neg", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [1197, 9], "def_end_pos": [1197, 18]}]], "state_before": "R : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Ring R\np q : R[X]\ndf : natDegree p < n\n\u22a2 coeff (p - q) n = -coeff q n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/FieldTheory/PerfectClosure.lean", "full_name": "PerfectClosure.induction_on", "start": [72, 1], "end": [74, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Homology/ComplexShape.lean", "full_name": "ComplexShape.down_mk", "start": [205, 1], "end": [207, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Field/Basic.lean", "full_name": "inv_add_inv", "start": [214, 1], "end": [215, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.VectorMeasure.AbsolutelyContinuous.ennrealToMeasure", "start": [1144, 1], "end": [1153, 35], "traced_tactics": [{"tactic": "constructor <;> intro h", "annotated_tactic": ["constructor <;> intro h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2075 : AddCommMonoid L\ninst\u271d\u2074 : TopologicalSpace L\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\n\u22a2 (\u2200 \u2983s : Set \u03b1\u2984, \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s = 0 \u2192 \u2191v s = 0) \u2194 v \u226a\u1d65 \u03bc", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2075 : AddCommMonoid L\ninst\u271d\u2074 : TopologicalSpace L\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : \u2200 \u2983s : Set \u03b1\u2984, \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s = 0 \u2192 \u2191v s = 0\n\u22a2 v \u226a\u1d65 \u03bc\n\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2075 : AddCommMonoid L\ninst\u271d\u2074 : TopologicalSpace L\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : v \u226a\u1d65 \u03bc\n\u22a2 \u2200 \u2983s : Set \u03b1\u2984, \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s = 0 \u2192 \u2191v s = 0"}, {"tactic": "refine' mk fun s hmeas hs => h _", "annotated_tactic": ["refine' <a>mk</a> fun s hmeas hs => h _", [{"full_name": "MeasureTheory.VectorMeasure.AbsolutelyContinuous.mk", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [1077, 9], "def_end_pos": [1077, 11]}]], "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2075 : AddCommMonoid L\ninst\u271d\u2074 : TopologicalSpace L\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : \u2200 \u2983s : Set \u03b1\u2984, \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s = 0 \u2192 \u2191v s = 0\n\u22a2 v \u226a\u1d65 \u03bc", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2075 : AddCommMonoid L\ninst\u271d\u2074 : TopologicalSpace L\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : \u2200 \u2983s : Set \u03b1\u2984, \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s = 0 \u2192 \u2191v s = 0\ns : Set \u03b1\nhmeas : MeasurableSet s\nhs : \u2191\u03bc s = 0\n\u22a2 \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s = 0"}, {"tactic": "rw [\u2190 hs, ennrealToMeasure_apply hmeas]", "annotated_tactic": ["rw [\u2190 hs, <a>ennrealToMeasure_apply</a> hmeas]", [{"full_name": "MeasureTheory.VectorMeasure.ennrealToMeasure_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [534, 9], "def_end_pos": [534, 31]}]], "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2075 : AddCommMonoid L\ninst\u271d\u2074 : TopologicalSpace L\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : \u2200 \u2983s : Set \u03b1\u2984, \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s = 0 \u2192 \u2191v s = 0\ns : Set \u03b1\nhmeas : MeasurableSet s\nhs : \u2191\u03bc s = 0\n\u22a2 \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s = 0", "state_after": "no goals"}, {"tactic": "intro s hs", "annotated_tactic": ["intro s hs", []], "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2075 : AddCommMonoid L\ninst\u271d\u2074 : TopologicalSpace L\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : v \u226a\u1d65 \u03bc\n\u22a2 \u2200 \u2983s : Set \u03b1\u2984, \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s = 0 \u2192 \u2191v s = 0", "state_after": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2075 : AddCommMonoid L\ninst\u271d\u2074 : TopologicalSpace L\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : v \u226a\u1d65 \u03bc\ns : Set \u03b1\nhs : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s = 0\n\u22a2 \u2191v s = 0"}, {"tactic": "by_cases hmeas : MeasurableSet s", "annotated_tactic": ["by_cases hmeas : <a>MeasurableSet</a> s", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}]], "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2075 : AddCommMonoid L\ninst\u271d\u2074 : TopologicalSpace L\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : v \u226a\u1d65 \u03bc\ns : Set \u03b1\nhs : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s = 0\n\u22a2 \u2191v s = 0", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2075 : AddCommMonoid L\ninst\u271d\u2074 : TopologicalSpace L\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : v \u226a\u1d65 \u03bc\ns : Set \u03b1\nhs : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s = 0\nhmeas : MeasurableSet s\n\u22a2 \u2191v s = 0\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2075 : AddCommMonoid L\ninst\u271d\u2074 : TopologicalSpace L\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : v \u226a\u1d65 \u03bc\ns : Set \u03b1\nhs : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s = 0\nhmeas : \u00acMeasurableSet s\n\u22a2 \u2191v s = 0"}, {"tactic": "rw [ennrealToMeasure_apply hmeas] at hs", "annotated_tactic": ["rw [<a>ennrealToMeasure_apply</a> hmeas] at hs", [{"full_name": "MeasureTheory.VectorMeasure.ennrealToMeasure_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [534, 9], "def_end_pos": [534, 31]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2075 : AddCommMonoid L\ninst\u271d\u2074 : TopologicalSpace L\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : v \u226a\u1d65 \u03bc\ns : Set \u03b1\nhs : \u2191\u2191(VectorMeasure.ennrealToMeasure \u03bc) s = 0\nhmeas : MeasurableSet s\n\u22a2 \u2191v s = 0", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2075 : AddCommMonoid L\ninst\u271d\u2074 : TopologicalSpace L\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : v \u226a\u1d65 \u03bc\ns : Set \u03b1\nhs : \u2191\u03bc s = 0\nhmeas : MeasurableSet s\n\u22a2 \u2191v s = 0"}, {"tactic": "exact h hs", "annotated_tactic": ["exact h hs", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2075 : AddCommMonoid L\ninst\u271d\u2074 : TopologicalSpace L\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : AddCommMonoid N\ninst\u271d : TopologicalSpace N\nv : VectorMeasure \u03b1 M\nw : VectorMeasure \u03b1 N\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\nh : v \u226a\u1d65 \u03bc\ns : Set \u03b1\nhs : \u2191\u03bc s = 0\nhmeas : MeasurableSet s\n\u22a2 \u2191v s = 0", "state_after": "no goals"}, {"tactic": "exact not_measurable v hmeas", "annotated_tactic": ["exact <a>not_measurable</a> v hmeas", [{"full_name": "MeasureTheory.VectorMeasure.not_measurable", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [102, 9], "def_end_pos": [102, 23]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2075 : AddCommMonoid L\ninst\u271d\u2074 : TopologicalSpace L\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : TopologicalSpace 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} s\n\u22a2 False", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d : BooleanAlgebra \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nx : \u03b1\nhx' : x \u2208 s\nhx : \u2191{ toFun := compl, inj' := (_ : Function.Injective compl) } x \u2208 s\nhxc :\n  \u2191{ toFun := compl, inj' := (_ : Function.Injective compl) } x \u2208\n    map { toFun := compl, inj' := (_ : Function.Injective compl) } s\n\u22a2 False"}, {"tactic": "exact hs.not_compl_mem hx' hx", "annotated_tactic": ["exact hs.not_compl_mem hx' hx", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d : BooleanAlgebra \u03b1\ns : Finset \u03b1\nhs : Intersecting \u2191s\nx : \u03b1\nhx' : x \u2208 s\nhx : \u2191{ toFun := compl, inj' := (_ : Function.Injective compl) } x \u2208 s\nhxc :\n  \u2191{ toFun := compl, inj' := (_ : Function.Injective compl) } x \u2208\n    map { toFun := compl, inj' := (_ : Function.Injective compl) } s\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Testing/SlimCheck/Functions.lean", "full_name": "SlimCheck.InjectiveFunction.applyId_injective", "start": [374, 1], "end": [398, 73], "traced_tactics": [{"tactic": "intro x y h", "annotated_tactic": ["intro x y h", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\n\u22a2 Injective (applyId (List.zip xs ys))", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\n\u22a2 x = y"}, {"tactic": "by_cases hx : x \u2208 xs <;> by_cases hy : y \u2208 xs", "annotated_tactic": ["by_cases hx : x \u2208 xs <;> by_cases hy : y \u2208 xs", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\n\u22a2 x = y", "state_after": "case pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : x \u2208 xs\nhy : y \u2208 xs\n\u22a2 x = y\n\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : x \u2208 xs\nhy : \u00acy \u2208 xs\n\u22a2 x = y\n\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : \u00acx \u2208 xs\nhy : y \u2208 xs\n\u22a2 x = y\n\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : \u00acx \u2208 xs\nhy : \u00acy \u2208 xs\n\u22a2 x = y"}, {"tactic": "rw [List.mem_iff_get?] at hx hy", "annotated_tactic": ["rw [<a>List.mem_iff_get?</a>] at hx hy", [{"full_name": "List.mem_iff_get?", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [626, 9], "def_end_pos": [626, 21]}]], "state_before": "case pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : x \u2208 xs\nhy : y \u2208 xs\n\u22a2 x = y", "state_after": "case pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : \u2203 n, List.get? xs n = some x\nhy : \u2203 n, List.get? xs n = some y\n\u22a2 x = y"}, {"tactic": "cases' hx with i hx", "annotated_tactic": ["cases' hx with i hx", []], "state_before": "case pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : \u2203 n, List.get? xs n = some x\nhy : \u2203 n, List.get? xs n = some y\n\u22a2 x = y", "state_after": "case pos.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhy : \u2203 n, List.get? xs n = some y\ni : \u2115\nhx : List.get? xs i = some x\n\u22a2 x = y"}, {"tactic": "cases' hy with j hy", "annotated_tactic": ["cases' hy with j hy", []], "state_before": "case pos.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhy : \u2203 n, List.get? xs n = some y\ni : \u2115\nhx : List.get? xs i = some x\n\u22a2 x = y", "state_after": "case pos.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\ni : \u2115\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\n\u22a2 x = y"}, {"tactic": "suffices some x = some y by injection this", "annotated_tactic": ["suffices <a>some</a> x = <a>some</a> y by injection this", [{"full_name": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}, {"full_name": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}]], "state_before": "case pos.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\ni : \u2115\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\n\u22a2 x = y", "state_after": "case pos.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\ni : \u2115\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\n\u22a2 some x = some y"}, {"tactic": "have h\u2082 := h\u2081.length_eq", "annotated_tactic": ["have h\u2082 := h\u2081.length_eq", []], "state_before": "case pos.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\ni : \u2115\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\n\u22a2 some x = some y", "state_after": "case pos.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\ni : \u2115\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 some x = some y"}, {"tactic": "rw [List.applyId_zip_eq h\u2080 h\u2082 _ _ _ hx] at h", "annotated_tactic": ["rw [<a>List.applyId_zip_eq</a> h\u2080 h\u2082 _ _ _ hx] at h", [{"full_name": "SlimCheck.InjectiveFunction.List.applyId_zip_eq", "def_path": "Mathlib/Testing/SlimCheck/Functions.lean", "def_pos": [306, 9], "def_end_pos": [306, 28]}]], "state_before": "case pos.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\ni : \u2115\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 some x = some y", "state_after": "case pos.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 some x = some y"}, {"tactic": "rw [\u2190 hx, \u2190 hy]", "annotated_tactic": ["rw [\u2190 hx, \u2190 hy]", []], "state_before": "case pos.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 some x = some y", "state_after": "case pos.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 List.get? xs i = List.get? xs j"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case pos.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 List.get? xs i = List.get? xs j", "state_after": "case pos.intro.intro.e_a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 i = j"}, {"tactic": "apply List.get?_injective _ (h\u2081.nodup_iff.1 h\u2080)", "annotated_tactic": ["apply <a>List.get?_injective</a> _ (h\u2081.nodup_iff.1 h\u2080)", [{"full_name": "List.get?_injective", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [1216, 9], "def_end_pos": [1216, 23]}]], "state_before": "case pos.intro.intro.e_a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 i = j", "state_after": "case pos.intro.intro.e_a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 List.get? ys i = List.get? ys j\n\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 i < List.length ys"}, {"tactic": "injection this", "annotated_tactic": ["injection this", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\ni : \u2115\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nthis : some x = some y\n\u22a2 x = y", "state_after": "no goals"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "case pos.intro.intro.e_a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 List.get? ys i = List.get? ys j", "state_after": "case pos.intro.intro.e_a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 List.get? ys j = List.get? ys i"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "case pos.intro.intro.e_a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 List.get? ys j = List.get? ys i", "state_after": "case pos.intro.intro.e_a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 List.get? ys j = some (applyId (List.zip xs ys) y)"}, {"tactic": "rw [\u2190 List.applyId_zip_eq] <;> assumption", "annotated_tactic": ["rw [\u2190 <a>List.applyId_zip_eq</a>] <;> assumption", [{"full_name": "SlimCheck.InjectiveFunction.List.applyId_zip_eq", "def_path": "Mathlib/Testing/SlimCheck/Functions.lean", "def_pos": [306, 9], "def_end_pos": [306, 28]}]], "state_before": "case pos.intro.intro.e_a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 List.get? ys j = some (applyId (List.zip xs ys) y)", "state_after": "no goals"}, {"tactic": "rw [\u2190 h\u2081.length_eq]", "annotated_tactic": ["rw [\u2190 h\u2081.length_eq]", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 i < List.length ys", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 i < List.length xs"}, {"tactic": "rw [List.get?_eq_some] at hx", "annotated_tactic": ["rw [<a>List.get?_eq_some</a>] at hx", [{"full_name": "List.get?_eq_some", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [600, 9], "def_end_pos": [600, 21]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : List.get? xs i = some x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 i < List.length xs", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : \u2203 h, List.get xs { val := i, isLt := h } = x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 i < List.length xs"}, {"tactic": "cases' hx with hx hx'", "annotated_tactic": ["cases' hx with hx hx'", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nhx : \u2203 h, List.get xs { val := i, isLt := h } = x\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\n\u22a2 i < List.length xs", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\nhx : i < List.length xs\nhx' : List.get xs { val := i, isLt := hx } = x\n\u22a2 i < List.length xs"}, {"tactic": "exact hx", "annotated_tactic": ["exact hx", []], "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\ni : \u2115\nh : List.get? ys i = some (applyId (List.zip xs ys) y)\nj : \u2115\nhy : List.get? xs j = some y\nh\u2082 : List.length xs = List.length ys\nhx : i < List.length xs\nhx' : List.get xs { val := i, isLt := hx } = x\n\u22a2 i < List.length xs", "state_after": 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\u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : \u00acapplyId (List.zip xs ys) x \u2208 ys\nhy : applyId (List.zip xs ys) y \u2208 ys\n\u22a2 x = y"}, {"tactic": "rw [h] at hx", "annotated_tactic": ["rw [h] at hx", []], "state_before": "case pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : \u00acapplyId (List.zip xs ys) x \u2208 ys\nhy : applyId (List.zip xs ys) y \u2208 ys\n\u22a2 x = y", "state_after": "case pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : \u00acapplyId (List.zip xs ys) y \u2208 ys\nhy : applyId (List.zip xs ys) y \u2208 ys\n\u22a2 x = y"}, {"tactic": "contradiction", "annotated_tactic": ["contradiction", []], "state_before": "case pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Sort w\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nh\u2080 : List.Nodup xs\nh\u2081 : xs ~ ys\nx y : \u03b1\nh : applyId (List.zip xs ys) x = applyId (List.zip xs ys) y\nhx : \u00acapplyId (List.zip xs ys) y \u2208 ys\nhy : applyId (List.zip xs ys) y \u2208 ys\n\u22a2 x = y", "state_after": "no goals"}, {"tactic": "rwa [List.applyId_eq_self, List.applyId_eq_self] at h <;> assumption", "annotated_tactic": ["rwa [<a>List.applyId_eq_self</a>, <a>List.applyId_eq_self</a>] at h <;> assumption", [{"full_name": "SlimCheck.InjectiveFunction.List.applyId_eq_self", "def_path": "Mathlib/Testing/SlimCheck/Functions.lean", "def_pos": [363, 9], "def_end_pos": [363, 29]}, {"full_name": "SlimCheck.InjectiveFunction.List.applyId_eq_self", "def_path": "Mathlib/Testing/SlimCheck/Functions.lean", "def_pos": [363, 9], "def_end_pos": 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"file_path": "Mathlib/Data/Finset/Pi.lean", "full_name": "Finset.pi_const_singleton", "start": [122, 1], "end": [124, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/RingQuot.lean", "full_name": "RingQuot.neg_quot", "start": [258, 1], "end": [262, 6], "traced_tactics": [{"tactic": "show neg r _ = _", "annotated_tactic": ["show <a>neg</a> r _ = _", [{"full_name": "_private.Mathlib.Algebra.RingQuot.0.RingQuot.neg", "def_path": "Mathlib/Algebra/RingQuot.lean", "def_pos": [176, 25], "def_end_pos": [176, 28]}]], "state_before": "R\u271d : Type uR\ninst\u271d\u2074 : Semiring R\u271d\nS : Type uS\ninst\u271d\u00b3 : CommSemiring S\nT : Type uT\nA : Type uA\ninst\u271d\u00b2 : Semiring A\ninst\u271d\u00b9 : Algebra S A\nr\u271d : R\u271d \u2192 R\u271d \u2192 Prop\nR : Type uR\ninst\u271d : Ring R\nr : R \u2192 R \u2192 Prop\na : R\n\u22a2 -{ toQuot := Quot.mk (Rel r) a } = { toQuot := Quot.mk (Rel r) (-a) }", "state_after": "R\u271d : Type uR\ninst\u271d\u2074 : Semiring R\u271d\nS : Type uS\ninst\u271d\u00b3 : CommSemiring S\nT : Type uT\nA : Type uA\ninst\u271d\u00b2 : Semiring A\ninst\u271d\u00b9 : Algebra S A\nr\u271d : R\u271d \u2192 R\u271d \u2192 Prop\nR : Type uR\ninst\u271d : Ring R\nr : R \u2192 R \u2192 Prop\na : R\n\u22a2 RingQuot.neg r { toQuot := Quot.mk (Rel r) a } = { toQuot := Quot.mk (Rel r) (-a) }"}, {"tactic": "rw [neg_def]", "annotated_tactic": ["rw [<a>neg_def</a>]", [{"full_name": "_private.Mathlib.Algebra.RingQuot.0.RingQuot.neg_def", "def_path": "Mathlib/Algebra/RingQuot.lean", "def_pos": [1, 1], "def_end_pos": [1, 1]}]], "state_before": "R\u271d : Type uR\ninst\u271d\u2074 : Semiring R\u271d\nS : Type uS\ninst\u271d\u00b3 : CommSemiring S\nT : Type uT\nA : Type uA\ninst\u271d\u00b2 : Semiring A\ninst\u271d\u00b9 : Algebra S A\nr\u271d : R\u271d \u2192 R\u271d \u2192 Prop\nR : Type uR\ninst\u271d : Ring R\nr : R \u2192 R \u2192 Prop\na : R\n\u22a2 RingQuot.neg r { toQuot := Quot.mk (Rel r) a } = { toQuot := Quot.mk (Rel r) (-a) }", "state_after": "R\u271d : Type uR\ninst\u271d\u2074 : Semiring R\u271d\nS : Type uS\ninst\u271d\u00b3 : CommSemiring S\nT : Type uT\nA : Type uA\ninst\u271d\u00b2 : Semiring A\ninst\u271d\u00b9 : Algebra S A\nr\u271d : R\u271d \u2192 R\u271d \u2192 Prop\nR : Type uR\ninst\u271d : Ring R\nr : R \u2192 R \u2192 Prop\na : R\n\u22a2 (match { toQuot := Quot.mk (Rel r) a } with\n    | { toQuot := a } => { toQuot := Quot.map (fun a => -a) (_ : \u2200 \u2983a b : R\u2984, Rel r a b \u2192 Rel r (-a) (-b)) a }) =\n    { toQuot := Quot.mk (Rel r) (-a) }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "R\u271d : Type uR\ninst\u271d\u2074 : Semiring R\u271d\nS : Type uS\ninst\u271d\u00b3 : CommSemiring S\nT : Type uT\nA : Type uA\ninst\u271d\u00b2 : Semiring A\ninst\u271d\u00b9 : Algebra S A\nr\u271d : R\u271d \u2192 R\u271d \u2192 Prop\nR : Type uR\ninst\u271d : Ring R\nr : R \u2192 R \u2192 Prop\na : R\n\u22a2 (match { toQuot := Quot.mk (Rel r) a } with\n    | { toQuot := a } => { toQuot := Quot.map (fun a => -a) (_ : \u2200 \u2983a b : R\u2984, Rel r a b \u2192 Rel r (-a) (-b)) a }) =\n    { toQuot := Quot.mk (Rel r) (-a) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/FieldTheory/Galois.lean", "full_name": "IsGalois.integral", "start": [74, 1], "end": [75, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "full_name": "intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le", "start": [1171, 1], "end": [1178, 84], "traced_tactics": [{"tactic": "refine' (NormedSpace.eq_iff_forall_dual_eq \u211d).2 fun g => _", "annotated_tactic": ["refine' (<a>NormedSpace.eq_iff_forall_dual_eq</a> \u211d).2 fun g => _", [{"full_name": "NormedSpace.eq_iff_forall_dual_eq", "def_path": "Mathlib/Analysis/NormedSpace/Dual.lean", "def_pos": [137, 9], "def_end_pos": [137, 30]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn f (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt f (f' x) (Ioi x) x\nf'int : IntervalIntegrable f' volume a b\n\u22a2 \u222b (y : \u211d) in a..b, f' y = f b - f a", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn f (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt f (f' x) (Ioi x) x\nf'int : IntervalIntegrable f' volume a b\ng : NormedSpace.Dual \u211d E\n\u22a2 \u2191g (\u222b (y : \u211d) in a..b, f' y) = \u2191g (f b - f a)"}, {"tactic": "rw [\u2190 g.intervalIntegral_comp_comm f'int, g.map_sub]", "annotated_tactic": ["rw [\u2190 g.intervalIntegral_comp_comm f'int, g.map_sub]", []], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn f (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt f (f' x) (Ioi x) x\nf'int : IntervalIntegrable f' volume a b\ng : NormedSpace.Dual \u211d E\n\u22a2 \u2191g (\u222b (y : \u211d) in a..b, f' y) = \u2191g (f b - f a)", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn f (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt f (f' x) (Ioi x) x\nf'int : IntervalIntegrable f' volume a b\ng : NormedSpace.Dual \u211d E\n\u22a2 \u222b (x : \u211d) in a..b, \u2191g (f' x) = \u2191g (f b) - \u2191g (f a)"}, {"tactic": "exact integral_eq_sub_of_hasDeriv_right_of_le_real hab (g.continuous.comp_continuousOn hcont)\n  (fun x hx => g.hasFDerivAt.comp_hasDerivWithinAt x (hderiv x hx))\n  (g.integrable_comp ((intervalIntegrable_iff_integrable_Icc_of_le hab).1 f'int))", "annotated_tactic": ["exact <a>integral_eq_sub_of_hasDeriv_right_of_le_real</a> hab (g.continuous.comp_continuousOn hcont)\n    (fun x hx => g.hasFDerivAt.comp_hasDerivWithinAt x (hderiv x hx))\n    (g.integrable_comp ((<a>intervalIntegrable_iff_integrable_Icc_of_le</a> hab).1 f'int))", [{"full_name": "intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le_real", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [1159, 9], "def_end_pos": [1159, 53]}, {"full_name": "intervalIntegrable_iff_integrable_Icc_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [101, 9], "def_end_pos": [101, 52]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf\u271d : \u211d \u2192 E\ng' g\u271d \u03c6 : \u211d \u2192 \u211d\nf f' : \u211d \u2192 E\na b : \u211d\nhab : a \u2264 b\nhcont : ContinuousOn f (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt f (f' x) (Ioi x) x\nf'int : IntervalIntegrable f' volume a b\ng : NormedSpace.Dual \u211d E\n\u22a2 \u222b (x : \u211d) in a..b, \u2191g (f' x) = \u2191g (f b) - \u2191g (f a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.stronglyMeasurable", "start": [207, 11], "end": [208, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.univ_le_of_forall_fin_meas_le", "start": [1625, 1], "end": [1636, 86], "traced_tactics": [{"tactic": "let S := @spanningSets _ m (\u03bc.trim hm) _", "annotated_tactic": ["let S := @<a>spanningSets</a> _ m (\u03bc.trim hm) _", [{"full_name": "MeasureTheory.spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3316, 5], "def_end_pos": [3316, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : Set \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C\nh_F_lim : \u2200 (S : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), MeasurableSet (S n)) \u2192 Monotone S \u2192 f (\u22c3 n, S n) \u2264 \u2a06 n, f (S n)\n\u22a2 f univ \u2264 C", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : Set \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C\nh_F_lim : \u2200 (S : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), MeasurableSet (S n)) \u2192 Monotone S \u2192 f (\u22c3 n, S n) \u2264 \u2a06 n, f (S n)\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\n\u22a2 f univ \u2264 C"}, {"tactic": "have hS_mono : Monotone S := @monotone_spanningSets _ m (\u03bc.trim hm) _", "annotated_tactic": ["have hS_mono : <a>Monotone</a> S := @<a>monotone_spanningSets</a> _ m (\u03bc.trim hm) _", [{"full_name": "Monotone", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [77, 5], "def_end_pos": [77, 13]}, {"full_name": "MeasureTheory.monotone_spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3320, 9], "def_end_pos": [3320, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : Set \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C\nh_F_lim : \u2200 (S : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), MeasurableSet (S n)) \u2192 Monotone S \u2192 f (\u22c3 n, S n) \u2264 \u2a06 n, f (S n)\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\n\u22a2 f univ \u2264 C", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : Set \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C\nh_F_lim : \u2200 (S : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), MeasurableSet (S n)) \u2192 Monotone S \u2192 f (\u22c3 n, S n) \u2264 \u2a06 n, f (S n)\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_mono : Monotone S\n\u22a2 f univ \u2264 C"}, {"tactic": "have hS_meas : \u2200 n, MeasurableSet[m] (S n) := @measurable_spanningSets _ m (\u03bc.trim hm) _", "annotated_tactic": ["have hS_meas : \u2200 n, MeasurableSet[m] (S n) := @<a>measurable_spanningSets</a> _ m (\u03bc.trim hm) _", [{"full_name": "MeasureTheory.measurable_spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3324, 9], "def_end_pos": [3324, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : Set \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C\nh_F_lim : \u2200 (S : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), MeasurableSet (S n)) \u2192 Monotone S \u2192 f (\u22c3 n, S n) \u2264 \u2a06 n, f (S n)\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_mono : Monotone S\n\u22a2 f univ \u2264 C", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : Set \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C\nh_F_lim : \u2200 (S : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), MeasurableSet (S n)) \u2192 Monotone S \u2192 f (\u22c3 n, S n) \u2264 \u2a06 n, f (S n)\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_mono : Monotone S\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\n\u22a2 f univ \u2264 C"}, {"tactic": "rw [\u2190 @iUnion_spanningSets _ m (\u03bc.trim hm)]", "annotated_tactic": ["rw [\u2190 @<a>iUnion_spanningSets</a> _ m (\u03bc.trim hm)]", [{"full_name": "MeasureTheory.iUnion_spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3334, 9], "def_end_pos": [3334, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : Set \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C\nh_F_lim : \u2200 (S : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), MeasurableSet (S n)) \u2192 Monotone S \u2192 f (\u22c3 n, S n) \u2264 \u2a06 n, f (S n)\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_mono : Monotone S\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\n\u22a2 f univ \u2264 C", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : Set \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C\nh_F_lim : \u2200 (S : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), MeasurableSet (S n)) \u2192 Monotone S \u2192 f (\u22c3 n, S n) \u2264 \u2a06 n, f (S n)\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_mono : Monotone S\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\n\u22a2 f (\u22c3 i, spanningSets (Measure.trim \u03bc hm) i) \u2264 C"}, {"tactic": "refine' (h_F_lim S hS_meas hS_mono).trans _", "annotated_tactic": ["refine' (h_F_lim S hS_meas hS_mono).<a>trans</a> _", [{"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : Set \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C\nh_F_lim : \u2200 (S : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), MeasurableSet (S n)) \u2192 Monotone S \u2192 f (\u22c3 n, S n) \u2264 \u2a06 n, f (S n)\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_mono : Monotone S\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\n\u22a2 f (\u22c3 i, spanningSets (Measure.trim \u03bc hm) i) \u2264 C", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : Set \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C\nh_F_lim : \u2200 (S : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), MeasurableSet (S n)) \u2192 Monotone S \u2192 f (\u22c3 n, S n) \u2264 \u2a06 n, f (S n)\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_mono : Monotone S\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\n\u22a2 \u2a06 n, f (S n) \u2264 C"}, {"tactic": "refine' iSup_le fun n => hf (S n) (hS_meas n) _", "annotated_tactic": ["refine' <a>iSup_le</a> fun n => hf (S n) (hS_meas n) _", [{"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : Set \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C\nh_F_lim : \u2200 (S : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), MeasurableSet (S n)) \u2192 Monotone S \u2192 f (\u22c3 n, S n) \u2264 \u2a06 n, f (S n)\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_mono : Monotone S\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\n\u22a2 \u2a06 n, f (S n) \u2264 C", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : Set \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C\nh_F_lim : \u2200 (S : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), MeasurableSet (S n)) \u2192 Monotone S \u2192 f (\u22c3 n, S n) \u2264 \u2a06 n, f (S n)\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_mono : Monotone S\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (S n) \u2260 \u22a4"}, {"tactic": "exact ((le_trim hm).trans_lt (@measure_spanningSets_lt_top _ m (\u03bc.trim hm) _ n)).ne", "annotated_tactic": ["exact ((<a>le_trim</a> hm).<a>trans_lt</a> (@<a>measure_spanningSets_lt_top</a> _ m (\u03bc.trim hm) _ n)).<a>ne</a>", [{"full_name": "MeasureTheory.le_trim", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [57, 9], "def_end_pos": [57, 16]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}, {"full_name": "MeasureTheory.measure_spanningSets_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3329, 9], "def_end_pos": [3329, 36]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm m0 : MeasurableSpace \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : OpensMeasurableSpace E\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nC : \u211d\u22650\u221e\nf : Set \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 f s \u2264 C\nh_F_lim : \u2200 (S : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), MeasurableSet (S n)) \u2192 Monotone S \u2192 f (\u22c3 n, S n) \u2264 \u2a06 n, f (S n)\nS : \u2115 \u2192 Set \u03b1 := spanningSets (Measure.trim \u03bc hm)\nhS_mono : Monotone S\nhS_meas : \u2200 (n : \u2115), MeasurableSet (S n)\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (S n) \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/CharP/Basic.lean", "full_name": "sub_pow_char_pow_of_commute", "start": [263, 1], "end": [269, 61], "traced_tactics": [{"tactic": "induction n with\n| zero => simp\n| succ n n_ih =>\n    rw [pow_succ', pow_mul, pow_mul, pow_mul, n_ih]\n    apply sub_pow_char_of_commute; apply Commute.pow_pow h", "annotated_tactic": ["induction n with\n  | <a>zero</a> => simp\n  | <a>succ</a> n n_ih =>\n      rw [<a>pow_succ'</a>, <a>pow_mul</a>, <a>pow_mul</a>, <a>pow_mul</a>, n_ih]\n      apply <a>sub_pow_char_of_commute</a>; apply <a>Commute.pow_pow</a> h", [{"full_name": "Nat.zero", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1041, 5], "def_end_pos": [1041, 9]}, {"full_name": "Nat.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}, {"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [213, 9], "def_end_pos": [213, 25]}, {"full_name": "pow_mul", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [125, 9], "def_end_pos": [125, 16]}, {"full_name": "pow_mul", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [125, 9], "def_end_pos": [125, 16]}, {"full_name": "pow_mul", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [125, 9], "def_end_pos": [125, 16]}, {"full_name": "sub_pow_char_of_commute", "def_path": "Mathlib/Algebra/CharP/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 32]}, {"full_name": "Commute.pow_pow", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [182, 9], "def_end_pos": [182, 16]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nn : \u2115\nx y : R\nh : Commute x y\n\u22a2 (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case zero\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\n\u22a2 (x - y) ^ p ^ Nat.zero = x ^ p ^ Nat.zero - y ^ p ^ Nat.zero", "state_after": "no goals"}, {"tactic": "rw [pow_succ', pow_mul, pow_mul, pow_mul, n_ih]", "annotated_tactic": ["rw [<a>pow_succ'</a>, <a>pow_mul</a>, <a>pow_mul</a>, <a>pow_mul</a>, n_ih]", [{"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [213, 9], "def_end_pos": [213, 25]}, {"full_name": "pow_mul", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [125, 9], "def_end_pos": [125, 16]}, {"full_name": "pow_mul", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [125, 9], "def_end_pos": [125, 16]}, {"full_name": "pow_mul", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [125, 9], "def_end_pos": [125, 16]}]], "state_before": "case succ\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\nn : \u2115\nn_ih : (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n\n\u22a2 (x - y) ^ p ^ Nat.succ n = x ^ p ^ Nat.succ n - y ^ p ^ Nat.succ n", "state_after": "case succ\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\nn : \u2115\nn_ih : (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n\n\u22a2 (x ^ p ^ n - y ^ p ^ n) ^ p = (x ^ p ^ n) ^ p - (y ^ p ^ n) ^ p"}, {"tactic": "apply sub_pow_char_of_commute", "annotated_tactic": ["apply <a>sub_pow_char_of_commute</a>", [{"full_name": "sub_pow_char_of_commute", "def_path": "Mathlib/Algebra/CharP/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 32]}]], "state_before": "case succ\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\nn : \u2115\nn_ih : (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n\n\u22a2 (x ^ p ^ n - y ^ p ^ n) ^ p = (x ^ p ^ n) ^ p - (y ^ p ^ n) ^ p", "state_after": "case succ.h\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\nn : \u2115\nn_ih : (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n\n\u22a2 Commute (x ^ p ^ n) (y ^ p ^ n)"}, {"tactic": "apply Commute.pow_pow h", "annotated_tactic": ["apply <a>Commute.pow_pow</a> h", [{"full_name": "Commute.pow_pow", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [182, 9], "def_end_pos": [182, 16]}]], "state_before": "case succ.h\nR : Type u_1\ninst\u271d\u00b2 : Ring R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nx y : R\nh : Commute x y\nn : \u2115\nn_ih : (x - y) ^ p ^ n = x ^ p ^ n - y ^ p ^ n\n\u22a2 Commute (x ^ p ^ n) (y ^ p ^ n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "interior_Ioc", "start": [2381, 1], "end": [2382, 82], "traced_tactics": [{"tactic": "rw [\u2190 Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]", "annotated_tactic": ["rw [\u2190 <a>Ioi_inter_Iic</a>, <a>interior_inter</a>, <a>interior_Ioi</a>, <a>interior_Iic</a>, <a>Ioi_inter_Iio</a>]", [{"full_name": "Set.Ioi_inter_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [626, 9], "def_end_pos": [626, 22]}, {"full_name": "interior_inter", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [360, 9], "def_end_pos": [360, 23]}, {"full_name": "interior_Ioi", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [341, 9], "def_end_pos": [341, 21]}, {"full_name": "interior_Iic", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [2366, 9], "def_end_pos": [2366, 21]}, {"full_name": "Set.Ioi_inter_Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [630, 9], "def_end_pos": [630, 22]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : DenselyOrdered \u03b1\na\u271d b\u271d : \u03b1\ns : Set \u03b1\ninst\u271d : NoMaxOrder \u03b1\na b : \u03b1\n\u22a2 interior (Ioc a b) = Ioo a b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Star/NonUnitalSubalgebra.lean", "full_name": "NonUnitalStarSubalgebra.toNonUnitalSubalgebra_inj", "start": [148, 1], "end": [150, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/RelIso/Set.lean", "full_name": "RelEmbedding.codRestrict_apply", "start": [98, 1], "end": [100, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Star/StarAlgHom.lean", "full_name": "NonUnitalStarAlgHom.fst_prod", "start": [552, 1], "end": [553, 11], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "R : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst\u271d\u2079 : Monoid R\ninst\u271d\u2078 : NonUnitalNonAssocSemiring A\ninst\u271d\u2077 : DistribMulAction R A\ninst\u271d\u2076 : Star A\ninst\u271d\u2075 : NonUnitalNonAssocSemiring B\ninst\u271d\u2074 : DistribMulAction R B\ninst\u271d\u00b3 : Star B\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring C\ninst\u271d\u00b9 : DistribMulAction R C\ninst\u271d : Star C\nf : A \u2192\u22c6\u2099\u2090[R] B\ng : A \u2192\u22c6\u2099\u2090[R] C\n\u22a2 comp (fst R B C) (prod f g) = f", "state_after": "case h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst\u271d\u2079 : Monoid R\ninst\u271d\u2078 : NonUnitalNonAssocSemiring A\ninst\u271d\u2077 : DistribMulAction R A\ninst\u271d\u2076 : Star A\ninst\u271d\u2075 : NonUnitalNonAssocSemiring B\ninst\u271d\u2074 : DistribMulAction R B\ninst\u271d\u00b3 : Star B\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring C\ninst\u271d\u00b9 : DistribMulAction R C\ninst\u271d : Star C\nf : A \u2192\u22c6\u2099\u2090[R] B\ng : A \u2192\u22c6\u2099\u2090[R] C\nx\u271d : A\n\u22a2 \u2191(comp (fst R B C) (prod f g)) x\u271d = \u2191f x\u271d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h\nR : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst\u271d\u2079 : Monoid R\ninst\u271d\u2078 : NonUnitalNonAssocSemiring A\ninst\u271d\u2077 : DistribMulAction R A\ninst\u271d\u2076 : Star A\ninst\u271d\u2075 : NonUnitalNonAssocSemiring B\ninst\u271d\u2074 : DistribMulAction R B\ninst\u271d\u00b3 : Star B\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring C\ninst\u271d\u00b9 : DistribMulAction R C\ninst\u271d : Star C\nf : A \u2192\u22c6\u2099\u2090[R] B\ng : A \u2192\u22c6\u2099\u2090[R] C\nx\u271d : A\n\u22a2 \u2191(comp (fst R B C) (prod f g)) x\u271d = \u2191f x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Compactness/Compact.lean", "full_name": "IsCompact.elim_nhds_subcover", "start": [176, 1], "end": [182, 46], "traced_tactics": [{"tactic": "rwa [Finset.set_biUnion_finset_image]", "annotated_tactic": ["rwa [<a>Finset.set_biUnion_finset_image</a>]", [{"full_name": "Finset.set_biUnion_finset_image", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [2146, 9], "def_end_pos": [2146, 33]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t\u271d : Set \u03b1\nhs : IsCompact s\nU : \u03b1 \u2192 Set \u03b1\nhU : \u2200 (x : \u03b1), x \u2208 s \u2192 U x \u2208 \ud835\udcdd x\nt : Finset \u2191s\nht : s \u2286 \u22c3 x \u2208 t, U \u2191x\n\u22a2 s \u2286 \u22c3 x \u2208 Finset.image Subtype.val t, U x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "Substring.Valid.next_stop", "start": [1008, 1], "end": [1009, 75], "traced_tactics": [{"tactic": "simp [h.bsize, h.next_stop]", "annotated_tactic": ["simp [h.bsize, h.next_stop]", []], "state_before": "x\u271d : Substring\nh\u271d : Valid x\u271d\nl m r : List Char\nh : ValidFor l m r x\u271d\n\u22a2 Substring.next x\u271d { byteIdx := Substring.bsize x\u271d } = { byteIdx := Substring.bsize x\u271d }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Sheaves/SheafCondition/Sites.lean", "full_name": "TopCat.Presheaf.covering_presieve_eq_self", "start": [90, 1], "end": [94, 94], "traced_tactics": [{"tactic": "funext Z", "annotated_tactic": ["funext Z", []], "state_before": "X : TopCat\nY : Opens \u2191X\nR : Presieve Y\n\u22a2 presieveOfCoveringAux (coveringOfPresieve Y R) Y = R", "state_after": "case h\nX : TopCat\nY : Opens \u2191X\nR : Presieve Y\nZ : Opens \u2191X\n\u22a2 presieveOfCoveringAux (coveringOfPresieve Y R) Y = R"}, {"tactic": "ext f", "annotated_tactic": ["ext f", []], "state_before": "case h\nX : TopCat\nY : Opens \u2191X\nR : Presieve Y\nZ : Opens \u2191X\n\u22a2 presieveOfCoveringAux (coveringOfPresieve Y R) Y = R", "state_after": "case h.h\nX : TopCat\nY : Opens \u2191X\nR : Presieve Y\nZ : Opens \u2191X\nf : Z \u27f6 Y\n\u22a2 f \u2208 presieveOfCoveringAux (coveringOfPresieve Y R) Y \u2194 f \u2208 R"}, {"tactic": "exact \u27e8fun \u27e8\u27e8_, f', h\u27e9, rfl\u27e9 => by rwa [Subsingleton.elim f f'], fun h => \u27e8\u27e8Z, f, h\u27e9, rfl\u27e9\u27e9", "annotated_tactic": ["exact \u27e8fun \u27e8\u27e8_, f', h\u27e9, <a>rfl</a>\u27e9 => by rwa [<a>Subsingleton.elim</a> f f'], fun h => \u27e8\u27e8Z, f, h\u27e9, <a>rfl</a>\u27e9\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Subsingleton.elim", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [873, 19], "def_end_pos": [873, 36]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case h.h\nX : TopCat\nY : Opens \u2191X\nR : Presieve Y\nZ : Opens \u2191X\nf : Z \u27f6 Y\n\u22a2 f \u2208 presieveOfCoveringAux (coveringOfPresieve Y R) Y \u2194 f \u2208 R", "state_after": "no goals"}, {"tactic": "rwa [Subsingleton.elim f f']", "annotated_tactic": ["rwa [<a>Subsingleton.elim</a> f f']", [{"full_name": "Subsingleton.elim", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [873, 19], "def_end_pos": [873, 36]}]], "state_before": "X : TopCat\nY : Opens \u2191X\nR : Presieve Y\nZ : Opens \u2191X\nf : Z \u27f6 Y\nx\u271d : f \u2208 presieveOfCoveringAux (coveringOfPresieve Y R) Y\nf' : Z \u27f6 Y\nh : R f'\n\u22a2 f \u2208 R", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Finrank.lean", "full_name": "FiniteDimensional.nontrivial_of_finrank_pos", "start": [110, 1], "end": [111, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean", "full_name": "ContDiff.rpow_const_of_ne", "start": [537, 1], "end": [539, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/Degree/Lemmas.lean", "full_name": "Polynomial.degree_sum_eq_of_disjoint", "start": [202, 1], "end": [225, 73], "traced_tactics": [{"tactic": "induction' s using Finset.induction_on with x s hx IH", "annotated_tactic": ["induction' s using <a>Finset.induction_on</a> with x s hx IH", [{"full_name": "Finset.induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1251, 19], "def_end_pos": [1251, 31]}]], "state_before": "R : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\n\u22a2 degree (Finset.sum s f) = sup s fun i => degree (f i)", "state_after": "case empty\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nh : Set.Pairwise {i | i \u2208 \u2205 \u2227 f i \u2260 0} (Ne on degree \u2218 f)\n\u22a2 degree (Finset.sum \u2205 f) = sup \u2205 fun i => degree (f i)\n\ncase insert\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nIH : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on degree \u2218 f) \u2192 degree (Finset.sum s f) = sup s fun i => degree (f i)\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\n\u22a2 degree (Finset.sum (insert x s) f) = sup (insert x s) fun i => degree (f i)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case empty\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nh : Set.Pairwise {i | i \u2208 \u2205 \u2227 f i \u2260 0} (Ne on degree \u2218 f)\n\u22a2 degree (Finset.sum \u2205 f) = sup \u2205 fun i => degree (f i)", "state_after": "no goals"}, {"tactic": "simp only [hx, Finset.sum_insert, not_false_iff, Finset.sup_insert]", "annotated_tactic": ["simp only [hx, <a>Finset.sum_insert</a>, <a>not_false_iff</a>, <a>Finset.sup_insert</a>]", [{"full_name": "Finset.sum_insert", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [316, 3], "def_end_pos": [316, 14]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "Finset.sup_insert", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [59, 9], "def_end_pos": [59, 19]}]], "state_before": "case insert\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nIH : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on degree \u2218 f) \u2192 degree (Finset.sum s f) = sup s fun i => degree (f i)\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\n\u22a2 degree (Finset.sum (insert x s) f) = sup (insert x s) fun i => degree (f i)", "state_after": "case insert\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nIH : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on degree \u2218 f) \u2192 degree (Finset.sum s f) = sup s fun i => degree (f i)\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)"}, {"tactic": "specialize IH (h.mono fun _ => by simp (config := { contextual := true }))", "annotated_tactic": ["specialize IH (h.mono fun _ => by simp (config := { contextual := <a>true</a> }))", [{"full_name": "Bool.true", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [549, 5], "def_end_pos": [549, 9]}]], "state_before": "case insert\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nIH : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on degree \u2218 f) \u2192 degree (Finset.sum s f) = sup s fun i => degree (f i)\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)", "state_after": "case insert\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)"}, {"tactic": "rcases lt_trichotomy (degree (f x)) (degree (s.sum f)) with (H | H | H)", "annotated_tactic": ["rcases <a>lt_trichotomy</a> (<a>degree</a> (f x)) (<a>degree</a> (s.sum f)) with (H | H | H)", [{"full_name": "lt_trichotomy", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [311, 9], "def_end_pos": [311, 22]}, {"full_name": "Polynomial.degree", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [53, 5], "def_end_pos": [53, 11]}, {"full_name": "Polynomial.degree", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [53, 5], "def_end_pos": [53, 11]}]], "state_before": "case insert\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)", "state_after": "case insert.inl\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nH : degree (f x) < degree (Finset.sum s f)\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)\n\ncase insert.inr.inl\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nH : degree (f x) = degree (Finset.sum s f)\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)\n\ncase insert.inr.inr\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nH : degree (Finset.sum s f) < degree (f x)\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)"}, {"tactic": "simp (config := { contextual := true })", "annotated_tactic": ["simp (config := { contextual := <a>true</a> })", [{"full_name": "Bool.true", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [549, 5], "def_end_pos": [549, 9]}]], "state_before": "R : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nIH : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on degree \u2218 f) \u2192 degree (Finset.sum s f) = sup s fun i => degree (f i)\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx\u271d : S\n\u22a2 x\u271d \u2208 {i | i \u2208 s \u2227 f i \u2260 0} \u2192 x\u271d \u2208 {i | i \u2208 insert x s \u2227 f i \u2260 0}", "state_after": "no goals"}, {"tactic": "rw [\u2190 IH, sup_eq_right.mpr H.le, degree_add_eq_right_of_degree_lt H]", "annotated_tactic": ["rw [\u2190 IH, sup_eq_right.mpr H.le, <a>degree_add_eq_right_of_degree_lt</a> H]", [{"full_name": "Polynomial.degree_add_eq_right_of_degree_lt", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [706, 9], "def_end_pos": [706, 41]}]], "state_before": "case insert.inl\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nH : degree (f x) < degree (Finset.sum s f)\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)", "state_after": "no goals"}, {"tactic": "rcases s.eq_empty_or_nonempty with (rfl | hs)", "annotated_tactic": ["rcases s.eq_empty_or_nonempty with (rfl | hs)", []], "state_before": "case insert.inr.inl\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nH : degree (f x) = degree (Finset.sum s f)\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)", "state_after": "case insert.inr.inl.inl\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\nhx : \u00acx \u2208 \u2205\nh : Set.Pairwise {i | i \u2208 insert x \u2205 \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum \u2205 f) = sup \u2205 fun i => degree (f i)\nH : degree (f x) = degree (Finset.sum \u2205 f)\n\u22a2 degree (f x + Finset.sum \u2205 fun x => f x) = degree (f x) \u2294 sup \u2205 fun i => degree (f i)\n\ncase insert.inr.inl.inr\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nH : degree (f x) = degree (Finset.sum s f)\nhs : Finset.Nonempty s\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)"}, {"tactic": "obtain \u27e8y, hy, hy'\u27e9 := Finset.exists_mem_eq_sup s hs fun i => degree (f i)", "annotated_tactic": ["obtain \u27e8y, hy, hy'\u27e9 := <a>Finset.exists_mem_eq_sup</a> s hs fun i => <a>degree</a> (f i)", [{"full_name": "Finset.exists_mem_eq_sup", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1221, 9], "def_end_pos": [1221, 26]}, {"full_name": "Polynomial.degree", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [53, 5], "def_end_pos": [53, 11]}]], "state_before": "case insert.inr.inl.inr\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nH : degree (f x) = degree (Finset.sum s f)\nhs : Finset.Nonempty s\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)", "state_after": "case insert.inr.inl.inr.intro.intro\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nH : degree (f x) = degree (Finset.sum s f)\nhs : Finset.Nonempty s\ny : S\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)"}, {"tactic": "rw [IH, hy'] at H", "annotated_tactic": ["rw [IH, hy'] at H", []], "state_before": "case insert.inr.inl.inr.intro.intro\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nH : degree (f x) = degree (Finset.sum s f)\nhs : Finset.Nonempty s\ny : S\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)", "state_after": "case insert.inr.inl.inr.intro.intro\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)"}, {"tactic": "by_cases hx0 : f x = 0", "annotated_tactic": ["by_cases hx0 : f x = 0", []], "state_before": "case insert.inr.inl.inr.intro.intro\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)", "state_after": "case pos\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : f x = 0\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)\n\ncase neg\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : \u00acf x = 0\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)"}, {"tactic": "have hy0 : f y \u2260 0 := by\n  contrapose! H\n  simpa [H, degree_eq_bot] using hx0", "annotated_tactic": ["have hy0 : f y \u2260 0 := by\n        contrapose! H\n        simpa [H, <a>degree_eq_bot</a>] using hx0", [{"full_name": "Polynomial.degree_eq_bot", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [116, 9], "def_end_pos": [116, 22]}]], "state_before": "case neg\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : \u00acf x = 0\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)", "state_after": "case neg\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : \u00acf x = 0\nhy0 : f y \u2260 0\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)"}, {"tactic": "refine' absurd H (h _ _ fun H => hx _)", "annotated_tactic": ["refine' <a>absurd</a> H (h _ _ fun H => hx _)", [{"full_name": "absurd", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [233, 21], "def_end_pos": [233, 27]}]], "state_before": "case neg\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : \u00acf x = 0\nhy0 : f y \u2260 0\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)", "state_after": "case neg.refine'_1\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : \u00acf x = 0\nhy0 : f y \u2260 0\n\u22a2 x \u2208 {i | i \u2208 insert x s \u2227 f i \u2260 0}\n\ncase neg.refine'_2\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : \u00acf x = 0\nhy0 : f y \u2260 0\n\u22a2 y \u2208 {i | i \u2208 insert x s \u2227 f i \u2260 0}\n\ncase neg.refine'_3\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH\u271d : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : \u00acf x = 0\nhy0 : f y \u2260 0\nH : x = y\n\u22a2 x \u2208 s"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case insert.inr.inl.inl\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\nhx : \u00acx \u2208 \u2205\nh : Set.Pairwise {i | i \u2208 insert x \u2205 \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum \u2205 f) = sup \u2205 fun i => degree (f i)\nH : degree (f x) = degree (Finset.sum \u2205 f)\n\u22a2 degree (f x + Finset.sum \u2205 fun x => f x) = degree (f x) \u2294 sup \u2205 fun i => degree (f i)", "state_after": "no goals"}, {"tactic": "simp [hx0, IH]", "annotated_tactic": ["simp [hx0, IH]", []], "state_before": "case pos\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : f x = 0\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)", "state_after": "no goals"}, {"tactic": "contrapose! H", "annotated_tactic": ["contrapose! H", []], "state_before": "R : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : \u00acf x = 0\n\u22a2 f y \u2260 0", "state_after": "R : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : \u00acf x = 0\nH : f y = 0\n\u22a2 degree (f x) \u2260 degree (f y)"}, {"tactic": "simpa [H, degree_eq_bot] using hx0", "annotated_tactic": ["simpa [H, <a>degree_eq_bot</a>] using hx0", [{"full_name": "Polynomial.degree_eq_bot", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [116, 9], "def_end_pos": [116, 22]}]], "state_before": "R : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : \u00acf x = 0\nH : f y = 0\n\u22a2 degree (f x) \u2260 degree (f y)", "state_after": "no goals"}, {"tactic": "simp [hx0]", "annotated_tactic": ["simp [hx0]", []], "state_before": "case neg.refine'_1\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : \u00acf x = 0\nhy0 : f y \u2260 0\n\u22a2 x \u2208 {i | i \u2208 insert x s \u2227 f i \u2260 0}", "state_after": "no goals"}, {"tactic": "simp [hy, hy0]", "annotated_tactic": ["simp [hy, hy0]", []], "state_before": "case neg.refine'_2\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : \u00acf x = 0\nhy0 : f y \u2260 0\n\u22a2 y \u2208 {i | i \u2208 insert x s \u2227 f i \u2260 0}", "state_after": "no goals"}, {"tactic": "exact H.symm \u25b8 hy", "annotated_tactic": ["exact H.symm \u25b8 hy", []], "state_before": "case neg.refine'_3\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nhs : Finset.Nonempty s\ny : S\nH\u271d : degree (f x) = degree (f y)\nhy : y \u2208 s\nhy' : (sup s fun i => degree (f i)) = degree (f y)\nhx0 : \u00acf x = 0\nhy0 : f y \u2260 0\nH : x = y\n\u22a2 x \u2208 s", "state_after": "no goals"}, {"tactic": "rw [\u2190 IH, sup_eq_left.mpr H.le, degree_add_eq_left_of_degree_lt H]", "annotated_tactic": ["rw [\u2190 IH, sup_eq_left.mpr H.le, <a>degree_add_eq_left_of_degree_lt</a> H]", [{"full_name": "Polynomial.degree_add_eq_left_of_degree_lt", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [699, 9], "def_end_pos": [699, 40]}]], "state_before": "case insert.inr.inr\nR : Type u\nS : Type v\n\u03b9 : Type w\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nf : S \u2192 R[X]\ns\u271d : Finset S\nh\u271d : Set.Pairwise {i | i \u2208 s\u271d \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nx : S\ns : Finset S\nhx : \u00acx \u2208 s\nh : Set.Pairwise {i | i \u2208 insert x s \u2227 f i \u2260 0} (Ne on degree \u2218 f)\nIH : degree (Finset.sum s f) = sup s fun i => degree (f i)\nH : degree (Finset.sum s f) < degree (f x)\n\u22a2 degree (f x + Finset.sum s fun x => f x) = degree (f x) \u2294 sup s fun i => degree (f i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/SuccPred/Relation.lean", "full_name": "transGen_of_succ_of_ne", "start": [77, 1], "end": [79, 92], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": 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"3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "full_name": "Metric.closure_subset_cthickening", "start": [1165, 1], "end": [1167, 45], "traced_tactics": [{"tactic": "rw [\u2190 cthickening_of_nonpos (min_le_right \u03b4 0)]", "annotated_tactic": ["rw [\u2190 <a>cthickening_of_nonpos</a> (<a>min_le_right</a> \u03b4 0)]", [{"full_name": "Metric.cthickening_of_nonpos", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1073, 9], "def_end_pos": [1073, 30]}, {"full_name": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}]], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\n\u22a2 closure E \u2286 cthickening \u03b4 E", "state_after": "\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\n\u22a2 cthickening (min \u03b4 0) E \u2286 cthickening \u03b4 E"}, {"tactic": "exact cthickening_mono (min_le_left \u03b4 0) E", "annotated_tactic": ["exact <a>cthickening_mono</a> (<a>min_le_left</a> \u03b4 0) E", [{"full_name": "Metric.cthickening_mono", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [1090, 9], "def_end_pos": [1090, 25]}, {"full_name": "min_le_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 20]}]], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : PseudoEMetricSpace \u03b1\n\u03b4\u271d \u03b5 : \u211d\ns t : Set \u03b1\nx : \u03b1\n\u03b4 : \u211d\nE : Set \u03b1\n\u22a2 cthickening (min \u03b4 0) E \u2286 cthickening \u03b4 E", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Ordinal/NaturalOps.lean", "full_name": "NatOrdinal.toOrdinal_symm_eq", "start": [82, 1], "end": [83, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/MonoidAlgebra/Division.lean", "full_name": "AddMonoidAlgebra.mul_of'_modOf", "start": [162, 1], "end": [168, 34], "traced_tactics": [{"tactic": "refine Finsupp.ext fun g' => ?_", "annotated_tactic": ["refine <a>Finsupp.ext</a> fun g' => ?_", [{"full_name": "Finsupp.ext", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [134, 9], "def_end_pos": [134, 12]}]], "state_before": "k : Type u_1\nG : Type u_2\ninst\u271d\u00b9 : Semiring k\ninst\u271d : AddCancelCommMonoid G\nx : k[G]\ng : G\n\u22a2 x * of' k G g %\u1d52\u1da0 g = 0", "state_after": "k : Type u_1\nG : Type u_2\ninst\u271d\u00b9 : Semiring k\ninst\u271d : AddCancelCommMonoid G\nx : k[G]\ng g' : G\n\u22a2 \u2191(x * of' k G g %\u1d52\u1da0 g) g' = \u21910 g'"}, {"tactic": "rw [Finsupp.zero_apply]", "annotated_tactic": ["rw [<a>Finsupp.zero_apply</a>]", [{"full_name": "Finsupp.zero_apply", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [172, 9], "def_end_pos": [172, 19]}]], "state_before": "k : Type u_1\nG : Type u_2\ninst\u271d\u00b9 : Semiring k\ninst\u271d : AddCancelCommMonoid G\nx : k[G]\ng g' : G\n\u22a2 \u2191(x * of' k G g %\u1d52\u1da0 g) g' = \u21910 g'", "state_after": "k : Type u_1\nG : Type u_2\ninst\u271d\u00b9 : Semiring k\ninst\u271d : AddCancelCommMonoid G\nx : k[G]\ng g' : G\n\u22a2 \u2191(x * of' k G g %\u1d52\u1da0 g) g' = 0"}, {"tactic": "obtain \u27e8d, rfl\u27e9 | h := em (\u2203 d, g' = g + d)", "annotated_tactic": ["obtain \u27e8d, rfl\u27e9 | h := <a>em</a> (\u2203 d, g' = g + d)", [{"full_name": "em", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [194, 7], "def_end_pos": [194, 9]}]], "state_before": "k : Type u_1\nG : Type u_2\ninst\u271d\u00b9 : Semiring k\ninst\u271d : AddCancelCommMonoid G\nx : k[G]\ng g' : G\n\u22a2 \u2191(x * of' k G g %\u1d52\u1da0 g) g' = 0", "state_after": "case inl.intro\nk : Type u_1\nG : Type u_2\ninst\u271d\u00b9 : Semiring k\ninst\u271d : AddCancelCommMonoid G\nx : k[G]\ng d : G\n\u22a2 \u2191(x * of' k G g %\u1d52\u1da0 g) (g + d) = 0\n\ncase inr\nk : Type u_1\nG : Type u_2\ninst\u271d\u00b9 : Semiring k\ninst\u271d : AddCancelCommMonoid G\nx : k[G]\ng g' : G\nh : \u00ac\u2203 d, g' = g + d\n\u22a2 \u2191(x * of' k G g %\u1d52\u1da0 g) g' = 0"}, {"tactic": "rw [modOf_apply_self_add]", "annotated_tactic": ["rw [<a>modOf_apply_self_add</a>]", [{"full_name": "AddMonoidAlgebra.modOf_apply_self_add", "def_path": "Mathlib/Algebra/MonoidAlgebra/Division.lean", "def_pos": [150, 9], "def_end_pos": [150, 29]}]], "state_before": "case inl.intro\nk : Type u_1\nG : Type u_2\ninst\u271d\u00b9 : Semiring k\ninst\u271d : AddCancelCommMonoid G\nx : k[G]\ng d : G\n\u22a2 \u2191(x * of' k G g %\u1d52\u1da0 g) (g + d) = 0", "state_after": "no goals"}, {"tactic": "rw [modOf_apply_of_not_exists_add _ _ _ h, of'_apply, mul_single_apply_of_not_exists_add]", "annotated_tactic": ["rw [<a>modOf_apply_of_not_exists_add</a> _ _ _ h, <a>of'_apply</a>, <a>mul_single_apply_of_not_exists_add</a>]", [{"full_name": "AddMonoidAlgebra.modOf_apply_of_not_exists_add", "def_path": "Mathlib/Algebra/MonoidAlgebra/Division.lean", "def_pos": [133, 9], "def_end_pos": [133, 38]}, {"full_name": "AddMonoidAlgebra.of'_apply", "def_path": "Mathlib/Algebra/MonoidAlgebra/Basic.lean", "def_pos": [1625, 9], "def_end_pos": [1625, 18]}, {"full_name": "AddMonoidAlgebra.mul_single_apply_of_not_exists_add", "def_path": "Mathlib/Algebra/MonoidAlgebra/Basic.lean", "def_pos": [1660, 9], "def_end_pos": [1660, 43]}]], "state_before": "case inr\nk : Type u_1\nG : Type u_2\ninst\u271d\u00b9 : Semiring k\ninst\u271d : AddCancelCommMonoid G\nx : k[G]\ng g' : G\nh : \u00ac\u2203 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\u03b2)\n\u22a2 f \u207b\u00b9' \u22c2\u2080 s = \u22c2 t \u2208 s, f \u207b\u00b9' t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "continuousAt_id", "start": [1737, 1], "end": [1738, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "full_name": "circleMap_not_mem_ball", "start": [131, 1], "end": [132, 30], "traced_tactics": [{"tactic": "simp [dist_eq, le_abs_self]", "annotated_tactic": ["simp [<a>dist_eq</a>, <a>le_abs_self</a>]", [{"full_name": "Complex.dist_eq", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 16]}, {"full_name": "le_abs_self", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [57, 9], "def_end_pos": [57, 20]}]], "state_before": "E : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : \u2102\nR \u03b8 : \u211d\n\u22a2 \u00accircleMap c R \u03b8 \u2208 ball c R", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Ideal/Operations.lean", "full_name": "Ideal.basisSpanSingleton_apply", "start": [2016, 1], "end": [2023, 26], "traced_tactics": [{"tactic": "simp only [basisSpanSingleton, Basis.map_apply, LinearEquiv.trans_apply,\n  Submodule.restrictScalarsEquiv_apply, LinearEquiv.ofInjective_apply, LinearEquiv.coe_ofEq_apply,\n  LinearEquiv.restrictScalars_apply, Algebra.coe_lmul_eq_mul, LinearMap.mul_apply']", "annotated_tactic": ["simp only [<a>basisSpanSingleton</a>, <a>Basis.map_apply</a>, <a>LinearEquiv.trans_apply</a>,\n    <a>Submodule.restrictScalarsEquiv_apply</a>, <a>LinearEquiv.ofInjective_apply</a>, <a>LinearEquiv.coe_ofEq_apply</a>,\n    <a>LinearEquiv.restrictScalars_apply</a>, <a>Algebra.coe_lmul_eq_mul</a>, <a>LinearMap.mul_apply'</a>]", [{"full_name": "Ideal.basisSpanSingleton", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [2004, 19], "def_end_pos": [2004, 37]}, {"full_name": "Basis.map_apply", "def_path": "Mathlib/LinearAlgebra/Basis.lean", "def_pos": [354, 9], "def_end_pos": [354, 18]}, {"full_name": "LinearEquiv.trans_apply", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [364, 9], "def_end_pos": [364, 20]}, {"full_name": "Submodule.restrictScalarsEquiv_apply", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [490, 3], "def_end_pos": [490, 8]}, {"full_name": "LinearEquiv.ofInjective_apply", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [2106, 9], "def_end_pos": [2106, 26]}, {"full_name": "LinearEquiv.coe_ofEq_apply", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [1927, 9], "def_end_pos": [1927, 23]}, {"full_name": "LinearEquiv.restrictScalars_apply", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [614, 3], "def_end_pos": [614, 8]}, {"full_name": "Algebra.coe_lmul_eq_mul", "def_path": "Mathlib/Algebra/Algebra/Bilinear.lean", "def_pos": [194, 9], "def_end_pos": [194, 39]}, {"full_name": "LinearMap.mul_apply'", "def_path": "Mathlib/Algebra/Algebra/Bilinear.lean", "def_pos": [73, 9], "def_end_pos": [73, 19]}]], "state_before": "\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nb : Basis \u03b9 R S\nx : S\nhx : x \u2260 0\ni : \u03b9\n\u22a2 \u2191(\u2191(basisSpanSingleton b hx) i) = x * \u2191b i", "state_after": "\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nb : Basis \u03b9 R S\nx : S\nhx : x \u2260 0\ni : \u03b9\n\u22a2 \u2191(\u2191(LinearEquiv.ofEq (LinearMap.range (\u2191(Algebra.lmul R S) x)) (Submodule.restrictScalars R (span {x}))\n            (_ : LinearMap.range (\u2191(Algebra.lmul R S) x) = Submodule.restrictScalars R (span {x})))\n        (\u2191(LinearEquiv.ofInjective (\u2191(Algebra.lmul R S) x) (_ : Function.Injective \u2191(\u2191(LinearMap.mul R S) x)))\n          (\u2191b i))) =\n    x * \u2191b i"}, {"tactic": "erw [LinearEquiv.coe_ofEq_apply, LinearEquiv.ofInjective_apply, Algebra.coe_lmul_eq_mul,\n  LinearMap.mul_apply']", "annotated_tactic": ["erw [<a>LinearEquiv.coe_ofEq_apply</a>, <a>LinearEquiv.ofInjective_apply</a>, <a>Algebra.coe_lmul_eq_mul</a>,\n    <a>LinearMap.mul_apply'</a>]", [{"full_name": "LinearEquiv.coe_ofEq_apply", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [1927, 9], "def_end_pos": [1927, 23]}, {"full_name": "LinearEquiv.ofInjective_apply", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [2106, 9], "def_end_pos": [2106, 26]}, {"full_name": "Algebra.coe_lmul_eq_mul", "def_path": "Mathlib/Algebra/Algebra/Bilinear.lean", "def_pos": [194, 9], "def_end_pos": [194, 39]}, {"full_name": "LinearMap.mul_apply'", "def_path": "Mathlib/Algebra/Algebra/Bilinear.lean", "def_pos": [73, 9], "def_end_pos": [73, 19]}]], "state_before": "\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nb : Basis \u03b9 R S\nx : S\nhx : x \u2260 0\ni : \u03b9\n\u22a2 \u2191(\u2191(LinearEquiv.ofEq (LinearMap.range (\u2191(Algebra.lmul R S) x)) (Submodule.restrictScalars R (span {x}))\n            (_ : LinearMap.range (\u2191(Algebra.lmul R S) x) = Submodule.restrictScalars R (span {x})))\n        (\u2191(LinearEquiv.ofInjective (\u2191(Algebra.lmul R S) x) (_ : Function.Injective \u2191(\u2191(LinearMap.mul R S) x)))\n          (\u2191b i))) =\n    x * \u2191b i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/NormedSpace/Multilinear.lean", "full_name": "ContinuousMultilinearMap.norm_mkPiAlgebraFin", "start": [873, 1], "end": [881, 9], "traced_tactics": [{"tactic": "cases n", "annotated_tactic": ["cases n", []], "state_before": "\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nn : \u2115\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nEi : Fin (Nat.succ n) \u2192 Type wEi\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i)\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u00b2 : NormedRing A\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A\ninst\u271d : NormOneClass A\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c n A\u2016 = 1", "state_after": "case zero\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd5c A\ninst\u271d\u00b2 : NormOneClass A\nEi : Fin (Nat.succ Nat.zero) \u2192 Type wEi\ninst\u271d\u00b9 : (i : Fin (Nat.succ Nat.zero)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d : (i : Fin (Nat.succ Nat.zero)) \u2192 NormedSpace \ud835\udd5c (Ei i)\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c Nat.zero A\u2016 = 1\n\ncase succ\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd5c A\ninst\u271d\u00b2 : NormOneClass A\nn\u271d : \u2115\nEi : Fin (Nat.succ (Nat.succ n\u271d)) \u2192 Type wEi\ninst\u271d\u00b9 : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedSpace \ud835\udd5c (Ei i)\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c (Nat.succ n\u271d) A\u2016 = 1"}, {"tactic": "rw [norm_mkPiAlgebraFin_zero]", "annotated_tactic": ["rw [<a>norm_mkPiAlgebraFin_zero</a>]", [{"full_name": "ContinuousMultilinearMap.norm_mkPiAlgebraFin_zero", "def_path": "Mathlib/Analysis/NormedSpace/Multilinear.lean", "def_pos": [862, 9], "def_end_pos": [862, 33]}]], "state_before": "case zero\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd5c A\ninst\u271d\u00b2 : NormOneClass A\nEi : Fin (Nat.succ Nat.zero) \u2192 Type wEi\ninst\u271d\u00b9 : (i : Fin (Nat.succ Nat.zero)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d : (i : Fin (Nat.succ Nat.zero)) \u2192 NormedSpace \ud835\udd5c (Ei i)\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c Nat.zero A\u2016 = 1", "state_after": "case zero\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd5c A\ninst\u271d\u00b2 : NormOneClass A\nEi : Fin (Nat.succ Nat.zero) \u2192 Type wEi\ninst\u271d\u00b9 : (i : Fin (Nat.succ Nat.zero)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d : (i : Fin (Nat.succ Nat.zero)) \u2192 NormedSpace \ud835\udd5c (Ei i)\n\u22a2 \u20161\u2016 = 1"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case zero\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd5c A\ninst\u271d\u00b2 : NormOneClass A\nEi : Fin (Nat.succ Nat.zero) \u2192 Type wEi\ninst\u271d\u00b9 : (i : Fin (Nat.succ Nat.zero)) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d : (i : Fin (Nat.succ Nat.zero)) \u2192 NormedSpace \ud835\udd5c (Ei i)\n\u22a2 \u20161\u2016 = 1", "state_after": "no goals"}, {"tactic": "refine' le_antisymm norm_mkPiAlgebraFin_succ_le _", "annotated_tactic": ["refine' <a>le_antisymm</a> <a>norm_mkPiAlgebraFin_succ_le</a> _", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "ContinuousMultilinearMap.norm_mkPiAlgebraFin_succ_le", "def_path": "Mathlib/Analysis/NormedSpace/Multilinear.lean", "def_pos": [843, 9], "def_end_pos": [843, 36]}]], "state_before": "case succ\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd5c A\ninst\u271d\u00b2 : NormOneClass A\nn\u271d : \u2115\nEi : Fin (Nat.succ (Nat.succ n\u271d)) \u2192 Type wEi\ninst\u271d\u00b9 : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedSpace \ud835\udd5c (Ei i)\n\u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c (Nat.succ n\u271d) A\u2016 = 1", "state_after": "case succ\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd5c A\ninst\u271d\u00b2 : NormOneClass A\nn\u271d : \u2115\nEi : Fin (Nat.succ (Nat.succ n\u271d)) \u2192 Type wEi\ninst\u271d\u00b9 : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedSpace \ud835\udd5c (Ei i)\n\u22a2 1 \u2264 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c (Nat.succ n\u271d) A\u2016"}, {"tactic": "convert ratio_le_op_norm (ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c (Nat.succ _) A)\n  fun _ => 1", "annotated_tactic": ["convert <a>ratio_le_op_norm</a> (<a>ContinuousMultilinearMap.mkPiAlgebraFin</a> \ud835\udd5c (<a>Nat.succ</a> _) A)\n      fun _ => 1", [{"full_name": "ContinuousMultilinearMap.ratio_le_op_norm", "def_path": 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i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd5c A\ninst\u271d\u00b2 : NormOneClass A\nn\u271d : \u2115\nEi : Fin (Nat.succ (Nat.succ n\u271d)) \u2192 Type wEi\ninst\u271d\u00b9 : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedSpace \ud835\udd5c (Ei i)\n\u22a2 1 \u2264 \u2016ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c (Nat.succ n\u271d) A\u2016", "state_after": "case h.e'_3\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd5c A\ninst\u271d\u00b2 : NormOneClass A\nn\u271d : \u2115\nEi : Fin (Nat.succ (Nat.succ n\u271d)) \u2192 Type wEi\ninst\u271d\u00b9 : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedSpace \ud835\udd5c (Ei i)\n\u22a2 1 = \u2016\u2191(ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c (Nat.succ n\u271d) A) fun x => 1\u2016 / \u220f i : Fin (Nat.succ n\u271d), \u20161\u2016"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_3\n\ud835\udd5c : Type u\n\u03b9 : Type v\n\u03b9' : Type v'\nE : \u03b9 \u2192 Type wE\nE\u2081 : \u03b9 \u2192 Type wE\u2081\nE' : \u03b9' \u2192 Type wE'\nG : Type wG\nG' : Type wG'\ninst\u271d\u00b9\u2077 : Fintype \u03b9\ninst\u271d\u00b9\u2076 : Fintype \u03b9'\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i)\ninst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i)\ninst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i)\ninst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i)\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedSpace \ud835\udd5c G\ninst\u271d\u2076 : NormedAddCommGroup G'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G'\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd5c A\ninst\u271d\u00b2 : NormOneClass A\nn\u271d : \u2115\nEi : Fin (Nat.succ (Nat.succ n\u271d)) \u2192 Type wEi\ninst\u271d\u00b9 : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedAddCommGroup (Ei i)\ninst\u271d : (i : Fin (Nat.succ (Nat.succ n\u271d))) \u2192 NormedSpace \ud835\udd5c (Ei i)\n\u22a2 1 = \u2016\u2191(ContinuousMultilinearMap.mkPiAlgebraFin \ud835\udd5c (Nat.succ n\u271d) A) fun x => 1\u2016 / \u220f i : Fin (Nat.succ n\u271d), \u20161\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Bases.lean", "full_name": "Disjoint.exists_mem_filter_basis", "start": [652, 1], "end": [654, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/VonMangoldt.lean", "full_name": "Nat.ArithmeticFunction.vonMangoldt_eq_zero_iff", "start": [103, 1], "end": [104, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Cofinite.lean", "full_name": "Filter.le_cofinite_iff_compl_singleton_mem", "start": [94, 1], "end": [97, 23], "traced_tactics": [{"tactic": "refine' \u27e8fun h x => h (finite_singleton x).compl_mem_cofinite, fun h s (hs : s\u1d9c.Finite) => _\u27e9", "annotated_tactic": ["refine' \u27e8fun h x => h (<a>finite_singleton</a> x).<a>compl_mem_cofinite</a>, fun h s (hs : s\u1d9c.<a>Finite</a>) => _\u27e9", [{"full_name": "Set.finite_singleton", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [848, 9], "def_end_pos": [848, 25]}, {"full_name": "Set.Finite.compl_mem_cofinite", "def_path": "Mathlib/Order/Filter/Cofinite.lean", "def_pos": [72, 9], "def_end_pos": [72, 45]}, {"full_name": "Set.Finite", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [61, 11], "def_end_pos": [61, 17]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\n\u22a2 l \u2264 cofinite \u2194 \u2200 (x : \u03b1), {x}\u1d9c \u2208 l", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nl : Filter \u03b1\nh : \u2200 (x : \u03b1), {x}\u1d9c \u2208 l\ns : Set \u03b1\nhs : Set.Finite s\u1d9c\n\u22a2 s \u2208 l"}, 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"https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Hom/Basic.lean", "full_name": "codisjoint_map_orderIso_iff", "start": [1227, 1], "end": [1230, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Matrix/Basic.lean", "full_name": "Matrix.neg_mul", "start": [1170, 11], "end": [1172, 23], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o \u2192 Type u_5\nn' : o \u2192 Type u_6\nR : Type u_7\nS : Type u_8\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type u_9\ninst\u271d\u00b9 : NonUnitalNonAssocRing \u03b1\ninst\u271d : Fintype n\nM : Matrix m n \u03b1\nN : Matrix n o \u03b1\n\u22a2 -M * N = -(M * N)", "state_after": "case a.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o \u2192 Type u_5\nn' : o \u2192 Type u_6\nR : Type u_7\nS : Type u_8\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type u_9\ninst\u271d\u00b9 : NonUnitalNonAssocRing \u03b1\ninst\u271d : Fintype n\nM : Matrix m n \u03b1\nN : Matrix n o \u03b1\ni\u271d : m\nx\u271d : o\n\u22a2 (-M * N) i\u271d x\u271d = (-(M * N)) i\u271d x\u271d"}, {"tactic": "apply neg_dotProduct", "annotated_tactic": ["apply <a>neg_dotProduct</a>", [{"full_name": "Matrix.neg_dotProduct", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [858, 9], "def_end_pos": [858, 23]}]], "state_before": "case a.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o \u2192 Type u_5\nn' : o \u2192 Type u_6\nR : Type u_7\nS : Type u_8\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type u_9\ninst\u271d\u00b9 : NonUnitalNonAssocRing \u03b1\ninst\u271d : Fintype n\nM : Matrix m n \u03b1\nN : Matrix n o \u03b1\ni\u271d : m\nx\u271d : o\n\u22a2 (-M * N) i\u271d x\u271d = (-(M * N)) i\u271d x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.preimage_pi", "start": [798, 1], "end": [800, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/Rotate.lean", "full_name": "List.rotate'_nil", "start": [47, 1], "end": [47, 81], "traced_tactics": [{"tactic": "cases n <;> rfl", "annotated_tactic": ["cases n <;> rfl", []], "state_before": "\u03b1 : Type u\nn : \u2115\n\u22a2 rotate' [] n = []", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Ordinal/NaturalOps.lean", "full_name": "Ordinal.toNatOrdinal_zero", "start": [163, 1], "end": [164, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Alternating/Basic.lean", "full_name": "AlternatingMap.map_eq_zero_of_eq", "start": [209, 1], "end": [210, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Field/Basic.lean", "full_name": "neg_div", "start": [113, 1], "end": [114, 63], "traced_tactics": [{"tactic": "rw [neg_eq_neg_one_mul, mul_div_assoc, \u2190 neg_eq_neg_one_mul]", "annotated_tactic": ["rw [<a>neg_eq_neg_one_mul</a>, <a>mul_div_assoc</a>, \u2190 <a>neg_eq_neg_one_mul</a>]", [{"full_name": "neg_eq_neg_one_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 27]}, {"full_name": "mul_div_assoc", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 22]}, {"full_name": "neg_eq_neg_one_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9 : DivisionMonoid K\ninst\u271d : HasDistribNeg K\na\u271d b\u271d a b : K\n\u22a2 -b / a = -(b / a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "full_name": "MeasureTheory.integrableOn_condexpL2_of_measure_ne_top", "start": [84, 1], "end": [87, 8], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/WithZero.lean", "full_name": "mul_lt_mul\u2080", "start": [189, 1], "end": [190, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/ZeroAndBoundedAtFilter.lean", "full_name": "Filter.BoundedAtFilter.smul", "start": [101, 1], "end": [103, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Lipschitz.lean", "full_name": "LipschitzWith.const_min", "start": [481, 1], "end": [482, 45], "traced_tactics": [{"tactic": "simpa only [min_comm] using hf.min_const a", "annotated_tactic": ["simpa only [<a>min_comm</a>] using hf.min_const a", [{"full_name": "min_comm", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [80, 9], "def_end_pos": [80, 17]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : PseudoEMetricSpace \u03b1\nf g : \u03b1 \u2192 \u211d\nKf Kg : \u211d\u22650\nhf : LipschitzWith Kf f\na : \u211d\n\u22a2 LipschitzWith Kf fun x => min a (f x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Complement.lean", "full_name": "Subgroup.mem_rightTransversals_iff_existsUnique_quotient_mk''_eq", "start": [250, 1], "end": [255, 79], "traced_tactics": [{"tactic": "simp_rw [mem_rightTransversals_iff_existsUnique_mul_inv_mem, SetLike.mem_coe, \u2190\n  QuotientGroup.rightRel_apply, \u2190 Quotient.eq'']", "annotated_tactic": ["simp_rw [<a>mem_rightTransversals_iff_existsUnique_mul_inv_mem</a>, <a>SetLike.mem_coe</a>, \u2190\n    <a>QuotientGroup.rightRel_apply</a>, \u2190 <a>Quotient.eq''</a>]", [{"full_name": "Subgroup.mem_rightTransversals_iff_existsUnique_mul_inv_mem", "def_path": "Mathlib/GroupTheory/Complement.lean", "def_pos": [224, 9], "def_end_pos": [224, 59]}, {"full_name": "SetLike.mem_coe", "def_path": "Mathlib/Data/SetLike/Basic.lean", "def_pos": [163, 9], "def_end_pos": [163, 16]}, {"full_name": "QuotientGroup.rightRel_apply", "def_path": "Mathlib/GroupTheory/Coset.lean", "def_pos": [374, 9], "def_end_pos": [374, 23]}, {"full_name": "Quotient.eq''", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [783, 19], "def_end_pos": [783, 23]}]], "state_before": "G : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\n\u22a2 S \u2208 rightTransversals \u2191H \u2194 \u2200 (q : Quotient (QuotientGroup.rightRel H)), \u2203! s, Quotient.mk'' \u2191s = q", "state_after": "G : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\n\u22a2 (\u2200 (g : G), \u2203! s, Quotient.mk'' \u2191s = Quotient.mk'' g) \u2194\n    \u2200 (q : Quotient (QuotientGroup.rightRel H)), \u2203! s, Quotient.mk'' \u2191s = q"}, {"tactic": "exact \u27e8fun h q => Quotient.inductionOn' q h, fun h g => h (Quotient.mk'' g)\u27e9", "annotated_tactic": ["exact \u27e8fun h q => <a>Quotient.inductionOn'</a> q h, fun h g => h (<a>Quotient.mk''</a> g)\u27e9", [{"full_name": "Quotient.inductionOn'", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [671, 19], "def_end_pos": [671, 31]}, {"full_name": "Quotient.mk''", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [608, 15], "def_end_pos": [608, 19]}]], "state_before": "G : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\n\u22a2 (\u2200 (g : G), \u2203! s, Quotient.mk'' \u2191s = Quotient.mk'' g) \u2194\n    \u2200 (q : Quotient (QuotientGroup.rightRel H)), \u2203! s, Quotient.mk'' \u2191s = q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Subsemiring/Pointwise.lean", "full_name": "Subsemiring.le_pointwise_smul_iff\u2080", "start": [169, 1], "end": [171, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Module/Submodule/Lattice.lean", "full_name": "Submodule.restrictScalars_top", "start": [169, 1], "end": [170, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.eq_empty_of_isEmpty", "start": [594, 1], "end": [595, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "full_name": "CategoryTheory.Sieve.pullback_pushforward_le", "start": [532, 1], "end": [533, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Sum/Order.lean", "full_name": "Sum.denselyOrdered_iff", "start": [266, 1], "end": [277, 50], "traced_tactics": [{"tactic": "obtain \u27e8c | c, ha, hb\u27e9 := @exists_between (Sum \u03b1 \u03b2) _ _ _ _ (inl_lt_inl_iff.2 h)", "annotated_tactic": ["obtain \u27e8c | c, ha, hb\u27e9 := @<a>exists_between</a> (<a>Sum</a> \u03b1 \u03b2) _ _ _ _ (<a>inl_lt_inl_iff</a>.2 h)", [{"full_name": "exists_between", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [1294, 9], "def_end_pos": [1294, 23]}, {"full_name": "Sum", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [102, 11], "def_end_pos": [102, 14]}, {"full_name": "Sum.inl_lt_inl_iff", "def_path": "Mathlib/Data/Sum/Order.lean", "def_pos": [139, 9], "def_end_pos": [139, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : DenselyOrdered (\u03b1 \u2295 \u03b2)\na b : \u03b1\nh : a < b\n\u22a2 \u2203 a_1, a < a_1 \u2227 a_1 < b", "state_after": "case intro.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : DenselyOrdered (\u03b1 \u2295 \u03b2)\na b : \u03b1\nh : a < b\nc : \u03b1\nha : inl a < inl c\nhb : inl c < inl b\n\u22a2 \u2203 a_1, a < a_1 \u2227 a_1 < b\n\ncase intro.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : DenselyOrdered (\u03b1 \u2295 \u03b2)\na b : \u03b1\nh : a < b\nc : \u03b2\nha : inl a < inr c\nhb : inr c < inl b\n\u22a2 \u2203 a_1, a < a_1 \u2227 a_1 < b"}, {"tactic": "exact \u27e8c, inl_lt_inl_iff.1 ha, inl_lt_inl_iff.1 hb\u27e9", "annotated_tactic": ["exact \u27e8c, <a>inl_lt_inl_iff</a>.1 ha, <a>inl_lt_inl_iff</a>.1 hb\u27e9", [{"full_name": "Sum.inl_lt_inl_iff", "def_path": "Mathlib/Data/Sum/Order.lean", "def_pos": [139, 9], "def_end_pos": [139, 23]}, {"full_name": "Sum.inl_lt_inl_iff", "def_path": "Mathlib/Data/Sum/Order.lean", "def_pos": [139, 9], "def_end_pos": [139, 23]}]], "state_before": "case intro.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : DenselyOrdered (\u03b1 \u2295 \u03b2)\na b : \u03b1\nh : a < b\nc : \u03b1\nha : inl a < inl c\nhb : inl c < inl b\n\u22a2 \u2203 a_1, a < a_1 \u2227 a_1 < b", "state_after": "no goals"}, {"tactic": "exact (not_inl_lt_inr ha).elim", "annotated_tactic": ["exact (<a>not_inl_lt_inr</a> ha).<a>elim</a>", [{"full_name": "Sum.not_inl_lt_inr", "def_path": "Mathlib/Data/Sum/Order.lean", "def_pos": [154, 9], "def_end_pos": [154, 23]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case intro.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : DenselyOrdered (\u03b1 \u2295 \u03b2)\na b : \u03b1\nh : a < b\nc : \u03b2\nha : inl a < inr c\nhb : inr c < inl b\n\u22a2 \u2203 a_1, a < a_1 \u2227 a_1 < b", "state_after": "no goals"}, {"tactic": "obtain \u27e8c | c, ha, hb\u27e9 := @exists_between (Sum \u03b1 \u03b2) _ _ _ _ (inr_lt_inr_iff.2 h)", "annotated_tactic": ["obtain \u27e8c | c, ha, hb\u27e9 := @<a>exists_between</a> (<a>Sum</a> \u03b1 \u03b2) _ _ _ _ (<a>inr_lt_inr_iff</a>.2 h)", [{"full_name": "exists_between", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [1294, 9], "def_end_pos": [1294, 23]}, {"full_name": "Sum", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [102, 11], "def_end_pos": [102, 14]}, {"full_name": "Sum.inr_lt_inr_iff", "def_path": "Mathlib/Data/Sum/Order.lean", "def_pos": [144, 9], "def_end_pos": [144, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : DenselyOrdered (\u03b1 \u2295 \u03b2)\na b : \u03b2\nh : a < b\n\u22a2 \u2203 a_1, a < a_1 \u2227 a_1 < b", "state_after": "case intro.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : DenselyOrdered (\u03b1 \u2295 \u03b2)\na b : \u03b2\nh : a < b\nc : \u03b1\nha : inr a < inl c\nhb : inl c < inr b\n\u22a2 \u2203 a_1, a < a_1 \u2227 a_1 < b\n\ncase intro.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : DenselyOrdered (\u03b1 \u2295 \u03b2)\na b : \u03b2\nh : a < b\nc : \u03b2\nha : inr a < inr c\nhb : inr c < inr b\n\u22a2 \u2203 a_1, a < a_1 \u2227 a_1 < b"}, {"tactic": "exact (not_inl_lt_inr hb).elim", "annotated_tactic": ["exact (<a>not_inl_lt_inr</a> hb).<a>elim</a>", [{"full_name": "Sum.not_inl_lt_inr", "def_path": "Mathlib/Data/Sum/Order.lean", "def_pos": [154, 9], "def_end_pos": [154, 23]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case intro.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : DenselyOrdered (\u03b1 \u2295 \u03b2)\na b : \u03b2\nh : a < b\nc : \u03b1\nha : inr a < inl c\nhb : inl c < inr b\n\u22a2 \u2203 a_1, a < a_1 \u2227 a_1 < b", "state_after": "no goals"}, {"tactic": "exact \u27e8c, inr_lt_inr_iff.1 ha, inr_lt_inr_iff.1 hb\u27e9", "annotated_tactic": ["exact \u27e8c, <a>inr_lt_inr_iff</a>.1 ha, <a>inr_lt_inr_iff</a>.1 hb\u27e9", [{"full_name": "Sum.inr_lt_inr_iff", "def_path": "Mathlib/Data/Sum/Order.lean", "def_pos": [144, 9], "def_end_pos": [144, 23]}, {"full_name": "Sum.inr_lt_inr_iff", "def_path": "Mathlib/Data/Sum/Order.lean", "def_pos": [144, 9], "def_end_pos": [144, 23]}]], "state_before": "case intro.inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : LT \u03b1\ninst\u271d : LT \u03b2\nx\u271d : DenselyOrdered (\u03b1 \u2295 \u03b2)\na b : \u03b2\nh : a < b\nc : \u03b2\nha : inr a < inr c\nhb : inr c < inr b\n\u22a2 \u2203 a_1, a < a_1 \u2227 a_1 < b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/UniformSpace/Basic.lean", "full_name": "map_uniformity_set_coe", "start": [1487, 1], "end": [1489, 71], "traced_tactics": [{"tactic": "rw [uniformity_setCoe, map_comap, range_prod_map, Subtype.range_val]", "annotated_tactic": ["rw [<a>uniformity_setCoe</a>, <a>map_comap</a>, <a>range_prod_map</a>, <a>Subtype.range_val</a>]", [{"full_name": "uniformity_setCoe", "def_path": "Mathlib/Topology/UniformSpace/Basic.lean", "def_pos": [1481, 9], "def_end_pos": [1481, 26]}, {"full_name": "Filter.map_comap", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2323, 9], "def_end_pos": [2323, 18]}, {"full_name": "Set.range_prod_map", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [314, 9], "def_end_pos": [314, 23]}, {"full_name": "Subtype.range_val", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1417, 9], "def_end_pos": [1417, 18]}]], "state_before": "\u03b1 : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort u_1\ns : Set \u03b1\ninst\u271d : UniformSpace \u03b1\n\u22a2 map (Prod.map Subtype.val Subtype.val) (\ud835\udce4 \u2191s) = \ud835\udce4 \u03b1 \u2293 \ud835\udcdf (s \u00d7\u02e2 s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Homeomorph.lean", "full_name": "Homeomorph.isPreconnected_image", "start": [287, 1], "end": [291, 51], "traced_tactics": [{"tactic": "simpa only [image_symm, preimage_image]\nusing hs.image _ h.symm.continuous.continuousOn", "annotated_tactic": ["simpa only [<a>image_symm</a>, <a>preimage_image</a>]\n    using hs.image _ h.symm.continuous.continuousOn", [{"full_name": "Homeomorph.image_symm", "def_path": "Mathlib/Topology/Homeomorph.lean", "def_pos": [210, 9], "def_end_pos": [210, 19]}, {"full_name": "Homeomorph.preimage_image", "def_path": "Mathlib/Topology/Homeomorph.lean", "def_pos": [224, 9], "def_end_pos": [224, 23]}]], "state_before": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : TopologicalSpace Z\nX' : Type u_4\nY' : Type u_5\ninst\u271d\u00b9 : TopologicalSpace X'\ninst\u271d : TopologicalSpace Y'\ns : Set X\nh : X \u2243\u209c Y\nhs : IsPreconnected (\u2191h '' s)\n\u22a2 IsPreconnected s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Instances/TrivSqZeroExt.lean", "full_name": "TrivSqZeroExt.nhds_def", "start": [45, 1], "end": [47, 21], "traced_tactics": [{"tactic": "cases x", "annotated_tactic": ["cases x", []], "state_before": "\u03b1 : Type u_1\nS : Type u_2\nR : Type u_3\nM : Type u_4\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalSpace M\nx : tsze R M\n\u22a2 nhds x = Filter.prod (nhds (fst x)) (nhds (snd x))", "state_after": "case mk\n\u03b1 : Type u_1\nS : Type u_2\nR : Type u_3\nM : Type u_4\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalSpace M\nfst\u271d : R\nsnd\u271d : M\n\u22a2 nhds (fst\u271d, snd\u271d) = Filter.prod (nhds (fst (fst\u271d, snd\u271d))) (nhds (snd (fst\u271d, snd\u271d)))"}, {"tactic": "exact nhds_prod_eq", "annotated_tactic": ["exact <a>nhds_prod_eq</a>", [{"full_name": "nhds_prod_eq", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [513, 9], "def_end_pos": [513, 21]}]], "state_before": "case mk\n\u03b1 : Type u_1\nS : Type u_2\nR : Type u_3\nM : Type u_4\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalSpace M\nfst\u271d : R\nsnd\u271d : M\n\u22a2 nhds (fst\u271d, snd\u271d) = Filter.prod (nhds (fst (fst\u271d, snd\u271d))) (nhds (snd (fst\u271d, snd\u271d)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Connected/PathConnected.lean", "full_name": "Path.coe_mk", "start": [143, 1], "end": [144, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Matrix/Trace.lean", "full_name": "Matrix.trace_fin_two", "start": [201, 1], "end": [202, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "full_name": "Polynomial.degree_sub_le", "start": [1354, 1], "end": [1355, 55], "traced_tactics": [{"tactic": "simpa only [degree_neg q] using degree_add_le p (-q)", "annotated_tactic": ["simpa only [<a>degree_neg</a> q] using <a>degree_add_le</a> p (-q)", [{"full_name": "Polynomial.degree_neg", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [537, 9], "def_end_pos": [537, 19]}, {"full_name": "Polynomial.degree_add_le", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [637, 9], "def_end_pos": [637, 22]}]], "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Ring R\np\u271d q\u271d p q : R[X]\n\u22a2 degree (p - q) \u2264 max (degree p) (degree q)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Group/Basic.lean", "full_name": "mul_one_eq_id", "start": [84, 1], "end": [85, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Field/Basic.lean", "full_name": "inv_mul_lt_iff'", "start": [199, 1], "end": [199, 100], "traced_tactics": [{"tactic": "rw [inv_mul_lt_iff h, mul_comm]", "annotated_tactic": ["rw [<a>inv_mul_lt_iff</a> h, <a>mul_comm</a>]", [{"full_name": "inv_mul_lt_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [194, 9], "def_end_pos": [194, 23]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh : 0 < b\n\u22a2 b\u207b\u00b9 * a < c \u2194 a < c * b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "Iic_mem_nhds", "start": [367, 1], "end": [368, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/LinearPMap.lean", "full_name": "LinearPMap.inverse_range", "start": [1111, 1], "end": [1114, 6], "traced_tactics": [{"tactic": "rw [inverse, Submodule.toLinearPMap_range _ (mem_inverse_graph_snd_eq_zero hf),\n  \u2190 graph_map_fst_eq_domain, \u2190 LinearEquiv.snd_comp_prodComm, Submodule.map_comp]", "annotated_tactic": ["rw [<a>inverse</a>, <a>Submodule.toLinearPMap_range</a> _ (<a>mem_inverse_graph_snd_eq_zero</a> hf),\n    \u2190 <a>graph_map_fst_eq_domain</a>, \u2190 <a>LinearEquiv.snd_comp_prodComm</a>, <a>Submodule.map_comp</a>]", [{"full_name": "LinearPMap.inverse", "def_path": "Mathlib/LinearAlgebra/LinearPMap.lean", "def_pos": [1082, 19], "def_end_pos": [1082, 26]}, {"full_name": "Submodule.toLinearPMap_range", "def_path": "Mathlib/LinearAlgebra/LinearPMap.lean", "def_pos": [1068, 9], "def_end_pos": [1068, 27]}, {"full_name": "LinearPMap.mem_inverse_graph_snd_eq_zero", "def_path": "Mathlib/LinearAlgebra/LinearPMap.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 38]}, {"full_name": "LinearPMap.graph_map_fst_eq_domain", "def_path": "Mathlib/LinearAlgebra/LinearPMap.lean", "def_pos": [786, 9], "def_end_pos": [786, 32]}, {"full_name": "LinearEquiv.snd_comp_prodComm", "def_path": "Mathlib/LinearAlgebra/Prod.lean", "def_pos": [756, 9], "def_end_pos": [756, 26]}, {"full_name": "Submodule.map_comp", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [615, 9], "def_end_pos": [615, 17]}]], "state_before": "R : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nhf : LinearMap.ker f.toFun = \u22a5\n\u22a2 LinearMap.range (inverse f).toFun = f.domain", "state_after": "R : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nhf : LinearMap.ker f.toFun = \u22a5\n\u22a2 Submodule.map (LinearMap.snd R F E) (Submodule.map (LinearEquiv.prodComm R E F) (graph f)) =\n    Submodule.map (LinearMap.snd R F E) (Submodule.map (\u2191(LinearEquiv.prodComm R E F)) (graph f))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "R : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\nf : E \u2192\u2097.[R] F\nhf : LinearMap.ker f.toFun = \u22a5\n\u22a2 Submodule.map (LinearMap.snd R F E) (Submodule.map (LinearEquiv.prodComm R E F) (graph f)) =\n    Submodule.map (LinearMap.snd R F E) (Submodule.map (\u2191(LinearEquiv.prodComm R E F)) (graph f))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "full_name": "Zsqrtd.norm_int_cast", "start": [532, 1], "end": [532, 73], "traced_tactics": [{"tactic": "simp [norm]", "annotated_tactic": ["simp [<a>norm</a>]", [{"full_name": "Zsqrtd.norm", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [515, 5], "def_end_pos": [515, 9]}]], "state_before": "d n : \u2124\n\u22a2 norm \u2191n = n * n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.eventually_ne_atBot", "start": [229, 1], "end": [230, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Contraction.lean", "full_name": "lTensorHomEquivHomLTensor_apply", "start": [288, 1], "end": [290, 76], "traced_tactics": [{"tactic": "rw [\u2190 LinearEquiv.coe_toLinearMap, lTensorHomEquivHomLTensor_toLinearMap]", "annotated_tactic": ["rw [\u2190 <a>LinearEquiv.coe_toLinearMap</a>, <a>lTensorHomEquivHomLTensor_toLinearMap</a>]", [{"full_name": "LinearEquiv.coe_toLinearMap", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [221, 9], "def_end_pos": [221, 24]}, {"full_name": "lTensorHomEquivHomLTensor_toLinearMap", "def_path": "Mathlib/LinearAlgebra/Contraction.lean", "def_pos": [256, 9], "def_end_pos": [256, 46]}]], "state_before": "\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\nx : P \u2297[R] (M \u2192\u2097[R] Q)\n\u22a2 \u2191(lTensorHomEquivHomLTensor R M P Q) x = \u2191(lTensorHomToHomLTensor R M P Q) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Fintype/Basic.lean", "full_name": "Set.disjoint_toFinset", "start": [700, 1], "end": [701, 97], "traced_tactics": [{"tactic": "simp only [\u2190 disjoint_coe, coe_toFinset]", "annotated_tactic": ["simp only [\u2190 <a>disjoint_coe</a>, <a>coe_toFinset</a>]", [{"full_name": "Finset.disjoint_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1004, 9], "def_end_pos": [1004, 21]}, {"full_name": "Set.coe_toFinset", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [649, 9], "def_end_pos": [649, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns t : Set \u03b1\ninst\u271d\u00b9 : Fintype \u2191s\ninst\u271d : Fintype \u2191t\n\u22a2 Disjoint (toFinset s) (toFinset t) \u2194 Disjoint s t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Order/IntermediateValue.lean", "full_name": "intermediate_value_Ioc", "start": [562, 1], "end": [567, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Order/ExtendFrom.lean", "full_name": "continuousOn_Ioc_extendFrom_Ioo", "start": [67, 1], "end": [73, 19], "traced_tactics": [{"tactic": "have := @continuousOn_Ico_extendFrom_Ioo \u03b1\u1d52\u1d48 _ _ _ _ _ _ _ f _ _ lb hab", "annotated_tactic": ["have := @<a>continuousOn_Ico_extendFrom_Ioo</a> \u03b1\u1d52\u1d48 _ _ _ _ _ _ _ f _ _ lb hab", [{"full_name": "continuousOn_Ico_extendFrom_Ioo", "def_path": "Mathlib/Topology/Algebra/Order/ExtendFrom.lean", "def_pos": [53, 9], "def_end_pos": [53, 40]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : DenselyOrdered \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : RegularSpace \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nlb : \u03b2\nhab : a < b\nhf : ContinuousOn f (Ioo a b)\nhb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lb)\n\u22a2 ContinuousOn (extendFrom (Ioo a b) f) (Ioc a b)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : DenselyOrdered \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : RegularSpace \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nlb : \u03b2\nhab : a < b\nhf : ContinuousOn f (Ioo a b)\nhb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lb)\nthis : ContinuousOn f (Ioo b a) \u2192 Tendsto f (\ud835\udcdd[Ioi b] b) (\ud835\udcdd lb) \u2192 ContinuousOn (extendFrom (Ioo b a) f) (Ico b a)\n\u22a2 ContinuousOn (extendFrom (Ioo a b) f) (Ioc a b)"}, {"tactic": "erw [dual_Ico, dual_Ioi, dual_Ioo] at this", "annotated_tactic": ["erw [<a>dual_Ico</a>, <a>dual_Ioi</a>, <a>dual_Ioo</a>] at this", [{"full_name": "Set.dual_Ico", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [256, 9], "def_end_pos": [256, 17]}, {"full_name": "Set.dual_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [236, 9], "def_end_pos": [236, 17]}, {"full_name": "Set.dual_Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [261, 9], "def_end_pos": [261, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : DenselyOrdered \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : RegularSpace \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nlb : \u03b2\nhab : a < b\nhf : ContinuousOn f (Ioo a b)\nhb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lb)\nthis : ContinuousOn f (Ioo b a) \u2192 Tendsto f (\ud835\udcdd[Ioi b] b) (\ud835\udcdd lb) \u2192 ContinuousOn (extendFrom (Ioo b a) f) (Ico b a)\n\u22a2 ContinuousOn (extendFrom (Ioo a b) f) (Ioc a b)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : DenselyOrdered \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : RegularSpace \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nlb : \u03b2\nhab : a < b\nhf : ContinuousOn f (Ioo a b)\nhb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lb)\nthis :\n  ContinuousOn f (\u2191OrderDual.ofDual \u207b\u00b9' Ioo a b) \u2192\n    Tendsto f (\ud835\udcdd[\u2191OrderDual.ofDual \u207b\u00b9' Iio b] b) (\ud835\udcdd lb) \u2192\n      ContinuousOn (extendFrom (\u2191OrderDual.ofDual \u207b\u00b9' Ioo a b) f) (\u2191OrderDual.ofDual \u207b\u00b9' Ioc a b)\n\u22a2 ContinuousOn (extendFrom (Ioo a b) f) (Ioc a b)"}, {"tactic": "exact this hf hb", "annotated_tactic": ["exact this hf hb", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : DenselyOrdered \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : RegularSpace \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nlb : \u03b2\nhab : a < b\nhf : ContinuousOn f (Ioo a b)\nhb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lb)\nthis :\n  ContinuousOn f (\u2191OrderDual.ofDual \u207b\u00b9' Ioo a b) \u2192\n    Tendsto f (\ud835\udcdd[\u2191OrderDual.ofDual \u207b\u00b9' Iio b] b) (\ud835\udcdd lb) \u2192\n      ContinuousOn (extendFrom (\u2191OrderDual.ofDual \u207b\u00b9' Ioo a b) f) (\u2191OrderDual.ofDual \u207b\u00b9' Ioc a b)\n\u22a2 ContinuousOn (extendFrom (Ioo a b) f) (Ioc a b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Order/Hom/Basic.lean", "full_name": "ContinuousOrderHom.id_comp", "start": [185, 1], "end": [186, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/GroupPower/Order.lean", "full_name": "sq_lt_sq", "start": [686, 1], "end": [688, 78], "traced_tactics": [{"tactic": "simpa only [sq_abs] using\n  (@strictMonoOn_pow R _ _ two_pos).lt_iff_lt (abs_nonneg x) (abs_nonneg y)", "annotated_tactic": ["simpa only [<a>sq_abs</a>] using\n    (@<a>strictMonoOn_pow</a> R _ _ <a>two_pos</a>).<a>lt_iff_lt</a> (<a>abs_nonneg</a> x) (<a>abs_nonneg</a> y)", [{"full_name": "sq_abs", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [680, 9], "def_end_pos": [680, 15]}, {"full_name": "strictMonoOn_pow", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [478, 9], "def_end_pos": [478, 25]}, {"full_name": "two_pos", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [113, 7], "def_end_pos": [113, 14]}, {"full_name": "StrictMonoOn.lt_iff_lt", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [831, 9], "def_end_pos": [831, 31]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}]], "state_before": "\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d : LinearOrderedRing R\nx y : R\n\u22a2 x ^ 2 < y ^ 2 \u2194 |x| < |y|", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "MeasureTheory.Measure.IsComplete.out", "start": [455, 1], "end": [456, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Disjoint.lean", "full_name": "disjoint_assoc", "start": [138, 1], "end": [139, 59], "traced_tactics": [{"tactic": "rw [disjoint_iff_inf_le, disjoint_iff_inf_le, inf_assoc]", "annotated_tactic": ["rw [<a>disjoint_iff_inf_le</a>, <a>disjoint_iff_inf_le</a>, <a>inf_assoc</a>]", [{"full_name": "disjoint_iff_inf_le", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [122, 9], "def_end_pos": [122, 28]}, {"full_name": "disjoint_iff_inf_le", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [122, 9], "def_end_pos": [122, 28]}, {"full_name": "inf_assoc", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [507, 9], "def_end_pos": [507, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\na b c d : \u03b1\n\u22a2 Disjoint (a \u2293 b) c \u2194 Disjoint a (b \u2293 c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/LocalHomeomorph.lean", "full_name": "LocalHomeomorph.preimage_interior", "start": [633, 1], "end": [635, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Complex/CauchyIntegral.lean", "full_name": "Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable", "start": [502, 1], "end": [508, 36], "traced_tactics": [{"tactic": "rw [\u2190 two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable\n  hs hw hc hd, smul_inv_smul\u2080]", "annotated_tactic": ["rw [\u2190 <a>two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable</a>\n    hs hw hc hd, <a>smul_inv_smul\u2080</a>]", [{"full_name": "Complex.two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable", "def_path": "Mathlib/Analysis/Complex/CauchyIntegral.lean", "def_pos": [463, 9], "def_end_pos": [463, 89]}, {"full_name": "smul_inv_smul\u2080", "def_path": "Mathlib/GroupTheory/GroupAction/Group.lean", "def_pos": [197, 9], "def_end_pos": [197, 23]}]], "state_before": "E : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 f z) = (2 * \u2191\u03c0 * I) \u2022 f w", "state_after": "case hc\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\n\u22a2 2 * \u2191\u03c0 * I \u2260 0"}, {"tactic": "simp [Real.pi_ne_zero, I_ne_zero]", "annotated_tactic": ["simp [<a>Real.pi_ne_zero</a>, <a>I_ne_zero</a>]", [{"full_name": "Real.pi_ne_zero", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "def_pos": [178, 9], "def_end_pos": [178, 19]}, {"full_name": "Complex.I_ne_zero", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [321, 9], "def_end_pos": [321, 18]}]], "state_before": "case hc\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\n\u22a2 2 * \u2191\u03c0 * I \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "full_name": "LatticeOrderedGroup.sup_div_inf_eq_abs_div", "start": [421, 1], "end": [435, 52], "traced_tactics": [{"tactic": "rw [div_eq_mul_inv, \u2190 inv_inf_eq_sup_inv]", "annotated_tactic": ["rw [<a>div_eq_mul_inv</a>, \u2190 <a>inv_inf_eq_sup_inv</a>]", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}, {"full_name": "inv_inf_eq_sup_inv", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [114, 9], "def_end_pos": [114, 27]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 (a \u2294 b) / (a \u2293 b) = (a \u2294 b) * (a\u207b\u00b9 \u2294 b\u207b\u00b9)", "state_after": "no goals"}, {"tactic": "rw [mul_sup, sup_mul, sup_mul]", "annotated_tactic": ["rw [<a>mul_sup</a>, <a>sup_mul</a>, <a>sup_mul</a>]", [{"full_name": "mul_sup", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [77, 9], "def_end_pos": [77, 16]}, {"full_name": "sup_mul", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [83, 9], "def_end_pos": [83, 16]}, {"full_name": "sup_mul", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [83, 9], "def_end_pos": [83, 16]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 (a \u2294 b) * (a\u207b\u00b9 \u2294 b\u207b\u00b9) = a * a\u207b\u00b9 \u2294 b * a\u207b\u00b9 \u2294 (a * b\u207b\u00b9 \u2294 b * b\u207b\u00b9)", "state_after": "no goals"}, {"tactic": "rw [mul_right_inv, mul_right_inv, \u2190div_eq_mul_inv, \u2190div_eq_mul_inv]", "annotated_tactic": ["rw [<a>mul_right_inv</a>, <a>mul_right_inv</a>, \u2190<a>div_eq_mul_inv</a>, \u2190<a>div_eq_mul_inv</a>]", [{"full_name": "mul_right_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1135, 9], "def_end_pos": [1135, 22]}, {"full_name": "mul_right_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1135, 9], "def_end_pos": [1135, 22]}, {"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}, {"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a * a\u207b\u00b9 \u2294 b * a\u207b\u00b9 \u2294 (a * b\u207b\u00b9 \u2294 b * b\u207b\u00b9) = 1 \u2294 b / a \u2294 (a / b \u2294 1)", "state_after": "no goals"}, {"tactic": "rw [one_div_div]", "annotated_tactic": ["rw [<a>one_div_div</a>]", [{"full_name": "one_div_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [406, 9], "def_end_pos": [406, 20]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 1 \u2294 b / a \u2294 (a / b \u2294 1) = 1 \u2294 b / a \u2294 (1 / (b / a) \u2294 1)", "state_after": "no goals"}, {"tactic": "rw [inv_eq_one_div]", "annotated_tactic": ["rw [<a>inv_eq_one_div</a>]", [{"full_name": "inv_eq_one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [295, 9], "def_end_pos": [295, 23]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 1 \u2294 b / a \u2294 (1 / (b / a) \u2294 1) = 1 \u2294 b / a \u2294 ((b / a)\u207b\u00b9 \u2294 1)", "state_after": "no goals"}, {"tactic": "rw [sup_assoc, sup_assoc]", "annotated_tactic": ["rw [<a>sup_assoc</a>, <a>sup_assoc</a>]", [{"full_name": "sup_assoc", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [256, 9], "def_end_pos": [256, 18]}, {"full_name": "sup_assoc", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [256, 9], "def_end_pos": [256, 18]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 1 \u2294 b / a \u2294 ((b / a)\u207b\u00b9 \u2294 1) = 1 \u2294 (b / a \u2294 (b / a)\u207b\u00b9 \u2294 1)", "state_after": "no goals"}, {"tactic": "rw [abs_eq_sup_inv]", "annotated_tactic": ["rw [<a>abs_eq_sup_inv</a>]", [{"full_name": "abs_eq_sup_inv", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [34, 9], "def_end_pos": [34, 23]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 1 \u2294 (b / a \u2294 (b / a)\u207b\u00b9 \u2294 1) = 1 \u2294 (|b / a| \u2294 1)", "state_after": "no goals"}, {"tactic": "rw [\u2190 m_pos_part_def, m_pos_abs]", "annotated_tactic": ["rw [\u2190 <a>m_pos_part_def</a>, <a>m_pos_abs</a>]", [{"full_name": "LatticeOrderedGroup.m_pos_part_def", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [137, 9], "def_end_pos": [137, 23]}, {"full_name": "LatticeOrderedGroup.m_pos_abs", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [401, 9], "def_end_pos": [401, 18]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 1 \u2294 (|b / a| \u2294 1) = 1 \u2294 |b / a|", "state_after": "no goals"}, {"tactic": "rw [sup_comm]", "annotated_tactic": ["rw [<a>sup_comm</a>]", [{"full_name": "sup_comm", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [250, 9], "def_end_pos": [250, 17]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 1 \u2294 |b / a| = |b / a| \u2294 1", "state_after": "no goals"}, {"tactic": "rw [\u2190 m_pos_part_def, m_pos_abs]", "annotated_tactic": ["rw [\u2190 <a>m_pos_part_def</a>, <a>m_pos_abs</a>]", [{"full_name": "LatticeOrderedGroup.m_pos_part_def", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [137, 9], "def_end_pos": [137, 23]}, {"full_name": "LatticeOrderedGroup.m_pos_abs", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [401, 9], "def_end_pos": [401, 18]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 |b / a| \u2294 1 = |b / a|", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/Dist.lean", "full_name": "Nat.dist_eq_max_sub_min", "start": [108, 1], "end": [111, 78], "traced_tactics": [{"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "i j : \u2115\n\u22a2 i < j \u2192 dist i j = max i j - min i j", "state_after": "i j : \u2115\nh : i < j\n\u22a2 dist i j = max i j - min i j"}, {"tactic": "rw [max_eq_right_of_lt h, min_eq_left_of_lt h, dist_eq_sub_of_le (Nat.le_of_lt h)]", "annotated_tactic": ["rw [<a>max_eq_right_of_lt</a> h, <a>min_eq_left_of_lt</a> h, <a>dist_eq_sub_of_le</a> (<a>Nat.le_of_lt</a> h)]", [{"full_name": "max_eq_right_of_lt", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [154, 9], "def_end_pos": [154, 27]}, {"full_name": "min_eq_left_of_lt", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [142, 9], "def_end_pos": [142, 26]}, {"full_name": "Nat.dist_eq_sub_of_le", "def_path": "Mathlib/Data/Nat/Dist.lean", "def_pos": [46, 9], "def_end_pos": [46, 26]}, {"full_name": "Nat.le_of_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [283, 19], "def_end_pos": [283, 27]}]], "state_before": "i j : \u2115\nh : i < j\n\u22a2 dist i j = max i j - min i j", "state_after": "no goals"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "i j : \u2115\n\u22a2 i \u2265 j \u2192 dist i j = max i j - min i j", "state_after": "i j : \u2115\nh : i \u2265 j\n\u22a2 dist i j = max i j - min i j"}, {"tactic": "rw [max_eq_left h, min_eq_right h, dist_eq_sub_of_le_right h]", "annotated_tactic": ["rw [<a>max_eq_left</a> h, <a>min_eq_right</a> h, <a>dist_eq_sub_of_le_right</a> h]", [{"full_name": "max_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [133, 9], "def_end_pos": [133, 20]}, {"full_name": "min_eq_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [104, 9], "def_end_pos": [104, 21]}, {"full_name": "Nat.dist_eq_sub_of_le_right", "def_path": "Mathlib/Data/Nat/Dist.lean", "def_pos": [50, 9], "def_end_pos": [50, 32]}]], "state_before": "i j : \u2115\nh : i \u2265 j\n\u22a2 dist i j = max i j - min i j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Manifold/Diffeomorph.lean", "full_name": "ModelWithCorners.coe_extChartAt_transDiffeomorph", "start": [533, 1], "end": [535, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Lipschitz.lean", "full_name": "LipschitzWith.mul_end", "start": [302, 11], "end": [304, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Ordmap/Ordset.lean", "full_name": "Ordnode.Valid'.node", "start": [1065, 1], "end": [1068, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean", "full_name": "HasStrictFDerivAt.cos", "start": [920, 1], "end": [922, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Logic/Equiv/Defs.lean", "full_name": "Equiv.symm_apply_eq", "start": [327, 1], "end": [328, 52], "traced_tactics": [{"tactic": "simp [H.symm]", "annotated_tactic": ["simp [H.symm]", []], "state_before": "\u03b1\u271d : Sort u\n\u03b2\u271d : Sort v\n\u03b3 : Sort w\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ne : \u03b1 \u2243 \u03b2\nx : \u03b2\ny : (fun x => \u03b1) x\nH : \u2191e.symm x = y\n\u22a2 x = \u2191e y", "state_after": "no goals"}, {"tactic": "simp [H]", "annotated_tactic": ["simp [H]", []], "state_before": "\u03b1\u271d : Sort u\n\u03b2\u271d : Sort v\n\u03b3 : Sort w\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ne : \u03b1 \u2243 \u03b2\nx : \u03b2\ny : (fun x => \u03b1) x\nH : x = \u2191e y\n\u22a2 \u2191e.symm x = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/QuadraticForm/Basic.lean", "full_name": "QuadraticForm.polar_sub_right", "start": [313, 1], "end": [314, 72], "traced_tactics": [{"tactic": "rw [sub_eq_add_neg, sub_eq_add_neg, polar_add_right, polar_neg_right]", "annotated_tactic": ["rw [<a>sub_eq_add_neg</a>, <a>sub_eq_add_neg</a>, <a>polar_add_right</a>, <a>polar_neg_right</a>]", [{"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "QuadraticForm.polar_add_right", "def_path": "Mathlib/LinearAlgebra/QuadraticForm/Basic.lean", "def_pos": [298, 9], "def_end_pos": [298, 24]}, {"full_name": "QuadraticForm.polar_neg_right", "def_path": "Mathlib/LinearAlgebra/QuadraticForm/Basic.lean", "def_pos": [308, 9], "def_end_pos": [308, 24]}]], "state_before": "S : Type u_1\nT : Type u_2\nR : Type u_3\nM : Type u_4\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx y y' : M\n\u22a2 polar (\u2191Q) x (y - y') = polar (\u2191Q) x y - polar (\u2191Q) x y'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Rel.lean", "full_name": "Rel.dom_inv", "start": [85, 1], "end": [87, 6], "traced_tactics": [{"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\n\u22a2 dom (inv r) = codom r", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\nx : \u03b2\n\u22a2 x \u2208 dom (inv r) \u2194 x \u2208 codom r"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\nx : \u03b2\n\u22a2 x \u2208 dom (inv r) \u2194 x \u2208 codom r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Real/Pi/Wallis.lean", "full_name": "Real.Wallis.W_pos", "start": [57, 1], "end": [61, 76], "traced_tactics": [{"tactic": "induction' k with k hk", "annotated_tactic": ["induction' k with k hk", []], "state_before": "k : \u2115\n\u22a2 0 < W k", "state_after": "case zero\n\n\u22a2 0 < W Nat.zero\n\ncase succ\nk : \u2115\nhk : 0 < W k\n\u22a2 0 < W (Nat.succ k)"}, {"tactic": "unfold W", "annotated_tactic": ["unfold <a>W</a>", [{"full_name": "Real.Wallis.W", "def_path": "Mathlib/Data/Real/Pi/Wallis.lean", "def_pos": [48, 19], "def_end_pos": [48, 20]}]], "state_before": "case zero\n\n\u22a2 0 < W Nat.zero", "state_after": "case zero\n\n\u22a2 0 < \u220f i in range Nat.zero, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3))"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case zero\n\n\u22a2 0 < \u220f i in range Nat.zero, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3))", "state_after": "no goals"}, {"tactic": "rw [W_succ]", "annotated_tactic": ["rw [<a>W_succ</a>]", [{"full_name": "Real.Wallis.W_succ", "def_path": "Mathlib/Data/Real/Pi/Wallis.lean", "def_pos": [52, 9], "def_end_pos": [52, 15]}]], "state_before": "case succ\nk : \u2115\nhk : 0 < W k\n\u22a2 0 < W (Nat.succ k)", "state_after": "case succ\nk : \u2115\nhk : 0 < W k\n\u22a2 0 < W k * ((2 * \u2191k + 2) / (2 * \u2191k + 1) * ((2 * \u2191k + 2) / (2 * \u2191k + 3)))"}, {"tactic": "refine' mul_pos hk (mul_pos (div_pos _ _) (div_pos _ _)) <;> positivity", "annotated_tactic": ["refine' <a>mul_pos</a> hk (<a>mul_pos</a> (<a>div_pos</a> _ _) (<a>div_pos</a> _ _)) <;> positivity", [{"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [345, 7], "def_end_pos": [345, 14]}, {"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [345, 7], "def_end_pos": [345, 14]}, {"full_name": "div_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 16]}, {"full_name": "div_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 16]}]], "state_before": "case succ\nk : \u2115\nhk : 0 < W k\n\u22a2 0 < W k * ((2 * \u2191k + 2) / (2 * \u2191k + 1) * ((2 * \u2191k + 2) / (2 * \u2191k + 3)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/Div.lean", "full_name": "Polynomial.multiplicity_finite_of_degree_pos_of_monic", "start": [65, 1], "end": [86, 14], "traced_tactics": [{"tactic": "have hp0 : p \u2260 0 := fun hp0 => by simp [hp0] at hp", "annotated_tactic": ["have hp0 : p \u2260 0 := fun hp0 => by simp [hp0] at hp", []], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\n\u22a2 False", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p \u2260 0\n\u22a2 False"}, {"tactic": "have hr0 : r \u2260 0 := fun hr0 => by subst hr0; simp [hq] at hr", "annotated_tactic": ["have hr0 : r \u2260 0 := fun hr0 => by subst hr0; simp [hq] at hr", []], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p \u2260 0\n\u22a2 False", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p \u2260 0\nhr0 : r \u2260 0\n\u22a2 False"}, {"tactic": "have hpn1 : leadingCoeff p ^ (natDegree q + 1) = 1 := by simp [show _ = _ from hmp]", "annotated_tactic": ["have hpn1 : <a>leadingCoeff</a> p ^ (<a>natDegree</a> q + 1) = 1 := by simp [show _ = _ from hmp]", [{"full_name": "Polynomial.leadingCoeff", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [70, 5], "def_end_pos": [70, 17]}, {"full_name": "Polynomial.natDegree", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [65, 5], "def_end_pos": [65, 14]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p \u2260 0\nhr0 : r \u2260 0\n\u22a2 False", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p \u2260 0\nhr0 : r \u2260 0\nhpn1 : leadingCoeff p ^ (natDegree q + 1) = 1\n\u22a2 False"}, {"tactic": "have hpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0 := hpn1.symm \u25b8 zn0.symm", "annotated_tactic": ["have hpn0' : <a>leadingCoeff</a> p ^ (<a>natDegree</a> q + 1) \u2260 0 := hpn1.symm \u25b8 zn0.symm", [{"full_name": "Polynomial.leadingCoeff", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [70, 5], "def_end_pos": [70, 17]}, {"full_name": "Polynomial.natDegree", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [65, 5], "def_end_pos": [65, 14]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p \u2260 0\nhr0 : r \u2260 0\nhpn1 : leadingCoeff p ^ (natDegree q + 1) = 1\n\u22a2 False", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p \u2260 0\nhr0 : r \u2260 0\nhpn1 : leadingCoeff p ^ (natDegree q + 1) = 1\nhpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0\n\u22a2 False"}, {"tactic": "have hpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r \u2260 0 := by\n  simp only [leadingCoeff_pow' hpn0', leadingCoeff_eq_zero, hpn1, one_pow, one_mul, Ne.def,\n      hr0]", "annotated_tactic": ["have hpnr0 : <a>leadingCoeff</a> (p ^ (<a>natDegree</a> q + 1)) * <a>leadingCoeff</a> r \u2260 0 := by\n      simp only [<a>leadingCoeff_pow'</a> hpn0', <a>leadingCoeff_eq_zero</a>, hpn1, <a>one_pow</a>, <a>one_mul</a>, <a>Ne.def</a>,\n          hr0]", [{"full_name": "Polynomial.leadingCoeff", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [70, 5], "def_end_pos": [70, 17]}, {"full_name": "Polynomial.natDegree", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [65, 5], "def_end_pos": [65, 14]}, {"full_name": "Polynomial.leadingCoeff", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [70, 5], "def_end_pos": [70, 17]}, {"full_name": "Polynomial.leadingCoeff_pow'", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [993, 9], "def_end_pos": [993, 26]}, {"full_name": "Polynomial.leadingCoeff_eq_zero", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [670, 9], "def_end_pos": [670, 29]}, {"full_name": "one_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [90, 9], "def_end_pos": [90, 16]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p \u2260 0\nhr0 : r \u2260 0\nhpn1 : leadingCoeff p ^ (natDegree q + 1) = 1\nhpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0\n\u22a2 False", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p \u2260 0\nhr0 : r \u2260 0\nhpn1 : leadingCoeff p ^ (natDegree q + 1) = 1\nhpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0\nhpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r \u2260 0\n\u22a2 False"}, {"tactic": "have hnp : 0 < natDegree p := Nat.cast_lt.1 <| by\n  rw [\u2190 degree_eq_natDegree hp0]; exact hp", "annotated_tactic": ["have hnp : 0 < <a>natDegree</a> p := <a>Nat.cast_lt</a>.1 <| by\n      rw [\u2190 <a>degree_eq_natDegree</a> hp0]; exact hp", [{"full_name": "Polynomial.natDegree", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [65, 5], "def_end_pos": [65, 14]}, {"full_name": "Nat.cast_lt", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [96, 9], "def_end_pos": [96, 16]}, {"full_name": "Polynomial.degree_eq_natDegree", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [130, 9], "def_end_pos": [130, 28]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p \u2260 0\nhr0 : r \u2260 0\nhpn1 : leadingCoeff p ^ (natDegree q + 1) = 1\nhpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0\nhpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r \u2260 0\n\u22a2 False", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p \u2260 0\nhr0 : r \u2260 0\nhpn1 : leadingCoeff p ^ (natDegree q + 1) = 1\nhpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0\nhpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r \u2260 0\nhnp : 0 < natDegree p\n\u22a2 False"}, {"tactic": "have := congr_arg natDegree hr", "annotated_tactic": ["have := <a>congr_arg</a> <a>natDegree</a> hr", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "Polynomial.natDegree", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [65, 5], "def_end_pos": [65, 14]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p \u2260 0\nhr0 : r \u2260 0\nhpn1 : leadingCoeff p ^ (natDegree q + 1) = 1\nhpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0\nhpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r \u2260 0\nhnp : 0 < natDegree p\n\u22a2 False", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p \u2260 0\nhr0 : r \u2260 0\nhpn1 : leadingCoeff p ^ (natDegree q + 1) = 1\nhpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0\nhpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r \u2260 0\nhnp : 0 < natDegree p\nthis : natDegree q = natDegree (p ^ (natDegree q + 1) * r)\n\u22a2 False"}, {"tactic": "rw [natDegree_mul' hpnr0, natDegree_pow' hpn0', add_mul, add_assoc] at this", "annotated_tactic": ["rw [<a>natDegree_mul'</a> hpnr0, <a>natDegree_pow'</a> hpn0', <a>add_mul</a>, <a>add_assoc</a>] at this", [{"full_name": "Polynomial.natDegree_mul'", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [967, 9], "def_end_pos": [967, 23]}, {"full_name": "Polynomial.natDegree_pow'", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [1009, 9], "def_end_pos": [1009, 23]}, {"full_name": "add_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [91, 7], "def_end_pos": [91, 14]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p \u2260 0\nhr0 : r \u2260 0\nhpn1 : leadingCoeff p ^ (natDegree q + 1) = 1\nhpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0\nhpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r \u2260 0\nhnp : 0 < natDegree p\nthis : natDegree q = natDegree (p ^ (natDegree q + 1) * r)\n\u22a2 False", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p \u2260 0\nhr0 : r \u2260 0\nhpn1 : leadingCoeff p ^ (natDegree q + 1) = 1\nhpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0\nhpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r \u2260 0\nhnp : 0 < natDegree p\nthis : natDegree q = natDegree q * natDegree p + (1 * natDegree p + natDegree r)\n\u22a2 False"}, {"tactic": "exact\n  ne_of_lt\n    (lt_add_of_le_of_pos (le_mul_of_one_le_right (Nat.zero_le _) hnp)\n      (add_pos_of_pos_of_nonneg (by rwa [one_mul]) (Nat.zero_le _)))\n    this", "annotated_tactic": ["exact\n      <a>ne_of_lt</a>\n        (<a>lt_add_of_le_of_pos</a> (<a>le_mul_of_one_le_right</a> (<a>Nat.zero_le</a> _) hnp)\n          (<a>add_pos_of_pos_of_nonneg</a> (by rwa [<a>one_mul</a>]) (<a>Nat.zero_le</a> _)))\n        this", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "lt_add_of_le_of_pos", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [724, 3], "def_end_pos": [724, 14]}, {"full_name": "le_mul_of_one_le_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [678, 9], "def_end_pos": [678, 31]}, {"full_name": "Nat.zero_le", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 20]}, {"full_name": "add_pos_of_pos_of_nonneg", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [1099, 24], "def_end_pos": [1099, 48]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "Nat.zero_le", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 20]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p \u2260 0\nhr0 : r \u2260 0\nhpn1 : leadingCoeff p ^ (natDegree q + 1) = 1\nhpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0\nhpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r \u2260 0\nhnp : 0 < natDegree p\nthis : natDegree q = natDegree q * natDegree p + (1 * natDegree p + natDegree r)\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simp [hp0] at hp", "annotated_tactic": ["simp [hp0] at hp", []], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p = 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "subst hr0", "annotated_tactic": ["subst hr0", []], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p \u2260 0\nhr0 : r = 0\n\u22a2 False", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nhp0 : p \u2260 0\nhr : q = p ^ (natDegree q + 1) * 0\n\u22a2 False"}, {"tactic": "simp [hq] at hr", "annotated_tactic": ["simp [hq] at hr", []], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nhp0 : p \u2260 0\nhr : q = p ^ (natDegree q + 1) * 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simp [show _ = _ from hmp]", "annotated_tactic": ["simp [show _ = _ from hmp]", []], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p \u2260 0\nhr0 : r \u2260 0\n\u22a2 leadingCoeff p ^ (natDegree q + 1) = 1", "state_after": "no goals"}, {"tactic": "simp only [leadingCoeff_pow' hpn0', leadingCoeff_eq_zero, hpn1, one_pow, one_mul, Ne.def,\n    hr0]", "annotated_tactic": ["simp only [<a>leadingCoeff_pow'</a> hpn0', <a>leadingCoeff_eq_zero</a>, hpn1, <a>one_pow</a>, <a>one_mul</a>, <a>Ne.def</a>,\n          hr0]", [{"full_name": "Polynomial.leadingCoeff_pow'", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [993, 9], "def_end_pos": [993, 26]}, {"full_name": "Polynomial.leadingCoeff_eq_zero", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [670, 9], "def_end_pos": [670, 29]}, {"full_name": "one_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [90, 9], "def_end_pos": [90, 16]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p \u2260 0\nhr0 : r \u2260 0\nhpn1 : leadingCoeff p ^ (natDegree q + 1) = 1\nhpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0\n\u22a2 leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r \u2260 0", "state_after": "no goals"}, {"tactic": "rw [\u2190 degree_eq_natDegree hp0]", "annotated_tactic": ["rw [\u2190 <a>degree_eq_natDegree</a> hp0]", [{"full_name": "Polynomial.degree_eq_natDegree", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [130, 9], "def_end_pos": [130, 28]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p \u2260 0\nhr0 : r \u2260 0\nhpn1 : leadingCoeff p ^ (natDegree q + 1) = 1\nhpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0\nhpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r \u2260 0\n\u22a2 \u21910 < \u2191(natDegree p)", "state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p \u2260 0\nhr0 : r \u2260 0\nhpn1 : leadingCoeff p ^ (natDegree q + 1) = 1\nhpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0\nhpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r \u2260 0\n\u22a2 \u21910 < degree p\n\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommSemiring R\np q : R[X]\nhp : 0 < degree p\nhmp : Monic p\nhq : q \u2260 0\nzn0 : 0 \u2260 1\nx\u271d : p ^ (natDegree q + 1) \u2223 q\nr : R[X]\nhr : q = p ^ (natDegree q + 1) * r\nhp0 : p \u2260 0\nhr0 : r \u2260 0\nhpn1 : leadingCoeff p ^ (natDegree q + 1) = 1\nhpn0' : leadingCoeff p ^ (natDegree q + 1) \u2260 0\nhpnr0 : leadingCoeff (p ^ (natDegree q + 1)) * leadingCoeff r \u2260 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u_3\ninst\u271d\u00b3\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3\u00b3 : TopologicalSpace M\ninst\u271d\u00b3\u00b2 : ChartedSpace H M\ninst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b3\u2070 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b2\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2\u2077 : TopologicalSpace M'\ninst\u271d\u00b2\u2076 : ChartedSpace H' M'\ninst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2\u00b9 : TopologicalSpace M''\ninst\u271d\u00b2\u2070 : ChartedSpace H'' M''\nF : Type u_11\ninst\u271d\u00b9\u2079 : NormedAddCommGroup F\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_12\ninst\u271d\u00b9\u2077 : TopologicalSpace G\nJ : ModelWithCorners \ud835\udd5c F G\nN : Type u_13\ninst\u271d\u00b9\u2076 : TopologicalSpace N\ninst\u271d\u00b9\u2075 : ChartedSpace G N\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N\nF' : Type u_14\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\nG' : Type u_15\ninst\u271d\u00b9\u00b9 : TopologicalSpace G'\nJ' : ModelWithCorners \ud835\udd5c F' G'\nN' : Type u_16\ninst\u271d\u00b9\u2070 : TopologicalSpace N'\ninst\u271d\u2079 : ChartedSpace G' N'\ninst\u271d\u2078 : SmoothManifoldWithCorners J' N'\nF\u2081 : Type u_17\ninst\u271d\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081\nF\u2082 : Type u_18\ninst\u271d\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082\nF\u2083 : Type u_19\ninst\u271d\u00b3 : NormedAddCommGroup F\u2083\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083\nF\u2084 : Type u_20\ninst\u271d\u00b9 : NormedAddCommGroup F\u2084\ninst\u271d : NormedSpace \ud835\udd5c F\u2084\ne : LocalHomeomorph M H\ne' : LocalHomeomorph M' H'\nf f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx\u271d : M\nm n : \u2115\u221e\nhe : e \u2208 maximalAtlas I M\nhs : s \u2286 e.source\nx : M\nhx : x \u2208 s\n\u22a2 ContMDiffWithinAt I I' n f s x \u2194\n    ContMDiffWithinAt \ud835\udcd8(\ud835\udd5c, E) I' n (f \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend e I)))\n      (\u2191(LocalHomeomorph.extend e I) '' s) (\u2191(LocalHomeomorph.extend e I) x)"}, {"tactic": "rw [contMDiffWithinAt_iff_source_of_mem_maximalAtlas he (hs hx)]", "annotated_tactic": ["rw [<a>contMDiffWithinAt_iff_source_of_mem_maximalAtlas</a> he (hs hx)]", [{"full_name": "contMDiffWithinAt_iff_source_of_mem_maximalAtlas", "def_path": "Mathlib/Geometry/Manifold/ContMDiff.lean", "def_pos": [477, 9], "def_end_pos": [477, 57]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3\u2076 : NormedAddCommGroup E\ninst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b3\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3\u00b3 : TopologicalSpace M\ninst\u271d\u00b3\u00b2 : ChartedSpace H M\ninst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b3\u2070 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b2\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2\u2077 : TopologicalSpace M'\ninst\u271d\u00b2\u2076 : ChartedSpace H' M'\ninst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2\u00b9 : TopologicalSpace M''\ninst\u271d\u00b2\u2070 : ChartedSpace H'' M''\nF : Type u_11\ninst\u271d\u00b9\u2079 : NormedAddCommGroup F\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_12\ninst\u271d\u00b9\u2077 : TopologicalSpace G\nJ : ModelWithCorners \ud835\udd5c F G\nN : Type u_13\ninst\u271d\u00b9\u2076 : TopologicalSpace N\ninst\u271d\u00b9\u2075 : ChartedSpace G N\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N\nF' : Type u_14\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\nG' : Type u_15\ninst\u271d\u00b9\u00b9 : TopologicalSpace G'\nJ' : ModelWithCorners \ud835\udd5c F' G'\nN' : Type u_16\ninst\u271d\u00b9\u2070 : TopologicalSpace N'\ninst\u271d\u2079 : ChartedSpace G' N'\ninst\u271d\u2078 : SmoothManifoldWithCorners J' N'\nF\u2081 : Type u_17\ninst\u271d\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081\nF\u2082 : Type u_18\ninst\u271d\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082\nF\u2083 : Type u_19\ninst\u271d\u00b3 : NormedAddCommGroup F\u2083\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083\nF\u2084 : Type u_20\ninst\u271d\u00b9 : NormedAddCommGroup F\u2084\ninst\u271d : NormedSpace \ud835\udd5c F\u2084\ne : LocalHomeomorph M H\ne' : LocalHomeomorph M' H'\nf f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx\u271d : M\nm n : \u2115\u221e\nhe : e \u2208 maximalAtlas I M\nhs : s \u2286 e.source\nx : M\nhx : x \u2208 s\n\u22a2 ContMDiffWithinAt I I' n f s x \u2194\n    ContMDiffWithinAt \ud835\udcd8(\ud835\udd5c, E) I' n (f \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend e I)))\n      (\u2191(LocalHomeomorph.extend e I) '' s) (\u2191(LocalHomeomorph.extend e I) x)", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3\u2076 : NormedAddCommGroup E\ninst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b3\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3\u00b3 : TopologicalSpace M\ninst\u271d\u00b3\u00b2 : ChartedSpace H M\ninst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b3\u2070 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b2\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2\u2077 : TopologicalSpace M'\ninst\u271d\u00b2\u2076 : ChartedSpace H' M'\ninst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2\u00b9 : TopologicalSpace M''\ninst\u271d\u00b2\u2070 : ChartedSpace H'' M''\nF : Type u_11\ninst\u271d\u00b9\u2079 : NormedAddCommGroup F\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_12\ninst\u271d\u00b9\u2077 : TopologicalSpace G\nJ : ModelWithCorners \ud835\udd5c F G\nN : Type u_13\ninst\u271d\u00b9\u2076 : TopologicalSpace N\ninst\u271d\u00b9\u2075 : ChartedSpace G N\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N\nF' : Type u_14\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\nG' : Type u_15\ninst\u271d\u00b9\u00b9 : TopologicalSpace G'\nJ' : ModelWithCorners \ud835\udd5c F' G'\nN' : Type u_16\ninst\u271d\u00b9\u2070 : TopologicalSpace N'\ninst\u271d\u2079 : ChartedSpace G' N'\ninst\u271d\u2078 : SmoothManifoldWithCorners J' N'\nF\u2081 : Type u_17\ninst\u271d\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081\nF\u2082 : Type u_18\ninst\u271d\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082\nF\u2083 : Type u_19\ninst\u271d\u00b3 : NormedAddCommGroup F\u2083\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083\nF\u2084 : Type u_20\ninst\u271d\u00b9 : NormedAddCommGroup F\u2084\ninst\u271d : NormedSpace \ud835\udd5c F\u2084\ne : LocalHomeomorph M H\ne' : LocalHomeomorph M' H'\nf f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx\u271d : M\nm n : \u2115\u221e\nhe : e \u2208 maximalAtlas I M\nhs : s \u2286 e.source\nx : M\nhx : x \u2208 s\n\u22a2 ContMDiffWithinAt \ud835\udcd8(\ud835\udd5c, E) I' n (f \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend e I)))\n      (\u2191(LocalEquiv.symm (LocalHomeomorph.extend e I)) \u207b\u00b9' s \u2229 range \u2191I) (\u2191(LocalHomeomorph.extend e I) x) \u2194\n    ContMDiffWithinAt \ud835\udcd8(\ud835\udd5c, E) I' n (f \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend e I)))\n      (\u2191(LocalHomeomorph.extend e I) '' s) (\u2191(LocalHomeomorph.extend e I) x)"}, {"tactic": "apply contMDiffWithinAt_congr_nhds", "annotated_tactic": ["apply <a>contMDiffWithinAt_congr_nhds</a>", [{"full_name": "contMDiffWithinAt_congr_nhds", "def_path": "Mathlib/Geometry/Manifold/ContMDiff.lean", "def_pos": [731, 9], "def_end_pos": [731, 37]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3\u2076 : NormedAddCommGroup E\ninst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b3\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3\u00b3 : TopologicalSpace M\ninst\u271d\u00b3\u00b2 : ChartedSpace H M\ninst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b3\u2070 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b2\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2\u2077 : TopologicalSpace M'\ninst\u271d\u00b2\u2076 : ChartedSpace H' M'\ninst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2\u00b9 : TopologicalSpace M''\ninst\u271d\u00b2\u2070 : ChartedSpace H'' M''\nF : Type u_11\ninst\u271d\u00b9\u2079 : NormedAddCommGroup F\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_12\ninst\u271d\u00b9\u2077 : TopologicalSpace G\nJ : ModelWithCorners \ud835\udd5c F G\nN : Type u_13\ninst\u271d\u00b9\u2076 : TopologicalSpace N\ninst\u271d\u00b9\u2075 : ChartedSpace G N\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N\nF' : Type u_14\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\nG' : Type u_15\ninst\u271d\u00b9\u00b9 : TopologicalSpace G'\nJ' : ModelWithCorners \ud835\udd5c F' G'\nN' : Type u_16\ninst\u271d\u00b9\u2070 : TopologicalSpace N'\ninst\u271d\u2079 : ChartedSpace G' N'\ninst\u271d\u2078 : SmoothManifoldWithCorners J' N'\nF\u2081 : Type u_17\ninst\u271d\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081\nF\u2082 : Type u_18\ninst\u271d\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082\nF\u2083 : Type u_19\ninst\u271d\u00b3 : NormedAddCommGroup F\u2083\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083\nF\u2084 : Type u_20\ninst\u271d\u00b9 : NormedAddCommGroup F\u2084\ninst\u271d : NormedSpace \ud835\udd5c F\u2084\ne : LocalHomeomorph M H\ne' : LocalHomeomorph M' H'\nf f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx\u271d : M\nm n : \u2115\u221e\nhe : e \u2208 maximalAtlas I M\nhs : s \u2286 e.source\nx : M\nhx : x \u2208 s\n\u22a2 ContMDiffWithinAt \ud835\udcd8(\ud835\udd5c, E) I' n (f \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend e I)))\n      (\u2191(LocalEquiv.symm (LocalHomeomorph.extend e I)) \u207b\u00b9' s \u2229 range \u2191I) (\u2191(LocalHomeomorph.extend e I) x) \u2194\n    ContMDiffWithinAt \ud835\udcd8(\ud835\udd5c, E) I' n (f \u2218 \u2191(LocalEquiv.symm (LocalHomeomorph.extend e I)))\n      (\u2191(LocalHomeomorph.extend e I) '' s) (\u2191(LocalHomeomorph.extend e I) x)", "state_after": "case hst\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3\u2076 : NormedAddCommGroup E\ninst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b3\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3\u00b3 : TopologicalSpace M\ninst\u271d\u00b3\u00b2 : ChartedSpace H M\ninst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b3\u2070 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b2\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2\u2077 : TopologicalSpace M'\ninst\u271d\u00b2\u2076 : ChartedSpace H' M'\ninst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2\u00b9 : TopologicalSpace M''\ninst\u271d\u00b2\u2070 : ChartedSpace H'' M''\nF : Type u_11\ninst\u271d\u00b9\u2079 : NormedAddCommGroup F\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_12\ninst\u271d\u00b9\u2077 : TopologicalSpace G\nJ : ModelWithCorners \ud835\udd5c F G\nN : Type u_13\ninst\u271d\u00b9\u2076 : TopologicalSpace N\ninst\u271d\u00b9\u2075 : ChartedSpace G N\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N\nF' : Type u_14\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\nG' : Type u_15\ninst\u271d\u00b9\u00b9 : TopologicalSpace G'\nJ' : ModelWithCorners \ud835\udd5c F' G'\nN' : Type u_16\ninst\u271d\u00b9\u2070 : TopologicalSpace N'\ninst\u271d\u2079 : ChartedSpace G' N'\ninst\u271d\u2078 : SmoothManifoldWithCorners J' N'\nF\u2081 : Type u_17\ninst\u271d\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081\nF\u2082 : Type u_18\ninst\u271d\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082\nF\u2083 : Type u_19\ninst\u271d\u00b3 : NormedAddCommGroup F\u2083\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083\nF\u2084 : Type u_20\ninst\u271d\u00b9 : NormedAddCommGroup F\u2084\ninst\u271d : NormedSpace \ud835\udd5c F\u2084\ne : LocalHomeomorph M H\ne' : LocalHomeomorph M' H'\nf f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx\u271d : M\nm n : \u2115\u221e\nhe : e \u2208 maximalAtlas I M\nhs : s \u2286 e.source\nx : M\nhx : x \u2208 s\n\u22a2 \ud835\udcdd[\u2191(LocalEquiv.symm (LocalHomeomorph.extend e I)) \u207b\u00b9' s \u2229 range \u2191I] \u2191(LocalHomeomorph.extend e I) x =\n    \ud835\udcdd[\u2191(LocalHomeomorph.extend e I) '' s] \u2191(LocalHomeomorph.extend e I) x"}, {"tactic": "simp_rw [nhdsWithin_eq_iff_eventuallyEq,\n  e.extend_symm_preimage_inter_range_eventuallyEq I hs (hs hx)]", "annotated_tactic": ["simp_rw [<a>nhdsWithin_eq_iff_eventuallyEq</a>,\n    e.extend_symm_preimage_inter_range_eventuallyEq I hs (hs hx)]", [{"full_name": "nhdsWithin_eq_iff_eventuallyEq", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [130, 9], "def_end_pos": [130, 39]}]], "state_before": "case hst\n\ud835\udd5c : Type u_1\ninst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3\u2076 : NormedAddCommGroup E\ninst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b3\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3\u00b3 : TopologicalSpace M\ninst\u271d\u00b3\u00b2 : ChartedSpace H M\ninst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b3\u2070 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b2\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2\u2077 : TopologicalSpace M'\ninst\u271d\u00b2\u2076 : ChartedSpace H' M'\ninst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2\u00b9 : TopologicalSpace M''\ninst\u271d\u00b2\u2070 : ChartedSpace H'' M''\nF : Type u_11\ninst\u271d\u00b9\u2079 : NormedAddCommGroup F\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_12\ninst\u271d\u00b9\u2077 : TopologicalSpace G\nJ : ModelWithCorners \ud835\udd5c F G\nN : Type u_13\ninst\u271d\u00b9\u2076 : TopologicalSpace N\ninst\u271d\u00b9\u2075 : ChartedSpace G N\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N\nF' : Type u_14\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\nG' : Type u_15\ninst\u271d\u00b9\u00b9 : TopologicalSpace G'\nJ' : ModelWithCorners \ud835\udd5c F' G'\nN' : Type u_16\ninst\u271d\u00b9\u2070 : TopologicalSpace N'\ninst\u271d\u2079 : ChartedSpace G' N'\ninst\u271d\u2078 : SmoothManifoldWithCorners J' N'\nF\u2081 : Type u_17\ninst\u271d\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081\nF\u2082 : Type u_18\ninst\u271d\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082\nF\u2083 : Type u_19\ninst\u271d\u00b3 : NormedAddCommGroup F\u2083\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083\nF\u2084 : Type u_20\ninst\u271d\u00b9 : NormedAddCommGroup F\u2084\ninst\u271d : NormedSpace \ud835\udd5c F\u2084\ne : LocalHomeomorph M H\ne' : LocalHomeomorph M' H'\nf f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx\u271d : M\nm n : \u2115\u221e\nhe : e \u2208 maximalAtlas I M\nhs : s \u2286 e.source\nx : M\nhx : x \u2208 s\n\u22a2 \ud835\udcdd[\u2191(LocalEquiv.symm (LocalHomeomorph.extend e I)) \u207b\u00b9' s \u2229 range \u2191I] \u2191(LocalHomeomorph.extend e I) x =\n    \ud835\udcdd[\u2191(LocalHomeomorph.extend e I) '' s] \u2191(LocalHomeomorph.extend e I) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Composition.lean", "full_name": "ProbabilityTheory.kernel.prodMkLeft_apply'", "start": [689, 1], "end": [691, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Fintype/Basic.lean", "full_name": "Set.toFinset_congr", "start": [624, 1], "end": [625, 78], "traced_tactics": [{"tactic": "subst h", "annotated_tactic": ["subst h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d t\u271d s t : Set \u03b1\ninst\u271d\u00b9 : Fintype \u2191s\ninst\u271d : Fintype \u2191t\nh : s = t\n\u22a2 toFinset s = toFinset t", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d t s : Set \u03b1\ninst\u271d\u00b9 inst\u271d : Fintype \u2191s\n\u22a2 toFinset s = toFinset s"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d t s : Set \u03b1\ninst\u271d\u00b9 inst\u271d : Fintype \u2191s\n\u22a2 toFinset s = toFinset s", "state_after": "case h.e_3.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d t s : Set \u03b1\ninst\u271d\u00b9 inst\u271d : Fintype \u2191s\n\u22a2 inst\u271d\u00b9 = inst\u271d"}, {"tactic": "exact Subsingleton.elim _ _", "annotated_tactic": ["exact <a>Subsingleton.elim</a> _ _", [{"full_name": "Subsingleton.elim", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [873, 19], "def_end_pos": [873, 36]}]], "state_before": "case h.e_3.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns\u271d t s : Set \u03b1\ninst\u271d\u00b9 inst\u271d : Fintype \u2191s\n\u22a2 inst\u271d\u00b9 = inst\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Ultrafilter.lean", "full_name": "Filter.isAtom_pure", "start": [409, 1], "end": [410, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sigma.lean", "full_name": "Finset.card_sigmaLift", "start": [189, 1], "end": [192, 32], "traced_tactics": [{"tactic": "simp_rw [sigmaLift]", "annotated_tactic": ["simp_rw [<a>sigmaLift</a>]", [{"full_name": "Finset.sigmaLift", "def_path": "Mathlib/Data/Finset/Sigma.lean", "def_pos": [119, 5], "def_end_pos": [119, 14]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : (i : \u03b9) \u00d7 \u03b1 i\nb : (i : \u03b9) \u00d7 \u03b2 i\n\u22a2 card (sigmaLift f a b) = if h : a.fst = b.fst then card (f (h \u25b8 a.snd) b.snd) else 0", "state_after": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : (i : \u03b9) \u00d7 \u03b1 i\nb : (i : \u03b9) \u00d7 \u03b2 i\n\u22a2 card (if h : a.fst = b.fst then map (Embedding.sigmaMk b.fst) (f (h \u25b8 a.snd) b.snd) else \u2205) =\n    if h : a.fst = b.fst then card (f (h \u25b8 a.snd) b.snd) else 0"}, {"tactic": "split_ifs with h <;> simp [h]", "annotated_tactic": ["split_ifs with h <;> simp [h]", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\n\u03b3 : \u03b9 \u2192 Type u_4\ninst\u271d : DecidableEq \u03b9\nf g : \u2983i : \u03b9\u2984 \u2192 \u03b1 i \u2192 \u03b2 i \u2192 Finset (\u03b3 i)\na : (i : \u03b9) \u00d7 \u03b1 i\nb : (i : \u03b9) \u00d7 \u03b2 i\n\u22a2 card (if h : a.fst = b.fst then map (Embedding.sigmaMk b.fst) (f (h \u25b8 a.snd) b.snd) else \u2205) =\n    if h : a.fst = b.fst then card (f (h \u25b8 a.snd) b.snd) else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.pi_update_of_not_mem", "start": [824, 1], "end": [828, 31], "traced_tactics": [{"tactic": "rw [update_noteq]", "annotated_tactic": ["rw [<a>update_noteq</a>]", [{"full_name": "Function.update_noteq", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [560, 9], "def_end_pos": [560, 21]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\nhi : \u00aci \u2208 s\nf : (j : \u03b9) \u2192 \u03b1 j\na : \u03b1 i\nt : (j : \u03b9) \u2192 \u03b1 j \u2192 Set (\u03b2 j)\nj : \u03b9\nhj : j \u2208 s\n\u22a2 t j (update f i a j) = t j (f j)", "state_after": "case h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\nhi : \u00aci \u2208 s\nf : (j : \u03b9) \u2192 \u03b1 j\na : \u03b1 i\nt : (j : \u03b9) \u2192 \u03b1 j \u2192 Set (\u03b2 j)\nj : \u03b9\nhj : j \u2208 s\n\u22a2 j \u2260 i"}, {"tactic": "exact fun h => hi (h \u25b8 hj)", "annotated_tactic": ["exact fun h => hi (h \u25b8 hj)", []], "state_before": "case h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt\u271d t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\ninst\u271d : DecidableEq \u03b9\nhi : \u00aci \u2208 s\nf : (j : \u03b9) \u2192 \u03b1 j\na : \u03b1 i\nt : (j : \u03b9) \u2192 \u03b1 j \u2192 Set (\u03b2 j)\nj : \u03b9\nhj : j \u2208 s\n\u22a2 j \u2260 i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "full_name": "hasSum_single", "start": [213, 1], "end": [215, 48], "traced_tactics": [{"tactic": "simpa using this", "annotated_tactic": ["simpa using this", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na b\u271d : \u03b1\ns : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nhf : \u2200 (b' : \u03b2), b' \u2260 b \u2192 f b' = 0\nthis : HasSum f (\u2211 b' in {b}, f b')\n\u22a2 HasSum f (f b)", "state_after": "no goals"}, {"tactic": "simpa [hf]", "annotated_tactic": ["simpa [hf]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : AddCommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na b\u271d : \u03b1\ns : Finset \u03b2\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nhf : \u2200 (b' : \u03b2), b' \u2260 b \u2192 f b' = 0\n\u22a2 \u2200 (b_1 : \u03b2), \u00acb_1 \u2208 {b} \u2192 f b_1 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.SimpleFunc.integrable_approxOn_range", "start": [261, 1], "end": [264, 59], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\ninst\u271d : SeparableSpace \u2191(Set.range f \u222a {0})\nhf : Integrable f\nn : \u2115\n\u22a2 0 \u2208 Set.range f \u222a {0}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/Cycle.lean", "full_name": "Cycle.Chain.eq_nil_of_irrefl", "start": [1026, 1], "end": [1030, 67], "traced_tactics": [{"tactic": "induction' s using Cycle.induction_on with a l _ h", "annotated_tactic": ["induction' s using <a>Cycle.induction_on</a> with a l _ h", [{"full_name": "Cycle.induction_on", "def_path": "Mathlib/Data/List/Cycle.lean", "def_pos": [513, 9], "def_end_pos": [513, 21]}]], "state_before": "\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsIrrefl \u03b1 r\nh : Chain r s\n\u22a2 s = Cycle.nil", "state_after": "case H0\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsIrrefl \u03b1 r\nh\u271d : Chain r s\nh : Chain r Cycle.nil\n\u22a2 Cycle.nil = Cycle.nil\n\ncase HI\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsIrrefl \u03b1 r\nh\u271d : Chain r s\na : \u03b1\nl : List \u03b1\na\u271d : Chain r \u2191l \u2192 \u2191l = Cycle.nil\nh : Chain r \u2191(a :: l)\n\u22a2 \u2191(a :: l) = Cycle.nil"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case H0\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsIrrefl \u03b1 r\nh\u271d : Chain r s\nh : Chain r Cycle.nil\n\u22a2 Cycle.nil = Cycle.nil", "state_after": "no goals"}, {"tactic": "have ha := mem_cons_self a l", "annotated_tactic": ["have ha := <a>mem_cons_self</a> a l", [{"full_name": "List.mem_cons_self", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [66, 9], "def_end_pos": [66, 22]}]], "state_before": "case HI\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsIrrefl \u03b1 r\nh\u271d : Chain r s\na : \u03b1\nl : List \u03b1\na\u271d : Chain r \u2191l \u2192 \u2191l = Cycle.nil\nh : Chain r \u2191(a :: l)\n\u22a2 \u2191(a :: l) = Cycle.nil", "state_after": "case HI\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsIrrefl \u03b1 r\nh\u271d : Chain r s\na : \u03b1\nl : List \u03b1\na\u271d : Chain r \u2191l \u2192 \u2191l = Cycle.nil\nh : Chain r \u2191(a :: l)\nha : a \u2208 a :: l\n\u22a2 \u2191(a :: l) = Cycle.nil"}, {"tactic": "exact (irrefl_of r a <| chain_iff_pairwise.1 h a ha a ha).elim", "annotated_tactic": ["exact (<a>irrefl_of</a> r a <| <a>chain_iff_pairwise</a>.1 h a ha a ha).<a>elim</a>", [{"full_name": "irrefl_of", "def_path": "Mathlib/Init/Algebra/Classes.lean", "def_pos": [343, 9], "def_end_pos": [343, 18]}, {"full_name": "Cycle.chain_iff_pairwise", "def_path": "Mathlib/Data/List/Cycle.lean", "def_pos": [1010, 9], "def_end_pos": [1010, 27]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case HI\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Cycle \u03b1\ninst\u271d\u00b9 : IsTrans \u03b1 r\ninst\u271d : IsIrrefl \u03b1 r\nh\u271d : Chain r s\na : \u03b1\nl : List \u03b1\na\u271d : Chain r \u2191l \u2192 \u2191l = Cycle.nil\nh : Chain r \u2191(a :: l)\nha : a \u2208 a :: l\n\u22a2 \u2191(a :: l) = Cycle.nil", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/ContinuousOn.lean", "full_name": "nhds_bind_nhdsWithin", "start": [41, 1], "end": [42, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.integral_eq_zero_iff_of_nonneg_ae", "start": [1265, 1], "end": [1271, 93], "traced_tactics": [{"tactic": "cases' le_total a b with hab hab <;>\n  simp only [Ioc_eq_empty hab.not_lt, empty_union, union_empty] at hf \u22a2", "annotated_tactic": ["cases' <a>le_total</a> a b with hab hab <;>\n    simp only [<a>Ioc_eq_empty</a> hab.not_lt, <a>empty_union</a>, <a>union_empty</a>] at hf \u22a2", [{"full_name": "le_total", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [295, 9], "def_end_pos": [295, 17]}, {"full_name": "Set.Ioc_eq_empty", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [368, 9], "def_end_pos": [368, 21]}, {"full_name": "Set.empty_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [782, 9], "def_end_pos": [782, 20]}, {"full_name": "Set.union_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [777, 9], "def_end_pos": [777, 20]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf g : \u211d \u2192 \u211d\na b : \u211d\n\u03bc : Measure \u211d\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc (Ioc a b \u222a Ioc b a)] f\nhfi : IntervalIntegrable f \u03bc a b\n\u22a2 \u222b (x : \u211d) in a..b, f x \u2202\u03bc = 0 \u2194 f =\u1d50[Measure.restrict \u03bc (Ioc a b \u222a Ioc b a)] 0", "state_after": "case inl\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf g : \u211d \u2192 \u211d\na b : \u211d\n\u03bc : Measure \u211d\nhfi : IntervalIntegrable f \u03bc a b\nhab : a \u2264 b\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc (Ioc a b)] f\n\u22a2 \u222b (x : \u211d) in a..b, f x \u2202\u03bc = 0 \u2194 f =\u1d50[Measure.restrict \u03bc (Ioc a b)] 0\n\ncase inr\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf g : \u211d \u2192 \u211d\na b : \u211d\n\u03bc : Measure \u211d\nhfi : IntervalIntegrable f \u03bc a b\nhab : b \u2264 a\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc (Ioc b a)] f\n\u22a2 \u222b (x : \u211d) in a..b, f x \u2202\u03bc = 0 \u2194 f =\u1d50[Measure.restrict \u03bc (Ioc b a)] 0"}, {"tactic": "exact integral_eq_zero_iff_of_le_of_nonneg_ae hab hf hfi", "annotated_tactic": ["exact <a>integral_eq_zero_iff_of_le_of_nonneg_ae</a> hab hf hfi", [{"full_name": "intervalIntegral.integral_eq_zero_iff_of_le_of_nonneg_ae", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1260, 9], "def_end_pos": [1260, 48]}]], "state_before": "case inl\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf g : \u211d \u2192 \u211d\na b : \u211d\n\u03bc : Measure \u211d\nhfi : IntervalIntegrable f \u03bc a b\nhab : a \u2264 b\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc (Ioc a b)] f\n\u22a2 \u222b (x : \u211d) in a..b, f x \u2202\u03bc = 0 \u2194 f =\u1d50[Measure.restrict \u03bc (Ioc a b)] 0", "state_after": "no goals"}, {"tactic": "rw [integral_symm, neg_eq_zero, integral_eq_zero_iff_of_le_of_nonneg_ae hab hf hfi.symm]", "annotated_tactic": ["rw [<a>integral_symm</a>, <a>neg_eq_zero</a>, <a>integral_eq_zero_iff_of_le_of_nonneg_ae</a> hab hf hfi.symm]", [{"full_name": "intervalIntegral.integral_symm", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [474, 9], "def_end_pos": [474, 22]}, {"full_name": "neg_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [422, 3], "def_end_pos": [422, 14]}, {"full_name": "intervalIntegral.integral_eq_zero_iff_of_le_of_nonneg_ae", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [1260, 9], "def_end_pos": [1260, 48]}]], "state_before": "case inr\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : NormedSpace \u211d E\nf g : \u211d \u2192 \u211d\na b : \u211d\n\u03bc : Measure \u211d\nhfi : IntervalIntegrable f \u03bc a b\nhab : b \u2264 a\nhf : 0 \u2264\u1d50[Measure.restrict \u03bc (Ioc b a)] f\n\u22a2 \u222b (x : \u211d) in a..b, f x \u2202\u03bc = 0 \u2194 f =\u1d50[Measure.restrict \u03bc (Ioc b a)] 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Euclidean/Sphere/Basic.lean", "full_name": "EuclideanGeometry.cospherical_empty", "start": [180, 1], "end": [182, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean", "full_name": "Real.contDiffAt_rpow_const_of_ne", "start": [375, 1], "end": [377, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/Cycle.lean", "full_name": "Cycle.length_nontrivial", "start": [639, 1], "end": [646, 32], "traced_tactics": [{"tactic": "obtain \u27e8x, y, hxy, hx, hy\u27e9 := h", "annotated_tactic": ["obtain \u27e8x, y, hxy, hx, hy\u27e9 := h", []], "state_before": "\u03b1 : Type u_1\ns : Cycle \u03b1\nh : Nontrivial s\n\u22a2 2 \u2264 length s", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ns : Cycle \u03b1\nx y : \u03b1\nhxy : x \u2260 y\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 2 \u2264 length s"}, {"tactic": "induction' s using Quot.inductionOn with l", "annotated_tactic": ["induction' s using <a>Quot.inductionOn</a> with l", [{"full_name": "Quot.inductionOn", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1233, 19], "def_end_pos": [1233, 30]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ns : Cycle \u03b1\nx y : \u03b1\nhxy : x \u2260 y\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 2 \u2264 length s", "state_after": "case intro.intro.intro.intro.h\n\u03b1 : Type u_1\ns : Cycle \u03b1\nx y : \u03b1\nhxy : x \u2260 y\nhx\u271d : x \u2208 s\nhy\u271d : y \u2208 s\nl : List \u03b1\nhx : x \u2208 Quot.mk Setoid.r l\nhy : y \u2208 Quot.mk Setoid.r l\n\u22a2 2 \u2264 length (Quot.mk Setoid.r l)"}, {"tactic": "rcases l with (_ | \u27e8hd, _ | \u27e8hd', tl\u27e9\u27e9)", "annotated_tactic": ["rcases l with (_ | \u27e8hd, _ | \u27e8hd', tl\u27e9\u27e9)", []], "state_before": "case intro.intro.intro.intro.h\n\u03b1 : Type u_1\ns : Cycle \u03b1\nx y : \u03b1\nhxy : x \u2260 y\nhx\u271d : x \u2208 s\nhy\u271d : y \u2208 s\nl : List \u03b1\nhx : x \u2208 Quot.mk Setoid.r l\nhy : y \u2208 Quot.mk Setoid.r l\n\u22a2 2 \u2264 length (Quot.mk Setoid.r l)", "state_after": "case intro.intro.intro.intro.h.nil\n\u03b1 : Type u_1\ns : Cycle \u03b1\nx y : \u03b1\nhxy : x \u2260 y\nhx\u271d : x \u2208 s\nhy\u271d : y \u2208 s\nhx : x \u2208 Quot.mk Setoid.r []\nhy : y \u2208 Quot.mk Setoid.r []\n\u22a2 2 \u2264 length (Quot.mk Setoid.r [])\n\ncase intro.intro.intro.intro.h.cons.nil\n\u03b1 : Type u_1\ns : Cycle \u03b1\nx y : \u03b1\nhxy : x \u2260 y\nhx\u271d : x \u2208 s\nhy\u271d : y \u2208 s\nhd : \u03b1\nhx : x \u2208 Quot.mk Setoid.r [hd]\nhy : y \u2208 Quot.mk Setoid.r [hd]\n\u22a2 2 \u2264 length (Quot.mk Setoid.r [hd])\n\ncase intro.intro.intro.intro.h.cons.cons\n\u03b1 : Type u_1\ns : Cycle \u03b1\nx y : \u03b1\nhxy : x \u2260 y\nhx\u271d : x \u2208 s\nhy\u271d : y \u2208 s\nhd hd' : \u03b1\ntl : List \u03b1\nhx : x \u2208 Quot.mk Setoid.r (hd :: hd' :: tl)\nhy : y \u2208 Quot.mk Setoid.r (hd :: hd' :: tl)\n\u22a2 2 \u2264 length (Quot.mk Setoid.r (hd :: hd' :: tl))"}, {"tactic": "simp at hx", "annotated_tactic": ["simp at hx", []], "state_before": "case intro.intro.intro.intro.h.nil\n\u03b1 : Type u_1\ns : Cycle \u03b1\nx y : \u03b1\nhxy : x \u2260 y\nhx\u271d : x \u2208 s\nhy\u271d : y \u2208 s\nhx : x \u2208 Quot.mk Setoid.r []\nhy : y \u2208 Quot.mk Setoid.r []\n\u22a2 2 \u2264 length (Quot.mk Setoid.r [])", "state_after": "no goals"}, {"tactic": "simp only [mem_coe_iff, mk_eq_coe, mem_singleton] at hx hy", "annotated_tactic": ["simp only [<a>mem_coe_iff</a>, <a>mk_eq_coe</a>, <a>mem_singleton</a>] at hx hy", [{"full_name": "Cycle.mem_coe_iff", "def_path": "Mathlib/Data/List/Cycle.lean", "def_pos": [529, 9], "def_end_pos": [529, 20]}, {"full_name": "Cycle.mk_eq_coe", "def_path": "Mathlib/Data/List/Cycle.lean", "def_pos": [470, 9], "def_end_pos": [470, 18]}, {"full_name": "List.mem_singleton", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [75, 22], "def_end_pos": [75, 35]}]], "state_before": "case intro.intro.intro.intro.h.cons.nil\n\u03b1 : Type u_1\ns : Cycle \u03b1\nx y : \u03b1\nhxy : x \u2260 y\nhx\u271d : x \u2208 s\nhy\u271d : y \u2208 s\nhd : \u03b1\nhx : x \u2208 Quot.mk Setoid.r [hd]\nhy : y \u2208 Quot.mk Setoid.r [hd]\n\u22a2 2 \u2264 length (Quot.mk Setoid.r [hd])", "state_after": "case intro.intro.intro.intro.h.cons.nil\n\u03b1 : Type u_1\ns : Cycle \u03b1\nx y : \u03b1\nhxy : x \u2260 y\nhx\u271d : x \u2208 s\nhy\u271d : y \u2208 s\nhd : \u03b1\nhx : x = hd\nhy : y = hd\n\u22a2 2 \u2264 length (Quot.mk Setoid.r [hd])"}, {"tactic": "simp [hx, hy] at hxy", "annotated_tactic": ["simp [hx, hy] at hxy", []], "state_before": "case intro.intro.intro.intro.h.cons.nil\n\u03b1 : Type u_1\ns : Cycle \u03b1\nx y : \u03b1\nhxy : x \u2260 y\nhx\u271d : x \u2208 s\nhy\u271d : y \u2208 s\nhd : \u03b1\nhx : x = hd\nhy : y = hd\n\u22a2 2 \u2264 length (Quot.mk Setoid.r [hd])", "state_after": "no goals"}, {"tactic": "simp [Nat.succ_le_succ_iff]", "annotated_tactic": ["simp [<a>Nat.succ_le_succ_iff</a>]", [{"full_name": "Nat.succ_le_succ_iff", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [257, 9], "def_end_pos": [257, 25]}]], "state_before": "case intro.intro.intro.intro.h.cons.cons\n\u03b1 : Type u_1\ns : Cycle \u03b1\nx y : \u03b1\nhxy : x \u2260 y\nhx\u271d : x \u2208 s\nhy\u271d : y \u2208 s\nhd hd' : \u03b1\ntl : List \u03b1\nhx : x \u2208 Quot.mk Setoid.r (hd :: hd' :: tl)\nhy : y \u2208 Quot.mk Setoid.r (hd :: hd' :: tl)\n\u22a2 2 \u2264 length (Quot.mk Setoid.r (hd :: hd' :: tl))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/Div.lean", "full_name": "Polynomial.rootMultiplicity_zero", "start": [600, 1], "end": [601, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.vsub_subset_vsub_left", "start": [670, 1], "end": [671, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean", "full_name": "IsBoundedBilinearMap.map_sub_left", "start": [386, 1], "end": [388, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.iUnion_ge_eq_iUnion_nat_add", "start": [2350, 1], "end": [2351, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Archimedean.lean", "full_name": "AddSubgroup.exists_isLeast_pos", "start": [59, 1], "end": [86, 41], "traced_tactics": [{"tactic": "have : \u2203 n : \u2115, Set.Nonempty (H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a)) := by\n  rcases (bot_or_exists_ne_zero H).resolve_left hbot with \u27e8g, hgH, hg\u2080\u27e9\n  rcases hex |g| (abs_pos.2 hg\u2080) with \u27e8n, hn\u27e9\n  exact \u27e8n, _, (@abs_mem_iff (AddSubgroup G) G _ _).2 hgH, hn\u27e9", "annotated_tactic": ["have : \u2203 n : \u2115, <a>Set.Nonempty</a> (H \u2229 <a>Ioc</a> (n \u2022 a) ((n + 1) \u2022 a)) := by\n    rcases (<a>bot_or_exists_ne_zero</a> H).<a>resolve_left</a> hbot with \u27e8g, hgH, hg\u2080\u27e9\n    rcases hex |g| (<a>abs_pos</a>.2 hg\u2080) with \u27e8n, hn\u27e9\n    exact \u27e8n, _, (@<a>abs_mem_iff</a> (<a>AddSubgroup</a> G) G _ _).2 hgH, hn\u27e9", [{"full_name": "Set.Nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [439, 15], "def_end_pos": [439, 23]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "AddSubgroup.bot_or_exists_ne_zero", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [957, 3], "def_end_pos": [957, 14]}, {"full_name": "Or.resolve_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [268, 9], "def_end_pos": [268, 24]}, {"full_name": "abs_pos", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [133, 9], "def_end_pos": [133, 16]}, {"full_name": "abs_mem_iff", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [136, 17], "def_end_pos": [136, 28]}, {"full_name": "AddSubgroup", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [370, 11], "def_end_pos": [370, 22]}]], "state_before": "G : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\n\u22a2 \u2203 b, IsLeast {g | g \u2208 H \u2227 0 < g} b", "state_after": "G : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\n\u22a2 \u2203 b, IsLeast {g | g \u2208 H \u2227 0 < g} b"}, {"tactic": "classical rcases Nat.findX this with \u27e8n, \u27e8x, hxH, hnx, hxn\u27e9, hmin\u27e9", "annotated_tactic": ["classical rcases <a>Nat.findX</a> this with \u27e8n, \u27e8x, hxH, hnx, hxn\u27e9, hmin\u27e9", [{"full_name": "Nat.findX", "def_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "def_pos": [691, 15], "def_end_pos": [691, 20]}]], "state_before": "G : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\n\u22a2 \u2203 b, IsLeast {g | g \u2208 H \u2227 0 < g} b", "state_after": "case mk.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\n\u22a2 \u2203 b, IsLeast {g | g \u2208 H \u2227 0 < g} b"}, {"tactic": "by_contra hxmin", "annotated_tactic": ["by_contra hxmin", []], "state_before": "case mk.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\n\u22a2 \u2203 b, IsLeast {g | g \u2208 H \u2227 0 < g} b", "state_after": "case mk.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\nhxmin : \u00ac\u2203 b, IsLeast {g | g \u2208 H \u2227 0 < g} b\n\u22a2 False"}, {"tactic": "simp only [IsLeast, not_and, mem_setOf_eq, mem_lowerBounds, not_exists, not_forall,\n  not_le] at hxmin", "annotated_tactic": ["simp only [<a>IsLeast</a>, <a>not_and</a>, <a>mem_setOf_eq</a>, <a>mem_lowerBounds</a>, <a>not_exists</a>, <a>not_forall</a>,\n    <a>not_le</a>] at hxmin", [{"full_name": "IsLeast", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [66, 5], "def_end_pos": [66, 12]}, {"full_name": "not_and", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [316, 17], "def_end_pos": [316, 24]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "mem_lowerBounds", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 24]}, {"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"full_name": "Classical.not_forall", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [686, 9], "def_end_pos": [686, 19]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "case mk.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\nhxmin : \u00ac\u2203 b, IsLeast {g | g \u2208 H \u2227 0 < g} b\n\u22a2 False", "state_after": "case mk.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\nhxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x\n\u22a2 False"}, {"tactic": "rcases hxmin x \u27e8hxH, (nsmul_nonneg h\u2080.le _).trans_lt hnx\u27e9 with \u27e8y, \u27e8hyH, hy\u2080\u27e9, hxy\u27e9", "annotated_tactic": ["rcases hxmin x \u27e8hxH, (<a>nsmul_nonneg</a> h\u2080.le _).<a>trans_lt</a> hnx\u27e9 with \u27e8y, \u27e8hyH, hy\u2080\u27e9, hxy\u27e9", [{"full_name": "nsmul_nonneg", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [50, 15], "def_end_pos": [50, 27]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}]], "state_before": "case mk.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\nhxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x\n\u22a2 False", "state_after": "case mk.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\nhxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x\ny : G\nhxy : y < x\nhyH : y \u2208 H\nhy\u2080 : 0 < y\n\u22a2 False"}, {"tactic": "rcases hex y hy\u2080 with \u27e8m, hm\u27e9", "annotated_tactic": ["rcases hex y hy\u2080 with \u27e8m, hm\u27e9", []], "state_before": "case mk.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\nhxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x\ny : G\nhxy : y < x\nhyH : y \u2208 H\nhy\u2080 : 0 < y\n\u22a2 False", "state_after": "case mk.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\nhxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x\ny : G\nhxy : y < x\nhyH : y \u2208 H\nhy\u2080 : 0 < y\nm : \u2115\nhm : y \u2208 Ioc (m \u2022 a) ((m + 1) \u2022 a)\n\u22a2 False"}, {"tactic": "cases' lt_or_le m n with hmn hnm", "annotated_tactic": ["cases' <a>lt_or_le</a> m n with hmn hnm", [{"full_name": "lt_or_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [336, 9], "def_end_pos": [336, 17]}]], "state_before": "case mk.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\nhxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x\ny : G\nhxy : y < x\nhyH : y \u2208 H\nhy\u2080 : 0 < y\nm : \u2115\nhm : y \u2208 Ioc (m \u2022 a) ((m + 1) \u2022 a)\n\u22a2 False", "state_after": "case mk.intro.intro.intro.intro.intro.intro.intro.intro.inl\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\nhxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x\ny : G\nhxy : y < x\nhyH : y \u2208 H\nhy\u2080 : 0 < y\nm : \u2115\nhm : y \u2208 Ioc (m \u2022 a) ((m + 1) \u2022 a)\nhmn : m < n\n\u22a2 False\n\ncase mk.intro.intro.intro.intro.intro.intro.intro.intro.inr\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\nhxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x\ny : G\nhxy : y < x\nhyH : y \u2208 H\nhy\u2080 : 0 < y\nm : \u2115\nhm : y \u2208 Ioc (m \u2022 a) ((m + 1) \u2022 a)\nhnm : n \u2264 m\n\u22a2 False"}, {"tactic": "rcases existsUnique_add_zsmul_mem_Ico h\u2080 0 (g - a) with \u27e8m, \u27e8hm, hm'\u27e9, -\u27e9", "annotated_tactic": ["rcases <a>existsUnique_add_zsmul_mem_Ico</a> h\u2080 0 (g - a) with \u27e8m, \u27e8hm, hm'\u27e9, -\u27e9", [{"full_name": "existsUnique_add_zsmul_mem_Ico", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [94, 9], "def_end_pos": [94, 39]}]], "state_before": "G : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\ng : G\nhg : g > 0\n\u22a2 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)", "state_after": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\ng : G\nhg : g > 0\nm : \u2124\nhm : g - a \u2264 0 + m \u2022 a\nhm' : 0 + m \u2022 a < g - a + a\n\u22a2 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)"}, {"tactic": "simp only [zero_add, sub_le_iff_le_add, sub_add_cancel, \u2190 add_one_zsmul] at hm hm'", "annotated_tactic": ["simp only [<a>zero_add</a>, <a>sub_le_iff_le_add</a>, <a>sub_add_cancel</a>, \u2190 <a>add_one_zsmul</a>] at hm hm'", [{"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "sub_le_iff_le_add", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [750, 3], "def_end_pos": [750, 14]}, {"full_name": "sub_add_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [728, 30], "def_end_pos": [728, 44]}, {"full_name": "add_one_zsmul", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [195, 15], "def_end_pos": [195, 28]}]], "state_before": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\ng : G\nhg : g > 0\nm : \u2124\nhm : g - a \u2264 0 + m \u2022 a\nhm' : 0 + m \u2022 a < g - a + a\n\u22a2 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)", "state_after": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\ng : G\nhg : g > 0\nm : \u2124\nhm : g \u2264 (m + 1) \u2022 a\nhm' : m \u2022 a < g\n\u22a2 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)"}, {"tactic": "lift m to \u2115", "annotated_tactic": ["lift m to \u2115", []], "state_before": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\ng : G\nhg : g > 0\nm : \u2124\nhm : g \u2264 (m + 1) \u2022 a\nhm' : m \u2022 a < g\n\u22a2 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)", "state_after": "G : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\ng : G\nhg : g > 0\nm : \u2124\nhm : g \u2264 (m + 1) \u2022 a\nhm' : m \u2022 a < g\n\u22a2 0 \u2264 m\n\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\ng : G\nhg : g > 0\nm : \u2115\nhm : g \u2264 (\u2191m + 1) \u2022 a\nhm' : \u2191m \u2022 a < g\n\u22a2 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)"}, {"tactic": "rw [\u2190 Int.lt_add_one_iff, \u2190 zsmul_lt_zsmul_iff h\u2080, zero_zsmul]", "annotated_tactic": ["rw [\u2190 <a>Int.lt_add_one_iff</a>, \u2190 <a>zsmul_lt_zsmul_iff</a> h\u2080, <a>zero_zsmul</a>]", [{"full_name": "Int.lt_add_one_iff", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [851, 9], "def_end_pos": [851, 23]}, {"full_name": "zsmul_lt_zsmul_iff", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [364, 3], "def_end_pos": [364, 14]}, {"full_name": "zero_zsmul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [942, 30], "def_end_pos": [942, 40]}]], "state_before": "G : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\ng : G\nhg : g > 0\nm : \u2124\nhm : g \u2264 (m + 1) \u2022 a\nhm' : m \u2022 a < g\n\u22a2 0 \u2264 m", "state_after": "G : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\ng : G\nhg : g > 0\nm : \u2124\nhm : g \u2264 (m + 1) \u2022 a\nhm' : m \u2022 a < g\n\u22a2 0 < (m + 1) \u2022 a"}, {"tactic": "exact hg.trans_le hm", "annotated_tactic": ["exact hg.trans_le hm", []], "state_before": "G : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\ng : G\nhg : g > 0\nm : \u2124\nhm : g \u2264 (m + 1) \u2022 a\nhm' : m \u2022 a < g\n\u22a2 0 < (m + 1) \u2022 a", "state_after": "no goals"}, {"tactic": "simp only [\u2190 Nat.cast_succ, coe_nat_zsmul] at hm hm'", "annotated_tactic": ["simp only [\u2190 <a>Nat.cast_succ</a>, <a>coe_nat_zsmul</a>] at hm hm'", [{"full_name": "Nat.cast_succ", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 18]}, {"full_name": "coe_nat_zsmul", "def_path": "Mathlib/Data/Int/Basic.lean", "def_pos": [91, 41], "def_end_pos": [91, 54]}]], "state_before": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\ng : G\nhg : g > 0\nm : \u2115\nhm : g \u2264 (\u2191m + 1) \u2022 a\nhm' : \u2191m \u2022 a < g\n\u22a2 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)", "state_after": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\ng : G\nhg : g > 0\nm : \u2115\nhm : g \u2264 Nat.succ m \u2022 a\nhm' : m \u2022 a < g\n\u22a2 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)"}, {"tactic": "exact \u27e8m, hm', hm\u27e9", "annotated_tactic": ["exact \u27e8m, hm', hm\u27e9", []], "state_before": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\ng : G\nhg : g > 0\nm : \u2115\nhm : g \u2264 Nat.succ m \u2022 a\nhm' : m \u2022 a < g\n\u22a2 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)", "state_after": "no goals"}, {"tactic": "rcases (bot_or_exists_ne_zero H).resolve_left hbot with \u27e8g, hgH, hg\u2080\u27e9", "annotated_tactic": ["rcases (<a>bot_or_exists_ne_zero</a> H).<a>resolve_left</a> hbot with \u27e8g, hgH, hg\u2080\u27e9", [{"full_name": "AddSubgroup.bot_or_exists_ne_zero", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [957, 3], "def_end_pos": [957, 14]}, {"full_name": "Or.resolve_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [268, 9], "def_end_pos": [268, 24]}]], "state_before": "G : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\n\u22a2 \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))", "state_after": "case intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\ng : G\nhgH : g \u2208 H\nhg\u2080 : g \u2260 0\n\u22a2 \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))"}, {"tactic": "rcases hex |g| (abs_pos.2 hg\u2080) with \u27e8n, hn\u27e9", "annotated_tactic": ["rcases hex |g| (<a>abs_pos</a>.2 hg\u2080) with \u27e8n, hn\u27e9", [{"full_name": "abs_pos", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [133, 9], "def_end_pos": [133, 16]}]], "state_before": "case intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\ng : G\nhgH : g \u2208 H\nhg\u2080 : g \u2260 0\n\u22a2 \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))", "state_after": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\ng : G\nhgH : g \u2208 H\nhg\u2080 : g \u2260 0\nn : \u2115\nhn : |g| \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\n\u22a2 \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))"}, {"tactic": "exact \u27e8n, _, (@abs_mem_iff (AddSubgroup G) G _ _).2 hgH, hn\u27e9", "annotated_tactic": ["exact \u27e8n, _, (@<a>abs_mem_iff</a> (<a>AddSubgroup</a> G) G _ _).2 hgH, hn\u27e9", [{"full_name": "abs_mem_iff", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [136, 17], "def_end_pos": [136, 28]}, {"full_name": "AddSubgroup", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [370, 11], "def_end_pos": [370, 22]}]], "state_before": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\ng : G\nhgH : g \u2208 H\nhg\u2080 : g \u2260 0\nn : \u2115\nhn : |g| \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\n\u22a2 \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))", "state_after": "no goals"}, {"tactic": "rcases Nat.findX this with \u27e8n, \u27e8x, hxH, hnx, hxn\u27e9, hmin\u27e9", "annotated_tactic": ["rcases <a>Nat.findX</a> this with \u27e8n, \u27e8x, hxH, hnx, hxn\u27e9, hmin\u27e9", [{"full_name": "Nat.findX", "def_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "def_pos": [691, 15], "def_end_pos": [691, 20]}]], "state_before": "G : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\n\u22a2 \u2203 b, IsLeast {g | g \u2208 H \u2227 0 < g} b", "state_after": "case mk.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\n\u22a2 \u2203 b, IsLeast {g | g \u2208 H \u2227 0 < g} b"}, {"tactic": "exact hmin m hmn \u27e8y, hyH, hm\u27e9", "annotated_tactic": ["exact hmin m hmn \u27e8y, hyH, hm\u27e9", []], "state_before": "case mk.intro.intro.intro.intro.intro.intro.intro.intro.inl\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\nhxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x\ny : G\nhxy : y < x\nhyH : y \u2208 H\nhy\u2080 : 0 < y\nm : \u2115\nhm : y \u2208 Ioc (m \u2022 a) ((m + 1) \u2022 a)\nhmn : m < n\n\u22a2 False", "state_after": "no goals"}, {"tactic": "refine disjoint_left.1 hd (sub_mem hxH hyH) \u27e8sub_pos.2 hxy, sub_lt_iff_lt_add'.2 ?_\u27e9", "annotated_tactic": ["refine <a>disjoint_left</a>.1 hd (<a>sub_mem</a> hxH hyH) \u27e8<a>sub_pos</a>.2 hxy, <a>sub_lt_iff_lt_add'</a>.2 ?_\u27e9", [{"full_name": "Set.disjoint_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1546, 9], "def_end_pos": [1546, 22]}, {"full_name": "sub_mem", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [143, 3], "def_end_pos": [143, 14]}, {"full_name": "sub_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [883, 30], "def_end_pos": [883, 37]}, {"full_name": "sub_lt_iff_lt_add'", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [979, 3], "def_end_pos": [979, 14]}]], "state_before": "case mk.intro.intro.intro.intro.intro.intro.intro.intro.inr\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\nhxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x\ny : G\nhxy : y < x\nhyH : y \u2208 H\nhy\u2080 : 0 < y\nm : \u2115\nhm : y \u2208 Ioc (m \u2022 a) ((m + 1) \u2022 a)\nhnm : n \u2264 m\n\u22a2 False", "state_after": "case mk.intro.intro.intro.intro.intro.intro.intro.intro.inr\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\nhxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x\ny : G\nhxy : y < x\nhyH : y \u2208 H\nhy\u2080 : 0 < y\nm : \u2115\nhm : y \u2208 Ioc (m \u2022 a) ((m + 1) \u2022 a)\nhnm : n \u2264 m\n\u22a2 x < y + a"}, {"tactic": "calc x \u2264 (n + 1) \u2022 a := hxn\n_ \u2264 (m + 1) \u2022 a := nsmul_le_nsmul h\u2080.le (add_le_add_right hnm _)\n_ = m \u2022 a + a := succ_nsmul' _ _\n_ < y + a := add_lt_add_right hm.1 _", "annotated_tactic": ["calc x \u2264 (n + 1) \u2022 a := hxn\n    _ \u2264 (m + 1) \u2022 a := <a>nsmul_le_nsmul</a> h\u2080.le (<a>add_le_add_right</a> hnm _)\n    _ = m \u2022 a + a := <a>succ_nsmul'</a> _ _\n    _ < y + a := <a>add_lt_add_right</a> hm.1 _", [{"full_name": "nsmul_le_nsmul", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [65, 15], "def_end_pos": [65, 29]}, {"full_name": "add_le_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [66, 15], "def_end_pos": [66, 31]}, {"full_name": "succ_nsmul'", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [212, 15], "def_end_pos": [212, 26]}, {"full_name": "add_lt_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [135, 15], "def_end_pos": [135, 31]}]], "state_before": "case mk.intro.intro.intro.intro.intro.intro.intro.intro.inr\nG : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup G\nhbot : H \u2260 \u22a5\na : G\nh\u2080 : 0 < a\nhd : Disjoint (\u2191H) (Ioo 0 a)\nhex : \u2200 (g : G), g > 0 \u2192 \u2203 n, g \u2208 Ioc (n \u2022 a) ((n + 1) \u2022 a)\nthis : \u2203 n, Set.Nonempty (\u2191H \u2229 Ioc (n \u2022 a) ((n + 1) \u2022 a))\nn : \u2115\nhmin : \u2200 (m : \u2115), m < n \u2192 \u00acSet.Nonempty (\u2191H \u2229 Ioc (m \u2022 a) ((m + 1) \u2022 a))\nx : G\nhxH : x \u2208 \u2191H\nhnx : n \u2022 a < x\nhxn : x \u2264 (n + 1) \u2022 a\nhxmin : \u2200 (x : G), x \u2208 H \u2227 0 < x \u2192 \u2203 x_1 h, x_1 < x\ny : G\nhxy : y < x\nhyH : y \u2208 H\nhy\u2080 : 0 < y\nm : \u2115\nhm : y \u2208 Ioc (m \u2022 a) ((m + 1) \u2022 a)\nhnm : n \u2264 m\n\u22a2 x < y + a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Multiset.Nodup.toFinset_inj", "start": [3194, 1], "end": [3196, 54], "traced_tactics": [{"tactic": "simpa [\u2190 toFinset_eq hl, \u2190 toFinset_eq hl'] using h", "annotated_tactic": ["simpa [\u2190 <a>toFinset_eq</a> hl, \u2190 <a>toFinset_eq</a> hl'] using h", [{"full_name": "Multiset.toFinset_eq", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3190, 9], "def_end_pos": [3190, 20]}, {"full_name": "Multiset.toFinset_eq", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3190, 9], "def_end_pos": [3190, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns t l l' : Multiset \u03b1\nhl : Nodup l\nhl' : Nodup l'\nh : toFinset l = toFinset l'\n\u22a2 l = l'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Congruence.lean", "full_name": "Con.toSetoid_inj", "start": [199, 1], "end": [200, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Lie/OfAssociative.lean", "full_name": "AlgHom.toLieHom_comp", "start": [193, 1], "end": [194, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/ModelTheory/Algebra/Field/CharP.lean", "full_name": "FirstOrder.Field.charP_iff_model_fieldOfChar", "start": [63, 1], "end": [78, 57], "traced_tactics": [{"tactic": "simp only [Theory.fieldOfChar, Theory.model_union_iff,\n  (show (Theory.field.Model K) by infer_instance), true_and]", "annotated_tactic": ["simp only [<a>Theory.fieldOfChar</a>, <a>Theory.model_union_iff</a>,\n    (show (Theory.field.Model K) by infer_instance), <a>true_and</a>]", [{"full_name": "FirstOrder.Language.Theory.fieldOfChar", "def_path": "Mathlib/ModelTheory/Algebra/Field/CharP.lean", "def_pos": [43, 5], "def_end_pos": [43, 50]}, {"full_name": "FirstOrder.Language.Theory.model_union_iff", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [850, 9], "def_end_pos": [850, 24]}, {"full_name": "true_and", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [84, 17], "def_end_pos": [84, 25]}]], "state_before": "p : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\n\u22a2 K \u22a8 Theory.fieldOfChar p \u2194 CharP K p", "state_after": "p : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\n\u22a2 (K \u22a8 if p = 0 then (fun q => \u223c(eqZero q)) '' {q | Nat.Prime q} else if Nat.Prime p then {eqZero p} else {\u22a5}) \u2194\n    CharP K p"}, {"tactic": "split_ifs with hp0 hp", "annotated_tactic": ["split_ifs with hp0 hp", []], "state_before": "p : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\n\u22a2 (K \u22a8 if p = 0 then (fun q => \u223c(eqZero q)) '' {q | Nat.Prime q} else if Nat.Prime p then {eqZero p} else {\u22a5}) \u2194\n    CharP K p", "state_after": "case pos\np : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\nhp0 : p = 0\n\u22a2 K \u22a8 (fun q => \u223c(eqZero q)) '' {q | Nat.Prime q} \u2194 CharP K p\n\ncase pos\np : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\nhp0 : \u00acp = 0\nhp : Nat.Prime p\n\u22a2 K \u22a8 {eqZero p} \u2194 CharP K p\n\ncase neg\np : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\nhp0 : \u00acp = 0\nhp : \u00acNat.Prime p\n\u22a2 K \u22a8 {\u22a5} \u2194 CharP K p"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "p : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\n\u22a2 K \u22a8 Theory.field", "state_after": "no goals"}, {"tactic": "subst hp0", "annotated_tactic": ["subst hp0", []], "state_before": "case pos\np : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\nhp0 : p = 0\n\u22a2 K \u22a8 (fun q => \u223c(eqZero q)) '' {q | Nat.Prime q} \u2194 CharP K p", "state_after": "case pos\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\n\u22a2 K \u22a8 (fun q => \u223c(eqZero q)) '' {q | Nat.Prime q} \u2194 CharP K 0"}, {"tactic": "simp only [Theory.model_iff, Set.mem_image, Set.mem_setOf_eq, Sentence.Realize,\n  forall_exists_index, and_imp, forall_apply_eq_imp_iff\u2082, Formula.realize_not,\n  realize_eqZero, \u2190 CharZero.charZero_iff_forall_prime_ne_zero]", "annotated_tactic": ["simp only [<a>Theory.model_iff</a>, <a>Set.mem_image</a>, <a>Set.mem_setOf_eq</a>, <a>Sentence.Realize</a>,\n      <a>forall_exists_index</a>, <a>and_imp</a>, <a>forall_apply_eq_imp_iff\u2082</a>, <a>Formula.realize_not</a>,\n      <a>realize_eqZero</a>, \u2190 <a>CharZero.charZero_iff_forall_prime_ne_zero</a>]", [{"full_name": "FirstOrder.Language.Theory.model_iff", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [814, 9], "def_end_pos": [814, 25]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "FirstOrder.Language.Sentence.Realize", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [719, 12], "def_end_pos": [719, 28]}, {"full_name": "forall_exists_index", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [356, 17], "def_end_pos": [356, 36]}, {"full_name": "and_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [313, 17], "def_end_pos": [313, 24]}, {"full_name": "forall_apply_eq_imp_iff\u2082", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [505, 17], "def_end_pos": [505, 41]}, {"full_name": "FirstOrder.Language.Formula.realize_not", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [616, 9], "def_end_pos": [616, 20]}, {"full_name": "FirstOrder.Field.realize_eqZero", "def_path": "Mathlib/ModelTheory/Algebra/Field/CharP.lean", "def_pos": [38, 17], "def_end_pos": [38, 31]}, {"full_name": "CharZero.charZero_iff_forall_prime_ne_zero", "def_path": "Mathlib/Algebra/CharP/Basic.lean", "def_pos": [744, 9], "def_end_pos": [744, 42]}]], "state_before": "case pos\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\n\u22a2 K \u22a8 (fun q => \u223c(eqZero q)) '' {q | Nat.Prime q} \u2194 CharP K 0", "state_after": "case pos\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\n\u22a2 CharZero K \u2194 CharP K 0"}, {"tactic": "exact \u27e8fun _ => CharP.ofCharZero _, fun _ => CharP.charP_to_charZero K\u27e9", "annotated_tactic": ["exact \u27e8fun _ => <a>CharP.ofCharZero</a> _, fun _ => <a>CharP.charP_to_charZero</a> K\u27e9", [{"full_name": "CharP.ofCharZero", "def_path": "Mathlib/Algebra/CharP/Basic.lean", "def_pos": [157, 10], "def_end_pos": [157, 26]}, {"full_name": "CharP.charP_to_charZero", "def_path": "Mathlib/Algebra/CharP/Basic.lean", "def_pos": [465, 9], "def_end_pos": [465, 26]}]], "state_before": "case pos\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\n\u22a2 CharZero K \u2194 CharP K 0", "state_after": "no goals"}, {"tactic": "simp only [Theory.model_iff, Set.mem_singleton_iff, Sentence.Realize, forall_eq,\n  realize_eqZero, \u2190 CharP.charP_iff_prime_eq_zero hp]", "annotated_tactic": ["simp only [<a>Theory.model_iff</a>, <a>Set.mem_singleton_iff</a>, <a>Sentence.Realize</a>, <a>forall_eq</a>,\n      <a>realize_eqZero</a>, \u2190 <a>CharP.charP_iff_prime_eq_zero</a> hp]", [{"full_name": "FirstOrder.Language.Theory.model_iff", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [814, 9], "def_end_pos": [814, 25]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}, {"full_name": "FirstOrder.Language.Sentence.Realize", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [719, 12], "def_end_pos": [719, 28]}, {"full_name": "forall_eq", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [450, 17], "def_end_pos": [450, 26]}, {"full_name": "FirstOrder.Field.realize_eqZero", "def_path": "Mathlib/ModelTheory/Algebra/Field/CharP.lean", "def_pos": [38, 17], "def_end_pos": [38, 31]}, {"full_name": "CharP.charP_iff_prime_eq_zero", "def_path": "Mathlib/Algebra/CharP/Basic.lean", "def_pos": [614, 9], "def_end_pos": [614, 32]}]], "state_before": "case pos\np : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\nhp0 : \u00acp = 0\nhp : Nat.Prime p\n\u22a2 K \u22a8 {eqZero p} \u2194 CharP K p", "state_after": "no goals"}, {"tactic": "simp only [Theory.model_iff, Set.mem_singleton_iff, Sentence.Realize,\n  forall_eq, Formula.realize_bot, false_iff]", "annotated_tactic": ["simp only [<a>Theory.model_iff</a>, <a>Set.mem_singleton_iff</a>, <a>Sentence.Realize</a>,\n      <a>forall_eq</a>, <a>Formula.realize_bot</a>, <a>false_iff</a>]", [{"full_name": "FirstOrder.Language.Theory.model_iff", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [814, 9], "def_end_pos": [814, 25]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}, {"full_name": "FirstOrder.Language.Sentence.Realize", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [719, 12], "def_end_pos": [719, 28]}, {"full_name": "forall_eq", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [450, 17], "def_end_pos": [450, 26]}, {"full_name": "FirstOrder.Language.Formula.realize_bot", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [621, 9], "def_end_pos": [621, 20]}, {"full_name": "false_iff", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [96, 17], "def_end_pos": [96, 26]}]], "state_before": "case neg\np : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\nhp0 : \u00acp = 0\nhp : \u00acNat.Prime p\n\u22a2 K \u22a8 {\u22a5} \u2194 CharP K p", "state_after": "case neg\np : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\nhp0 : \u00acp = 0\nhp : \u00acNat.Prime p\n\u22a2 \u00acCharP K p"}, {"tactic": "intro H", "annotated_tactic": ["intro H", []], "state_before": "case neg\np : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\nhp0 : \u00acp = 0\nhp : \u00acNat.Prime p\n\u22a2 \u00acCharP K p", "state_after": "case neg\np : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\nhp0 : \u00acp = 0\nhp : \u00acNat.Prime p\nH : CharP K p\n\u22a2 False"}, {"tactic": "cases (CharP.char_is_prime_or_zero K p) <;> simp_all", "annotated_tactic": ["cases (<a>CharP.char_is_prime_or_zero</a> K p) <;> simp_all", [{"full_name": "CharP.char_is_prime_or_zero", "def_path": "Mathlib/Algebra/CharP/Basic.lean", "def_pos": [550, 9], "def_end_pos": [550, 30]}]], "state_before": "case neg\np : \u2115\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : CompatibleRing K\nhp0 : \u00acp = 0\nhp : \u00acNat.Prime p\nH : CharP K p\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Module/LocalizedModule.lean", "full_name": "IsLocalizedModule.fromLocalizedModule'_mk", "start": [727, 1], "end": [730, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Control/Monad/Cont.lean", "full_name": "OptionT.goto_mkLabel", "start": [158, 1], "end": [160, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/ContDiff.lean", "full_name": "contDiff_clm_apply_iff", "start": [1922, 1], "end": [1924, 52], "traced_tactics": [{"tactic": "simp_rw [\u2190 contDiffOn_univ, contDiffOn_clm_apply]", "annotated_tactic": ["simp_rw [\u2190 <a>contDiffOn_univ</a>, <a>contDiffOn_clm_apply</a>]", [{"full_name": "contDiffOn_univ", "def_path": "Mathlib/Analysis/Calculus/ContDiffDef.lean", "def_pos": [1420, 9], "def_end_pos": [1420, 24]}, {"full_name": "contDiffOn_clm_apply", "def_path": "Mathlib/Analysis/Calculus/ContDiff.lean", "def_pos": [1911, 9], "def_end_pos": [1911, 29]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup D\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup X\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf\u271d f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\nn : \u2115\u221e\nf : E \u2192 F \u2192L[\ud835\udd5c] G\ninst\u271d : FiniteDimensional \ud835\udd5c F\n\u22a2 ContDiff \ud835\udd5c n f \u2194 \u2200 (y : F), ContDiff \ud835\udd5c n fun x => \u2191(f x) y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Disjoint.lean", "full_name": "Complementeds.mk_top", "start": [746, 1], "end": [746, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.inf_apply", "start": [524, 1], "end": [525, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Sylow.lean", "full_name": "Sylow.coe_comapOfKerIsPGroup", "start": [111, 1], "end": [113, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Field/Basic.lean", "full_name": "one_div_lt_of_neg", "start": [821, 1], "end": [822, 34], "traced_tactics": [{"tactic": "simpa using inv_lt_of_neg ha hb", "annotated_tactic": ["simpa using <a>inv_lt_of_neg</a> ha hb", [{"full_name": "inv_lt_of_neg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [773, 9], "def_end_pos": [773, 22]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nha : a < 0\nhb : b < 0\n\u22a2 1 / a < b \u2194 1 / b < a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Manifold/ContMDiff.lean", "full_name": "Smooth.snd", "start": [1636, 1], "end": [1637, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.map_eq_foldr", "start": [1712, 1], "end": [1713, 27], "traced_tactics": [{"tactic": "induction l <;> simp [*]", "annotated_tactic": ["induction l <;> simp [*]", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\n\u22a2 map f l = foldr (fun a bs => f a :: bs) [] l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/EssentialImage.lean", "full_name": "CategoryTheory.Functor.essImage.ofNatIso", "start": [63, 1], "end": [65, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean", "full_name": "CategoryTheory.Limits.WidePullback.hom_eq_lift", "start": [385, 1], "end": [389, 6], "traced_tactics": [{"tactic": "aesop_cat", "annotated_tactic": ["aesop_cat", []], "state_before": "J : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type ?u.185654\ninst\u271d\u00b9 : Category.{v\u2082, ?u.185654} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng : X \u27f6 widePullback B (fun j => objs j) arrows\n\u22a2 \u2200 (j : J), (fun j => g \u226b \u03c0 arrows j) j \u226b arrows j = g \u226b base arrows", "state_after": "no goals"}, {"tactic": "apply eq_lift_of_comp_eq", "annotated_tactic": ["apply <a>eq_lift_of_comp_eq</a>", [{"full_name": "CategoryTheory.Limits.WidePullback.eq_lift_of_comp_eq", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean", "def_pos": [374, 9], "def_end_pos": [374, 27]}]], "state_before": "J : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng : X \u27f6 widePullback B (fun j => objs j) arrows\n\u22a2 g = lift (g \u226b base arrows) (fun j => g \u226b \u03c0 arrows j) (_ : \u2200 (j : J), (g \u226b \u03c0 arrows j) \u226b arrows j = g \u226b base arrows)", "state_after": "case a\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng : X \u27f6 widePullback B (fun j => objs j) arrows\n\u22a2 \u2200 (j : J), g \u226b \u03c0 arrows j = g \u226b \u03c0 arrows j\n\ncase a\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng : X \u27f6 widePullback B (fun j => objs j) arrows\n\u22a2 g \u226b base arrows = g \u226b base arrows"}, {"tactic": "aesop_cat", "annotated_tactic": ["aesop_cat", []], "state_before": "case a\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng : X \u27f6 widePullback B (fun j => objs j) arrows\n\u22a2 \u2200 (j : J), g \u226b \u03c0 arrows j = g \u226b \u03c0 arrows j\n\ncase a\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng : X \u27f6 widePullback B (fun j => objs j) arrows\n\u22a2 g \u226b base arrows = g \u226b base arrows", "state_after": "case a\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng : X \u27f6 widePullback B (fun j => objs j) arrows\n\u22a2 g \u226b base arrows = g \u226b base arrows"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case a\nJ : Type w\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9 : Category.{v\u2082, u_1} D\nB : D\nobjs : J \u2192 D\narrows : (j : J) \u2192 objs j \u27f6 B\ninst\u271d : HasWidePullback B objs arrows\nX : D\nf : X \u27f6 B\nfs : (j : J) \u2192 X \u27f6 objs j\nw : \u2200 (j : J), fs j \u226b arrows j = f\ng : X \u27f6 widePullback B (fun j => objs j) arrows\n\u22a2 g \u226b base arrows = g \u226b base arrows", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Antichain.lean", "full_name": "IsAntichain.image", "start": [89, 1], "end": [92, 47], "traced_tactics": [{"tactic": "rintro _ \u27e8b, hb, rfl\u27e9 _ \u27e8c, hc, rfl\u27e9 hbc hr", "annotated_tactic": ["rintro _ \u27e8b, hb, rfl\u27e9 _ \u27e8c, hc, rfl\u27e9 hbc hr", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\nr' : \u03b2 \u2192 \u03b2 \u2192 Prop\ns t : Set \u03b1\na b : \u03b1\nhs : IsAntichain r s\nf : \u03b1 \u2192 \u03b2\nh : \u2200 \u2983a b : \u03b1\u2984, r' (f a) (f b) \u2192 r a b\n\u22a2 IsAntichain r' (f '' s)", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\nr' : \u03b2 \u2192 \u03b2 \u2192 Prop\ns t : Set \u03b1\na b\u271d : \u03b1\nhs : IsAntichain r s\nf : \u03b1 \u2192 \u03b2\nh : \u2200 \u2983a b : \u03b1\u2984, r' (f a) (f b) \u2192 r a b\nb : \u03b1\nhb : b \u2208 s\nc : \u03b1\nhc : c \u2208 s\nhbc : f b \u2260 f c\nhr : r' (f b) (f c)\n\u22a2 False"}, {"tactic": "exact hs hb hc (ne_of_apply_ne _ hbc) (h hr)", "annotated_tactic": ["exact hs hb hc (<a>ne_of_apply_ne</a> _ hbc) (h hr)", [{"full_name": "ne_of_apply_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [520, 9], "def_end_pos": [520, 23]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop\nr' : \u03b2 \u2192 \u03b2 \u2192 Prop\ns t : Set \u03b1\na b\u271d : \u03b1\nhs : IsAntichain r s\nf : \u03b1 \u2192 \u03b2\nh : \u2200 \u2983a b : \u03b1\u2984, r' (f a) (f b) \u2192 r a b\nb : \u03b1\nhb : b \u2208 s\nc : \u03b1\nhc : c \u2208 s\nhbc : f b \u2260 f c\nhr : r' (f b) (f c)\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Basic.lean", "full_name": "dist_triangle4_left", "start": [214, 1], "end": [217, 23], "traced_tactics": [{"tactic": "rw [add_left_comm, dist_comm x\u2081, \u2190 add_assoc]", "annotated_tactic": ["rw [<a>add_left_comm</a>, <a>dist_comm</a> x\u2081, \u2190 <a>add_assoc</a>]", [{"full_name": "add_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}, {"full_name": "dist_comm", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [188, 9], "def_end_pos": [188, 18]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u2081 y\u2081 x\u2082 y\u2082 : \u03b1\n\u22a2 dist x\u2082 y\u2082 \u2264 dist x\u2081 y\u2081 + (dist x\u2081 x\u2082 + dist y\u2081 y\u2082)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u2081 y\u2081 x\u2082 y\u2082 : \u03b1\n\u22a2 dist x\u2082 y\u2082 \u2264 dist x\u2082 x\u2081 + dist x\u2081 y\u2081 + dist y\u2081 y\u2082"}, {"tactic": "apply dist_triangle4", "annotated_tactic": ["apply <a>dist_triangle4</a>", [{"full_name": "dist_triangle4", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 23]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx\u2081 y\u2081 x\u2082 y\u2082 : \u03b1\n\u22a2 dist x\u2082 y\u2082 \u2264 dist x\u2082 x\u2081 + dist x\u2081 y\u2081 + dist y\u2081 y\u2082", "state_after": "no 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Type ud\n\u03b9 : Sort u_1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nhf : UniformContinuous fun x => f \u2191x\nha : s \u2208 \ud835\udcdd a\n\u22a2 Tendsto f (\ud835\udcdd a) (\ud835\udcdd (f a))", "state_after": "\u03b1 : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort u_1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na : \u03b1\nhf : UniformContinuous fun x => f \u2191x\nha : s \u2208 \ud835\udcdd a\n\u22a2 Tendsto f (map Subtype.val (\ud835\udcdd { val := a, property := (_ : a \u2208 s) })) (\ud835\udcdd (f a))"}, {"tactic": "exact tendsto_map' hf.rst.imntinuous.continuousAt", "annotated_tactic": ["exact <a>tendsto_map'</a> hf.rst.imntinuous.continuousAt", [{"full_name": "Filter.tendsto_map'", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3061, 11], "def_end_pos": [3061, 23]}]], 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Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3\nx\u271d\u00b9 : \u03b1\nx\u271d : \u2115\n\u22a2 Option.map (f x\u271d\u00b9) (decode x\u271d) = Option.bind (decode x\u271d) (Option.some \u2218 f x\u271d\u00b9)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Hom/Set.lean", "full_name": "OrderIso.image_symm_image", "start": [37, 1], "end": [38, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Disjointed.lean", "full_name": "Monotone.disjointed_eq", "start": [105, 1], "end": [106, 89], "traced_tactics": [{"tactic": "rw [disjointed_succ, hf.partialSups_eq]", "annotated_tactic": ["rw [<a>disjointed_succ</a>, hf.partialSups_eq]", [{"full_name": "disjointed_succ", "def_path": "Mathlib/Order/Disjointed.lean", "def_pos": [59, 9], "def_end_pos": [59, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nf : \u2115 \u2192 \u03b1\nhf : Monotone f\nn : \u2115\n\u22a2 disjointed f (n + 1) = f (n + 1) \\ f n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/SuccPred/Limit.lean", "full_name": "Order.not_isSuccLimit_succ_of_not_isMax", "start": [75, 1], "end": [77, 17], "traced_tactics": [{"tactic": "contrapose! ha", "annotated_tactic": ["contrapose! ha", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\na : \u03b1\ninst\u271d : SuccOrder \u03b1\nha : \u00acIsMax a\n\u22a2 \u00acIsSuccLimit (succ a)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\na : \u03b1\ninst\u271d : SuccOrder \u03b1\nha : IsSuccLimit (succ a)\n\u22a2 IsMax a"}, {"tactic": "exact ha.isMax", "annotated_tactic": ["exact ha.isMax", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\na : \u03b1\ninst\u271d : SuccOrder \u03b1\nha : IsSuccLimit (succ a)\n\u22a2 IsMax a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.Sublist.length_le", "start": [463, 1], "end": [466, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.sup'_map", "start": [908, 1], "end": [911, 6], "traced_tactics": [{"tactic": "rw [\u2190 WithBot.coe_eq_coe, coe_sup', sup_map, coe_sup']", "annotated_tactic": ["rw [\u2190 <a>WithBot.coe_eq_coe</a>, <a>coe_sup'</a>, <a>sup_map</a>, <a>coe_sup'</a>]", [{"full_name": "WithBot.coe_eq_coe", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [130, 9], "def_end_pos": [130, 19]}, {"full_name": "Finset.coe_sup'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [772, 9], "def_end_pos": [772, 17]}, {"full_name": "Finset.sup_map", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [69, 9], "def_end_pos": [69, 16]}, {"full_name": "Finset.coe_sup'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [772, 9], "def_end_pos": [772, 17]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : SemilatticeSup \u03b1\ns\u271d : Finset \u03b2\nH : Finset.Nonempty s\u271d\nf\u271d : \u03b2 \u2192 \u03b1\ns : Finset \u03b3\nf : \u03b3 \u21aa \u03b2\ng : \u03b2 \u2192 \u03b1\nhs : Finset.Nonempty (map f s)\nhs' : optParam (Finset.Nonempty s) (_ : Finset.Nonempty s)\n\u22a2 sup' (map f s) hs g = sup' s hs' (g \u2218 \u2191f)", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : SemilatticeSup \u03b1\ns\u271d : Finset \u03b2\nH : Finset.Nonempty s\u271d\nf\u271d : \u03b2 \u2192 \u03b1\ns : Finset \u03b3\nf : \u03b3 \u21aa \u03b2\ng : \u03b2 \u2192 \u03b1\nhs : Finset.Nonempty (map f s)\nhs' : optParam (Finset.Nonempty s) (_ : Finset.Nonempty s)\n\u22a2 sup s ((WithBot.some \u2218 g) \u2218 \u2191f) = sup s (WithBot.some \u2218 g \u2218 \u2191f)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : SemilatticeSup \u03b1\ns\u271d : Finset \u03b2\nH : Finset.Nonempty s\u271d\nf\u271d : \u03b2 \u2192 \u03b1\ns : Finset \u03b3\nf : \u03b3 \u21aa \u03b2\ng : \u03b2 \u2192 \u03b1\nhs : Finset.Nonempty (map f s)\nhs' : optParam (Finset.Nonempty s) (_ : Finset.Nonempty s)\n\u22a2 sup s ((WithBot.some \u2218 g) \u2218 \u2191f) = sup s (WithBot.some \u2218 g \u2218 \u2191f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Algebra/Operations.lean", "full_name": "Submodule.comap_unop_one", "start": [152, 1], "end": [154, 45], "traced_tactics": [{"tactic": "rw [\u2190 map_equiv_eq_comap_symm, map_op_one]", "annotated_tactic": ["rw [\u2190 <a>map_equiv_eq_comap_symm</a>, <a>map_op_one</a>]", [{"full_name": "Submodule.map_equiv_eq_comap_symm", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [2401, 9], "def_end_pos": [2401, 32]}, {"full_name": "Submodule.map_op_one", "def_path": "Mathlib/Algebra/Algebra/Operations.lean", "def_pos": [131, 9], "def_end_pos": [131, 19]}]], "state_before": "\u03b9 : Sort u\u03b9\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nA : Type v\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n\u22a2 comap (\u2191(LinearEquiv.symm (opLinearEquiv R))) 1 = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/ConstMulAction.lean", "full_name": "ContinuousConstSMul.secondCountableTopology", "start": [540, 1], "end": [542, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/Choose/Basic.lean", "full_name": "Nat.succ_mul_choose_eq", "start": [117, 1], "end": [123, 87], "traced_tactics": [{"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "\u22a2 succ 0 * choose 0 0 = choose (succ 0) (succ 0) * succ 0", "state_after": "no goals"}, {"tactic": "simp [choose]", "annotated_tactic": ["simp [<a>choose</a>]", [{"full_name": "Nat.choose", "def_path": "Mathlib/Data/Nat/Choose/Basic.lean", "def_pos": [46, 5], "def_end_pos": [46, 11]}]], "state_before": "k : \u2115\n\u22a2 succ 0 * choose 0 (k + 1) = choose (succ 0) (succ (k + 1)) * succ (k + 1)", "state_after": "no goals"}, {"tactic": "simp [choose, mul_succ, succ_eq_add_one, add_comm]", "annotated_tactic": ["simp [<a>choose</a>, <a>mul_succ</a>, <a>succ_eq_add_one</a>, <a>add_comm</a>]", [{"full_name": "Nat.choose", "def_path": "Mathlib/Data/Nat/Choose/Basic.lean", "def_pos": [46, 5], "def_end_pos": [46, 11]}, {"full_name": "Nat.mul_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [162, 9], "def_end_pos": [162, 17]}, {"full_name": "Nat.succ_eq_add_one", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [128, 9], "def_end_pos": [128, 24]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "n : \u2115\n\u22a2 succ (n + 1) * choose (n + 1) 0 = choose (succ (n + 1)) (succ 0) * succ 0", "state_after": "no goals"}, {"tactic": "rw [choose_succ_succ (succ n) (succ k), add_mul, \u2190 succ_mul_choose_eq n, mul_succ, \u2190\n  succ_mul_choose_eq n, add_right_comm, \u2190 mul_add, \u2190 choose_succ_succ, \u2190 succ_mul]", "annotated_tactic": ["rw [<a>choose_succ_succ</a> (<a>succ</a> n) (<a>succ</a> k), <a>add_mul</a>, \u2190 succ_mul_choose_eq n, <a>mul_succ</a>, \u2190\n      succ_mul_choose_eq n, <a>add_right_comm</a>, \u2190 <a>mul_add</a>, \u2190 <a>choose_succ_succ</a>, \u2190 <a>succ_mul</a>]", [{"full_name": "Nat.choose_succ_succ", "def_path": "Mathlib/Data/Nat/Choose/Basic.lean", "def_pos": [61, 9], "def_end_pos": [61, 25]}, {"full_name": "Nat.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}, {"full_name": "Nat.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}, {"full_name": "add_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [91, 7], "def_end_pos": [91, 14]}, {"full_name": "Nat.mul_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [162, 9], "def_end_pos": [162, 17]}, {"full_name": "add_right_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [101, 3], "def_end_pos": [101, 14]}, {"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}, {"full_name": "Nat.choose_succ_succ", "def_path": "Mathlib/Data/Nat/Choose/Basic.lean", "def_pos": [61, 9], "def_end_pos": [61, 25]}, {"full_name": "Nat.succ_mul", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [169, 9], "def_end_pos": [169, 17]}]], "state_before": "n k : \u2115\n\u22a2 succ (n + 1) * choose (n + 1) (k + 1) = choose (succ (n + 1)) (succ (k + 1)) * succ (k + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sum.lean", "full_name": "Finset.inr_mem_disjSum", "start": [78, 1], "end": [79, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean", "full_name": "Real.rpow_eq_nhds_of_pos", "start": [209, 1], "end": [216, 81], "traced_tactics": [{"tactic": "suffices : \u2200\u1da0 x : \u211d \u00d7 \u211d in \ud835\udcdd p, 0 < x.1", "annotated_tactic": ["suffices : \u2200\u1da0 x : \u211d \u00d7 \u211d in \ud835\udcdd p, 0 < x.1", []], "state_before": "p : \u211d \u00d7 \u211d\nhp_fst : 0 < p.1\n\u22a2 (fun x => x.1 ^ x.2) =\u1da0[\ud835\udcdd p] fun x => rexp (log x.1 * x.2)", "state_after": "p : \u211d \u00d7 \u211d\nhp_fst : 0 < p.1\nthis : \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, 0 < x.1\n\u22a2 (fun x => x.1 ^ x.2) =\u1da0[\ud835\udcdd p] fun x => rexp (log x.1 * x.2)\n\ncase this\np : \u211d \u00d7 \u211d\nhp_fst : 0 < p.1\n\u22a2 \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, 0 < x.1"}, {"tactic": "exact\n  this.mono fun x hx => by\n    dsimp only\n    rw [rpow_def_of_pos hx]", "annotated_tactic": ["exact\n    this.mono fun x hx => by\n      dsimp only\n      rw [<a>rpow_def_of_pos</a> hx]", [{"full_name": "Real.rpow_def_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [54, 9], "def_end_pos": [54, 24]}]], "state_before": "p : \u211d \u00d7 \u211d\nhp_fst : 0 < p.1\nthis : \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, 0 < x.1\n\u22a2 (fun x => x.1 ^ x.2) =\u1da0[\ud835\udcdd p] fun x => rexp (log x.1 * x.2)\n\ncase this\np : \u211d \u00d7 \u211d\nhp_fst : 0 < p.1\n\u22a2 \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, 0 < x.1", "state_after": "case this\np : \u211d \u00d7 \u211d\nhp_fst : 0 < p.1\n\u22a2 \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, 0 < x.1"}, {"tactic": "exact IsOpen.eventually_mem (isOpen_lt continuous_const continuous_fst) hp_fst", "annotated_tactic": ["exact <a>IsOpen.eventually_mem</a> (<a>isOpen_lt</a> <a>continuous_const</a> <a>continuous_fst</a>) hp_fst", [{"full_name": "IsOpen.eventually_mem", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 30]}, {"full_name": "isOpen_lt", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [317, 9], "def_end_pos": [317, 18]}, {"full_name": "continuous_const", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1723, 9], "def_end_pos": [1723, 25]}, {"full_name": "continuous_fst", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [328, 9], "def_end_pos": [328, 23]}]], "state_before": "case this\np : \u211d \u00d7 \u211d\nhp_fst : 0 < p.1\n\u22a2 \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, 0 < x.1", "state_after": "no goals"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "state_before": "p : \u211d \u00d7 \u211d\nhp_fst : 0 < p.1\nthis : \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, 0 < x.1\nx : \u211d \u00d7 \u211d\nhx : 0 < x.1\n\u22a2 (fun x => x.1 ^ x.2) x = (fun x => rexp (log x.1 * x.2)) x", "state_after": "p : \u211d \u00d7 \u211d\nhp_fst : 0 < p.1\nthis : \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, 0 < x.1\nx : \u211d \u00d7 \u211d\nhx : 0 < x.1\n\u22a2 x.1 ^ x.2 = rexp (log x.1 * x.2)"}, {"tactic": "rw [rpow_def_of_pos hx]", "annotated_tactic": ["rw [<a>rpow_def_of_pos</a> hx]", [{"full_name": "Real.rpow_def_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [54, 9], "def_end_pos": [54, 24]}]], "state_before": "p : \u211d \u00d7 \u211d\nhp_fst : 0 < p.1\nthis : \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, 0 < x.1\nx : \u211d \u00d7 \u211d\nhx : 0 < x.1\n\u22a2 x.1 ^ x.2 = rexp (log x.1 * x.2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/IndicatorFunction.lean", "full_name": "Set.mulIndicator_comp_of_one", "start": [261, 1], "end": [265, 25], "traced_tactics": [{"tactic": "funext", "annotated_tactic": ["funext", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g\u271d : \u03b1 \u2192 M\na : \u03b1\ng : M \u2192 N\nhg : g 1 = 1\n\u22a2 mulIndicator s (g \u2218 f) = g \u2218 mulIndicator s f", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g\u271d : \u03b1 \u2192 M\na : \u03b1\ng : M \u2192 N\nhg : g 1 = 1\nx\u271d : \u03b1\n\u22a2 mulIndicator s (g \u2218 f) x\u271d = (g \u2218 mulIndicator s f) x\u271d"}, {"tactic": "simp only [mulIndicator]", "annotated_tactic": ["simp only [<a>mulIndicator</a>]", [{"full_name": "Set.mulIndicator", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [47, 19], "def_end_pos": [47, 31]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g\u271d : \u03b1 \u2192 M\na : \u03b1\ng : M \u2192 N\nhg : g 1 = 1\nx\u271d : \u03b1\n\u22a2 mulIndicator s (g \u2218 f) x\u271d = (g \u2218 mulIndicator s f) x\u271d", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g\u271d : \u03b1 \u2192 M\na : \u03b1\ng : M \u2192 N\nhg : g 1 = 1\nx\u271d : \u03b1\n\u22a2 (if x\u271d \u2208 s then (g \u2218 f) x\u271d else 1) = (g \u2218 mulIndicator s f) x\u271d"}, {"tactic": "split_ifs <;> simp [*]", "annotated_tactic": ["split_ifs <;> simp [*]", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b9 : One M\ninst\u271d : One N\ns t : Set \u03b1\nf g\u271d : \u03b1 \u2192 M\na : \u03b1\ng : M \u2192 N\nhg : g 1 = 1\nx\u271d : \u03b1\n\u22a2 (if x\u271d \u2208 s then (g \u2218 f) x\u271d else 1) = (g \u2218 mulIndicator s f) x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/NormedSpace/MatrixExponential.lean", "full_name": "Matrix.exp_transpose", "start": [108, 1], "end": [109, 71], "traced_tactics": [{"tactic": "simp_rw [exp_eq_tsum, transpose_tsum, transpose_smul, transpose_pow]", "annotated_tactic": ["simp_rw [<a>exp_eq_tsum</a>, <a>transpose_tsum</a>, <a>transpose_smul</a>, <a>transpose_pow</a>]", [{"full_name": "exp_eq_tsum", "def_path": "Mathlib/Analysis/NormedSpace/Exponential.lean", "def_pos": [108, 9], "def_end_pos": [108, 20]}, {"full_name": "Matrix.transpose_tsum", "def_path": "Mathlib/Topology/Instances/Matrix.lean", "def_pos": [308, 9], "def_end_pos": [308, 30]}, {"full_name": "Matrix.transpose_smul", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [2013, 9], "def_end_pos": [2013, 23]}, {"full_name": "Matrix.transpose_pow", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [2098, 9], "def_end_pos": [2098, 22]}]], "state_before": "\ud835\udd42 : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nn' : m \u2192 Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : Field \ud835\udd42\ninst\u271d\u2074 : CommRing \ud835\udd38\ninst\u271d\u00b3 : TopologicalSpace \ud835\udd38\ninst\u271d\u00b2 : TopologicalRing \ud835\udd38\ninst\u271d\u00b9 : Algebra \ud835\udd42 \ud835\udd38\ninst\u271d : T2Space \ud835\udd38\nA : Matrix m m \ud835\udd38\n\u22a2 exp \ud835\udd42 A\u1d40 = (exp \ud835\udd42 A)\u1d40", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean", "full_name": "SimpleGraph.exists_ne_odd_degree_of_exists_odd_degree", "start": [158, 1], "end": [167, 16], "traced_tactics": [{"tactic": "haveI := Classical.decEq V", "annotated_tactic": ["haveI := <a>Classical.decEq</a> V", [{"full_name": "Classical.decEq", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [983, 19], "def_end_pos": [983, 24]}]], "state_before": "V : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\n\u22a2 \u2203 w, w \u2260 v \u2227 Odd (degree G w)", "state_after": "V : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nthis : DecidableEq V\n\u22a2 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w, w \u2260 v \u2227 Odd (degree G w)", "state_after": "case intro\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nthis : DecidableEq V\nk : \u2115\nhg : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) = 2 * k + 1\nhg' : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) > 0\n\u22a2 \u2203 w, w \u2260 v \u2227 Odd (degree G w)"}, {"tactic": "rcases card_pos.mp hg' with \u27e8w, hw\u27e9", "annotated_tactic": ["rcases card_pos.mp hg' with \u27e8w, hw\u27e9", []], "state_before": "case intro\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nthis : DecidableEq V\nk : \u2115\nhg : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) = 2 * k + 1\nhg' : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) > 0\n\u22a2 \u2203 w, w \u2260 v \u2227 Odd (degree G w)", "state_after": "case intro.intro\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nthis : DecidableEq V\nk : \u2115\nhg : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) = 2 * k + 1\nhg' : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) > 0\nw : V\nhw : w \u2208 filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ\n\u22a2 \u2203 w, w \u2260 v \u2227 Odd (degree G w)"}, {"tactic": "simp only [true_and_iff, mem_filter, mem_univ, Ne.def] at hw", "annotated_tactic": ["simp only [<a>true_and_iff</a>, <a>mem_filter</a>, <a>mem_univ</a>, <a>Ne.def</a>] at hw", [{"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2708, 9], "def_end_pos": [2708, 19]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}]], "state_before": "case intro.intro\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nthis : DecidableEq V\nk : \u2115\nhg : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) = 2 * k + 1\nhg' : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) > 0\nw : V\nhw : w \u2208 filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ\n\u22a2 \u2203 w, w \u2260 v \u2227 Odd (degree G w)", "state_after": "case intro.intro\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nthis : DecidableEq V\nk : \u2115\nhg : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) = 2 * k + 1\nhg' : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) > 0\nw : V\nhw : \u00acw = v \u2227 Odd (degree G w)\n\u22a2 \u2203 w, w \u2260 v \u2227 Odd (degree G w)"}, {"tactic": "exact \u27e8w, hw\u27e9", "annotated_tactic": ["exact \u27e8w, hw\u27e9", []], "state_before": "case intro.intro\nV : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nthis : DecidableEq V\nk : \u2115\nhg : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) = 2 * k + 1\nhg' : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) > 0\nw : V\nhw : \u00acw = v \u2227 Odd (degree G w)\n\u22a2 \u2203 w, w \u2260 v \u2227 Odd (degree G w)", "state_after": "no goals"}, {"tactic": "rw [hg]", "annotated_tactic": ["rw [hg]", []], "state_before": "V : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nthis : DecidableEq V\nk : \u2115\nhg : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) = 2 * k + 1\n\u22a2 card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) > 0", "state_after": "V : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nthis : DecidableEq V\nk : \u2115\nhg : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) = 2 * k + 1\n\u22a2 2 * k + 1 > 0"}, {"tactic": "apply Nat.succ_pos", "annotated_tactic": ["apply <a>Nat.succ_pos</a>", [{"full_name": "Nat.succ_pos", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1608, 9], "def_end_pos": [1608, 21]}]], "state_before": "V : Type u\nG : SimpleGraph V\ninst\u271d\u00b9 : Fintype V\ninst\u271d : DecidableRel G.Adj\nv : V\nh : Odd (degree G v)\nthis : DecidableEq V\nk : \u2115\nhg : card (filter (fun w => w \u2260 v \u2227 Odd (degree G w)) univ) = 2 * k + 1\n\u22a2 2 * k + 1 > 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "full_name": "CategoryTheory.MonoidalCategory.tensor_inv_hom_id'", "start": [434, 1], "end": [436, 55], "traced_tactics": [{"tactic": "rw [\u2190 tensor_comp, IsIso.inv_hom_id, comp_tensor_id]", "annotated_tactic": ["rw [\u2190 <a>tensor_comp</a>, <a>IsIso.inv_hom_id</a>, <a>comp_tensor_id</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.tensor_comp", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [93, 3], "def_end_pos": [93, 14]}, {"full_name": "CategoryTheory.IsIso.inv_hom_id", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [278, 9], "def_end_pos": [278, 19]}, {"full_name": "CategoryTheory.MonoidalCategory.comp_tensor_id", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [264, 9], "def_end_pos": [264, 23]}]], "state_before": "C\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b3 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : MonoidalCategory C\nU V\u271d W\u271d X\u271d Y\u271d Z\u271d V W X Y Z : C\nf : V \u27f6 W\ninst\u271d : IsIso f\ng : X \u27f6 Y\nh : Y \u27f6 Z\n\u22a2 (g \u2297 inv f) \u226b (h \u2297 f) = (g \u2297 \ud835\udfd9 W) \u226b (h \u2297 \ud835\udfd9 W)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "full_name": "Asymptotics.isBigOWith_inv", "start": [276, 1], "end": [277, 63], "traced_tactics": [{"tactic": "simp only [IsBigOWith_def, \u2190 div_eq_inv_mul, le_div_iff' hc]", "annotated_tactic": ["simp only [<a>IsBigOWith_def</a>, \u2190 <a>div_eq_inv_mul</a>, <a>le_div_iff'</a> hc]", [{"full_name": "Asymptotics.IsBigOWith_def", "def_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "def_pos": [1, 1], "def_end_pos": [1, 1]}, {"full_name": "div_eq_inv_mul", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [492, 9], "def_end_pos": [492, 23]}, {"full_name": "le_div_iff'", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [141, 9], "def_end_pos": [141, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u00b3 : Norm E\ninst\u271d\u00b9\u00b2 : Norm F\ninst\u271d\u00b9\u00b9 : Norm G\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup E'\ninst\u271d\u2079 : SeminormedAddCommGroup F'\ninst\u271d\u2078 : SeminormedAddCommGroup G'\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedAddCommGroup F''\ninst\u271d\u2075 : NormedAddCommGroup G''\ninst\u271d\u2074 : SeminormedRing R\ninst\u271d\u00b3 : SeminormedAddGroup E'''\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\nhc : 0 < c\n\u22a2 IsBigOWith c\u207b\u00b9 l f g \u2194 \u2200\u1da0 (x : \u03b1) in l, c * \u2016f x\u2016 \u2264 \u2016g x\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/UniformSpace/Completion.lean", "full_name": "UniformSpace.Completion.uniformEmbedding_coe", "start": [450, 1], "end": [452, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean", 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0\n\u22a2 y - z = x", "state_after": "no goals"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "\u03b1 \u03b2 : Type\nn : \u2115\nf g : (\u03b1 \u2192 \u2115) \u2192 \u2115\ndf : DiophFn f\ndg : DiophFn g\nx y z : \u2115\n\u22a2 y - z = x \u2192 y = x + z \u2228 y \u2264 z \u2227 x = 0", "state_after": "\u03b1 \u03b2 : Type\nn : \u2115\nf g : (\u03b1 \u2192 \u2115) \u2192 \u2115\ndf : DiophFn f\ndg : DiophFn g\ny z : \u2115\n\u22a2 y = y - z + z \u2228 y \u2264 z \u2227 y - z = 0"}, {"tactic": "cases' le_total y z with yz zy", "annotated_tactic": ["cases' <a>le_total</a> y z with yz zy", [{"full_name": "le_total", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [295, 9], "def_end_pos": [295, 17]}]], "state_before": "\u03b1 \u03b2 : Type\nn : \u2115\nf g : (\u03b1 \u2192 \u2115) \u2192 \u2115\ndf : DiophFn f\ndg : DiophFn g\ny z : \u2115\n\u22a2 y = y - z + z \u2228 y \u2264 z \u2227 y - z = 0", "state_after": "case inl\n\u03b1 \u03b2 : Type\nn 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"annotated_tactic": ["exact <a>Or.inl</a> (<a>tsub_add_cancel_of_le</a> zy).<a>symm</a>", [{"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "tsub_add_cancel_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [30, 9], "def_end_pos": [30, 30]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case inr\n\u03b1 \u03b2 : Type\nn : \u2115\nf g : (\u03b1 \u2192 \u2115) \u2192 \u2115\ndf : DiophFn f\ndg : DiophFn g\ny z : \u2115\nzy : z \u2264 y\n\u22a2 y = y - z + z \u2228 y \u2264 z \u2227 y - z = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Compactness/Compact.lean", "full_name": "Filter.mem_coclosed_compact'", "start": [596, 1], "end": 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"3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "full_name": "MeasureTheory.mem_lpMeas_indicatorConstLp", "start": [262, 1], "end": [266, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/UniformSpace/Equicontinuity.lean", "full_name": "equicontinuousAt_finite", "start": [152, 1], "end": [155, 41], "traced_tactics": [{"tactic": "simp [EquicontinuousAt, ContinuousAt, (nhds_basis_uniformity' (\ud835\udce4 \u03b1).basis_sets).tendsto_right_iff,\n  UniformSpace.ball, @forall_swap _ \u03b9]", "annotated_tactic": ["simp [<a>EquicontinuousAt</a>, <a>ContinuousAt</a>, (<a>nhds_basis_uniformity'</a> (\ud835\udce4 \u03b1).<a>basis_sets</a>).<a>tendsto_right_iff</a>,\n    <a>UniformSpace.ball</a>, @<a>forall_swap</a> _ \u03b9]", [{"full_name": "EquicontinuousAt", "def_path": 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s\n\u22a2 ContDiffWithinAt \ud835\udd5c 0 f s x \u2194 \u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 ContinuousOn f (s \u2229 u)", "state_after": "case mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\n\u22a2 ContDiffWithinAt \ud835\udd5c 0 f s x \u2192 \u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 ContinuousOn f (s \u2229 u)\n\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\n\u22a2 (\u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 ContinuousOn f (s \u2229 u)) \u2192 ContDiffWithinAt \ud835\udd5c 0 f s x"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\n\u22a2 ContDiffWithinAt \ud835\udd5c 0 f s x \u2192 \u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 ContinuousOn f (s \u2229 u)", "state_after": "case mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nh : ContDiffWithinAt \ud835\udd5c 0 f s x\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 ContinuousOn f (s \u2229 u)"}, {"tactic": "obtain \u27e8u, H, p, hp\u27e9 := h 0 le_rfl", "annotated_tactic": ["obtain \u27e8u, H, p, hp\u27e9 := h 0 <a>le_rfl</a>", [{"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nh : ContDiffWithinAt \ud835\udd5c 0 f s x\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 ContinuousOn f (s \u2229 u)", "state_after": "case mp.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nh : ContDiffWithinAt \ud835\udd5c 0 f s x\nu : Set E\nH : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhp : HasFTaylorSeriesUpToOn (\u21910) f p u\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 ContinuousOn f (s \u2229 u)"}, {"tactic": "refine' \u27e8u, _, _\u27e9", "annotated_tactic": ["refine' \u27e8u, _, _\u27e9", []], "state_before": "case mp.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nh : ContDiffWithinAt \ud835\udd5c 0 f s x\nu : Set E\nH : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhp : HasFTaylorSeriesUpToOn (\u21910) f p u\n\u22a2 \u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 ContinuousOn f (s \u2229 u)", "state_after": "case mp.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nh : ContDiffWithinAt \ud835\udd5c 0 f s x\nu : Set E\nH : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhp : HasFTaylorSeriesUpToOn (\u21910) f p u\n\u22a2 u \u2208 \ud835\udcdd[s] x\n\ncase mp.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nh : ContDiffWithinAt \ud835\udd5c 0 f s x\nu : Set E\nH : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhp : HasFTaylorSeriesUpToOn (\u21910) f p u\n\u22a2 ContinuousOn f (s \u2229 u)"}, {"tactic": "simpa [hx] using H", "annotated_tactic": ["simpa [hx] using H", []], "state_before": "case mp.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nh : ContDiffWithinAt \ud835\udd5c 0 f s x\nu : Set E\nH : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhp : HasFTaylorSeriesUpToOn (\u21910) f p u\n\u22a2 u \u2208 \ud835\udcdd[s] x", "state_after": "no goals"}, {"tactic": "simp only [Nat.cast_zero, hasFTaylorSeriesUpToOn_zero_iff] at hp", "annotated_tactic": ["simp only [<a>Nat.cast_zero</a>, <a>hasFTaylorSeriesUpToOn_zero_iff</a>] at hp", [{"full_name": "Nat.cast_zero", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [114, 9], "def_end_pos": [114, 18]}, {"full_name": "hasFTaylorSeriesUpToOn_zero_iff", "def_path": "Mathlib/Analysis/Calculus/ContDiffDef.lean", "def_pos": [228, 9], "def_end_pos": [228, 40]}]], "state_before": "case mp.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nh : ContDiffWithinAt \ud835\udd5c 0 f s x\nu : Set E\nH : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhp : HasFTaylorSeriesUpToOn (\u21910) f p u\n\u22a2 ContinuousOn f (s \u2229 u)", "state_after": "case mp.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nh : ContDiffWithinAt \ud835\udd5c 0 f s x\nu : Set E\nH : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhp : ContinuousOn f u \u2227 \u2200 (x : E), x \u2208 u \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\n\u22a2 ContinuousOn f (s \u2229 u)"}, {"tactic": "exact hp.1.mono (inter_subset_right s u)", "annotated_tactic": ["exact hp.1.<a>mono</a> (<a>inter_subset_right</a> s u)", [{"full_name": "ContinuousOn.mono", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [934, 9], "def_end_pos": [934, 26]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}]], "state_before": "case mp.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np\u271d : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nh : ContDiffWithinAt \ud835\udd5c 0 f s x\nu : Set E\nH : u \u2208 \ud835\udcdd[insert x s] x\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhp : ContinuousOn f u \u2227 \u2200 (x : E), x \u2208 u \u2192 ContinuousMultilinearMap.uncurry0 (p x 0) = f x\n\u22a2 ContinuousOn f (s \u2229 u)", "state_after": "no goals"}, {"tactic": "rintro \u27e8u, H, hu\u27e9", "annotated_tactic": ["rintro \u27e8u, H, hu\u27e9", []], "state_before": "case mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\n\u22a2 (\u2203 u, u \u2208 \ud835\udcdd[s] x \u2227 ContinuousOn f (s \u2229 u)) \u2192 ContDiffWithinAt \ud835\udd5c 0 f s x", "state_after": "case mpr.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nu : Set E\nH : u \u2208 \ud835\udcdd[s] x\nhu : ContinuousOn f (s \u2229 u)\n\u22a2 ContDiffWithinAt \ud835\udd5c 0 f s x"}, {"tactic": "rw [\u2190 contDiffWithinAt_inter' H]", "annotated_tactic": ["rw [\u2190 <a>contDiffWithinAt_inter'</a> H]", [{"full_name": "contDiffWithinAt_inter'", "def_path": "Mathlib/Analysis/Calculus/ContDiffDef.lean", "def_pos": [506, 9], "def_end_pos": [506, 32]}]], "state_before": "case mpr.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nu : Set E\nH : u \u2208 \ud835\udcdd[s] x\nhu : ContinuousOn f (s \u2229 u)\n\u22a2 ContDiffWithinAt \ud835\udd5c 0 f s x", "state_after": "case mpr.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nu : Set E\nH : u \u2208 \ud835\udcdd[s] x\nhu : ContinuousOn f (s \u2229 u)\n\u22a2 ContDiffWithinAt \ud835\udd5c 0 f (s \u2229 u) x"}, {"tactic": "have h' : x \u2208 s \u2229 u := \u27e8hx, mem_of_mem_nhdsWithin hx H\u27e9", "annotated_tactic": ["have h' : x \u2208 s \u2229 u := \u27e8hx, <a>mem_of_mem_nhdsWithin</a> hx H\u27e9", [{"full_name": "mem_of_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [171, 9], "def_end_pos": [171, 30]}]], "state_before": "case mpr.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nu : Set E\nH : u \u2208 \ud835\udcdd[s] x\nhu : ContinuousOn f (s \u2229 u)\n\u22a2 ContDiffWithinAt \ud835\udd5c 0 f (s \u2229 u) x", "state_after": "case mpr.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nu : Set E\nH : u \u2208 \ud835\udcdd[s] x\nhu : ContinuousOn f (s \u2229 u)\nh' : x \u2208 s \u2229 u\n\u22a2 ContDiffWithinAt \ud835\udd5c 0 f (s \u2229 u) x"}, {"tactic": "exact (contDiffOn_zero.mpr hu).contDiffWithinAt h'", "annotated_tactic": ["exact (contDiffOn_zero.mpr hu).<a>contDiffWithinAt</a> h'", [{"full_name": "ContDiffOn.contDiffWithinAt", "def_path": "Mathlib/Analysis/Calculus/ContDiffDef.lean", "def_pos": [648, 9], "def_end_pos": [648, 36]}]], "state_before": "case mpr.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u\u271d : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhx : x \u2208 s\nu : Set E\nH : u \u2208 \ud835\udcdd[s] x\nhu : ContinuousOn f (s \u2229 u)\nh' : x \u2208 s \u2229 u\n\u22a2 ContDiffWithinAt \ud835\udd5c 0 f (s \u2229 u) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Ordinal/NaturalOps.lean", "full_name": "Ordinal.nmul_nadd_le\u2083'", "start": [683, 1], "end": [688, 74], "traced_tactics": [{"tactic": "simp only [nmul_comm _ (_ \u2a33 _)]", "annotated_tactic": ["simp only [<a>nmul_comm</a> _ (_ \u2a33 _)]", [{"full_name": "Ordinal.nmul_comm", "def_path": "Mathlib/SetTheory/Ordinal/NaturalOps.lean", "def_pos": [553, 9], "def_end_pos": [553, 18]}]], "state_before": "a b c d a' b' c' : Ordinal.{u}\nha : a' \u2264 a\nhb : b' \u2264 b\nhc : c' \u2264 c\n\u22a2 a' \u2a33 (b \u2a33 c) \u266f a \u2a33 (b' \u2a33 c) \u266f a \u2a33 (b \u2a33 c') \u266f a' \u2a33 (b' \u2a33 c') \u2264\n    a \u2a33 (b \u2a33 c) \u266f a' \u2a33 (b' \u2a33 c) \u266f a' \u2a33 (b \u2a33 c') \u266f a \u2a33 (b' \u2a33 c')", "state_after": "a b c d a' b' c' : Ordinal.{u}\nha : a' \u2264 a\nhb : b' \u2264 b\nhc : c' \u2264 c\n\u22a2 b \u2a33 c \u2a33 a' \u266f b' \u2a33 c \u2a33 a \u266f b \u2a33 c' \u2a33 a \u266f b' \u2a33 c' \u2a33 a' \u2264 b \u2a33 c \u2a33 a \u266f b' \u2a33 c \u2a33 a' \u266f b \u2a33 c' \u2a33 a' \u266f b' \u2a33 c' \u2a33 a"}, {"tactic": "simp only [nadd_eq_add, NatOrdinal.toOrdinal_toNatOrdinal]", "annotated_tactic": ["simp only [<a>nadd_eq_add</a>, <a>NatOrdinal.toOrdinal_toNatOrdinal</a>]", [{"full_name": "Ordinal.nadd_eq_add", "def_path": "Mathlib/SetTheory/Ordinal/NaturalOps.lean", "def_pos": [415, 9], "def_end_pos": [415, 20]}, {"full_name": "NatOrdinal.toOrdinal_toNatOrdinal", "def_path": "Mathlib/SetTheory/Ordinal/NaturalOps.lean", "def_pos": [88, 9], "def_end_pos": [88, 31]}]], "state_before": "case h.e'_4\na b c d a' b' c' : Ordinal.{u}\nha : a' \u2264 a\nhb : b' \u2264 b\nhc : c' \u2264 c\n\u22a2 b \u2a33 c \u2a33 a \u266f b' \u2a33 c \u2a33 a' \u266f b \u2a33 c' \u2a33 a' \u266f b' \u2a33 c' \u2a33 a = b \u2a33 c \u2a33 a \u266f b' \u2a33 c' \u2a33 a \u266f b' \u2a33 c \u2a33 a' \u266f b \u2a33 c' \u2a33 a'", "state_after": "case h.e'_4\na b c d a' b' c' : Ordinal.{u}\nha : a' \u2264 a\nhb : b' \u2264 b\nhc : c' \u2264 c\n\u22a2 \u2191toOrdinal\n      (\u2191toNatOrdinal (b \u2a33 c \u2a33 a) + \u2191toNatOrdinal (b' \u2a33 c \u2a33 a') + \u2191toNatOrdinal (b \u2a33 c' \u2a33 a') +\n        \u2191toNatOrdinal (b' \u2a33 c' \u2a33 a)) =\n    \u2191toOrdinal\n      (\u2191toNatOrdinal (b \u2a33 c \u2a33 a) + \u2191toNatOrdinal (b' \u2a33 c' \u2a33 a) + \u2191toNatOrdinal (b' \u2a33 c \u2a33 a') +\n        \u2191toNatOrdinal (b \u2a33 c' \u2a33 a'))"}, {"tactic": "abel_nf", "annotated_tactic": ["abel_nf", []], "state_before": "case h.e'_4\na b c d a' b' c' : Ordinal.{u}\nha : a' \u2264 a\nhb : b' \u2264 b\nhc : c' \u2264 c\n\u22a2 \u2191toOrdinal\n      (\u2191toNatOrdinal (b \u2a33 c \u2a33 a) + \u2191toNatOrdinal (b' \u2a33 c \u2a33 a') + \u2191toNatOrdinal (b \u2a33 c' \u2a33 a') +\n        \u2191toNatOrdinal (b' \u2a33 c' \u2a33 a)) =\n    \u2191toOrdinal\n      (\u2191toNatOrdinal (b \u2a33 c \u2a33 a) + \u2191toNatOrdinal (b' \u2a33 c' \u2a33 a) + \u2191toNatOrdinal (b' \u2a33 c \u2a33 a') +\n        \u2191toNatOrdinal (b \u2a33 c' \u2a33 a'))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/Padics/PadicVal.lean", "full_name": "padicValNat.prime_pow", "start": [532, 11], "end": [533, 80], "traced_tactics": [{"tactic": "rw [padicValNat.pow _ (@Fact.out p.Prime).ne_zero, padicValNat_self, mul_one]", "annotated_tactic": ["rw [<a>padicValNat.pow</a> _ (@<a>Fact.out</a> p.Prime).<a>ne_zero</a>, <a>padicValNat_self</a>, <a>mul_one</a>]", [{"full_name": "padicValNat.pow", "def_path": "Mathlib/NumberTheory/Padics/PadicVal.lean", "def_pos": [527, 19], "def_end_pos": [527, 22]}, {"full_name": "Fact.out", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [118, 3], "def_end_pos": [118, 6]}, {"full_name": "Nat.Prime.ne_zero", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [62, 9], "def_end_pos": [62, 22]}, {"full_name": "padicValNat_self", "def_path": "Mathlib/NumberTheory/Padics/PadicVal.lean", "def_pos": [289, 9], "def_end_pos": [289, 25]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "p a b : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 padicValNat p (p ^ n) = n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RepresentationTheory/Character.lean", "full_name": "FdRep.char_one", "start": [52, 1], "end": [53, 44], "traced_tactics": [{"tactic": "simp only [character, map_one, trace_one]", "annotated_tactic": ["simp only [<a>character</a>, <a>map_one</a>, <a>trace_one</a>]", [{"full_name": "FdRep.character", "def_path": "Mathlib/RepresentationTheory/Character.lean", "def_pos": [43, 5], "def_end_pos": [43, 14]}, {"full_name": "map_one", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [202, 9], "def_end_pos": [202, 16]}, {"full_name": "LinearMap.trace_one", "def_path": "Mathlib/LinearAlgebra/Trace.lean", "def_pos": [194, 9], "def_end_pos": [194, 18]}]], "state_before": "k : Type u\ninst\u271d\u00b9 : Field k\nG : Type u\ninst\u271d : Monoid G\nV : FdRep k G\n\u22a2 character V 1 = \u2191(finrank k (CoeSort.coe V))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "full_name": "lipschitzWith_iff_norm_div_le", "start": [834, 1], "end": [836, 62], "traced_tactics": [{"tactic": "simp only [lipschitzWith_iff_dist_le_mul, dist_eq_norm_div]", "annotated_tactic": ["simp only [<a>lipschitzWith_iff_dist_le_mul</a>, <a>dist_eq_norm_div</a>]", [{"full_name": "lipschitzWith_iff_dist_le_mul", "def_path": "Mathlib/Topology/MetricSpace/Lipschitz.lean", "def_pos": [59, 9], "def_end_pos": [59, 38]}, {"full_name": "dist_eq_norm_div", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [373, 9], "def_end_pos": [373, 25]}]], "state_before": "\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : E \u2192 F\nC : \u211d\u22650\n\u22a2 LipschitzWith C f \u2194 \u2200 (x y : E), \u2016f x / f y\u2016 \u2264 \u2191C * \u2016x / y\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/RootsOfUnity/Basic.lean", "full_name": "IsPrimitiveRoot.pow_sub_one_eq", "start": [675, 1], "end": [678, 68], "traced_tactics": [{"tactic": "rw [eq_neg_iff_add_eq_zero, add_comm, \u2190 sum_range_succ, \u2190 Nat.succ_eq_add_one,\n  Nat.succ_pred_eq_of_pos (pos_of_gt hk), h\u03b6.geom_sum_eq_zero hk]", "annotated_tactic": ["rw [<a>eq_neg_iff_add_eq_zero</a>, <a>add_comm</a>, \u2190 <a>sum_range_succ</a>, \u2190 <a>Nat.succ_eq_add_one</a>,\n    <a>Nat.succ_pred_eq_of_pos</a> (<a>pos_of_gt</a> hk), h\u03b6.geom_sum_eq_zero hk]", [{"full_name": "eq_neg_iff_add_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [666, 3], "def_end_pos": [666, 14]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "Finset.sum_range_succ", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1219, 3], "def_end_pos": [1219, 14]}, {"full_name": "Nat.succ_eq_add_one", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [128, 9], "def_end_pos": [128, 24]}, {"full_name": "Nat.succ_pred_eq_of_pos", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [232, 9], "def_end_pos": [232, 28]}, {"full_name": "pos_of_gt", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [305, 9], "def_end_pos": [305, 18]}]], "state_before": "M : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u2074 : CommMonoid M\ninst\u271d\u00b3 : CommMonoid N\ninst\u271d\u00b2 : DivisionCommMonoid G\nk l : \u2115\ninst\u271d\u00b9 : CommRing R\n\u03b6\u271d : R\u02e3\nh : IsPrimitiveRoot \u03b6\u271d k\ninst\u271d : IsDomain R\n\u03b6 : R\nh\u03b6 : IsPrimitiveRoot \u03b6 k\nhk : 1 < k\n\u22a2 \u03b6 ^ Nat.pred k = -\u2211 i in range (Nat.pred k), \u03b6 ^ i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/IndicatorFunction.lean", "full_name": "Antitone.mulIndicator_eventuallyEq_iInter", "start": [77, 1], "end": [80, 58], "traced_tactics": [{"tactic": "classical exact hs.piecewise_eventually_eq_iInter f 1 a", "annotated_tactic": ["classical exact hs.piecewise_eventually_eq_iInter f 1 a", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nE : Type u_4\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : One \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\n\u22a2 (fun i => mulIndicator (s i) f a) =\u1da0[atTop] fun x => mulIndicator (\u22c2 i, s i) f a", "state_after": "no goals"}, {"tactic": "exact hs.piecewise_eventually_eq_iInter f 1 a", "annotated_tactic": ["exact hs.piecewise_eventually_eq_iInter f 1 a", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\nE : Type u_4\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : One \u03b2\ns : \u03b9 \u2192 Set \u03b1\nhs : Antitone s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\n\u22a2 (fun i => mulIndicator (s i) f a) =\u1da0[atTop] fun x => mulIndicator (\u22c2 i, s i) f a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Manifold/VectorBundle/Basic.lean", "full_name": "VectorPrebundle.smoothCoordChange_apply", "start": [688, 1], "end": [691, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "full_name": "WithTop.add_coe_eq_top_iff", "start": [173, 1], "end": [174, 59], "traced_tactics": [{"tactic": "induction x using WithTop.recTopCoe <;> simp [\u2190 coe_add]", "annotated_tactic": ["induction x using <a>WithTop.recTopCoe</a> <;> simp [\u2190 <a>coe_add</a>]", [{"full_name": "WithTop.recTopCoe", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [634, 5], "def_end_pos": [634, 14]}, {"full_name": "WithTop.coe_add", "def_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "def_pos": [122, 9], "def_end_pos": [122, 16]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Add \u03b1\na b c d : WithTop \u03b1\nx\u271d y\u271d : \u03b1\nx : WithTop \u03b1\ny : \u03b1\n\u22a2 x + \u2191y = \u22a4 \u2194 x = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Cofinite.lean", "full_name": "Filter.comap_cofinite_le", "start": [109, 1], "end": [111, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Basic.lean", "full_name": "Metric.closedBall_subset_closedBall'", "start": [613, 1], "end": [618, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Valuation/Basic.lean", "full_name": "AddValuation.ext_iff", "start": [721, 1], "end": [722, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/Cast/Order.lean", "full_name": "Nat.one_lt_cast", "start": [101, 1], "end": [101, 73], "traced_tactics": [{"tactic": "rw [\u2190 cast_one, cast_lt]", "annotated_tactic": ["rw [\u2190 <a>cast_one</a>, <a>cast_lt</a>]", [{"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [141, 9], "def_end_pos": [141, 17]}, {"full_name": "Nat.cast_lt", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [96, 9], "def_end_pos": [96, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : AddCommMonoidWithOne \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : ZeroLEOneClass \u03b1\ninst\u271d : CharZero \u03b1\nm n : \u2115\n\u22a2 1 < \u2191n \u2194 1 < n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/UniformSpace/UniformEmbedding.lean", "full_name": "UniformEmbedding.comp", "start": [168, 1], "end": [170, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Disjoint.lean", "full_name": "Set.Ici_disjoint_Iic", "start": [54, 1], "end": [55, 72], "traced_tactics": [{"tactic": "rw [Set.disjoint_iff_inter_eq_empty, Ici_inter_Iic, Icc_eq_empty_iff]", "annotated_tactic": ["rw [<a>Set.disjoint_iff_inter_eq_empty</a>, <a>Ici_inter_Iic</a>, <a>Icc_eq_empty_iff</a>]", [{"full_name": "Set.disjoint_iff_inter_eq_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1538, 9], "def_end_pos": [1538, 36]}, {"full_name": "Set.Ici_inter_Iic", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [618, 9], "def_end_pos": [618, 22]}, {"full_name": "Set.Icc_eq_empty_iff", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 25]}]], "state_before": "\u03b9 : Sort u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d : Preorder \u03b1\na b c : \u03b1\n\u22a2 Disjoint (Ici a) (Iic b) \u2194 \u00aca \u2264 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RepresentationTheory/FdRep.lean", "full_name": "FdRep.Iso.conj_\u03c1", "start": [91, 1], "end": [96, 28], "traced_tactics": [{"tactic": "erw [FdRep.isoToLinearEquiv, \u2190 FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]", "annotated_tactic": ["erw [<a>FdRep.isoToLinearEquiv</a>, \u2190 <a>FGModuleCat.Iso.conj_eq_conj</a>, <a>Iso.conj_apply</a>]", [{"full_name": "FdRep.isoToLinearEquiv", "def_path": "Mathlib/RepresentationTheory/FdRep.lean", "def_pos": [87, 5], "def_end_pos": [87, 21]}, {"full_name": "FGModuleCat.Iso.conj_eq_conj", "def_path": "Mathlib/Algebra/Category/FGModuleCat/Basic.lean", "def_pos": [201, 9], "def_end_pos": [201, 25]}, {"full_name": "CategoryTheory.Iso.conj_apply", "def_path": "Mathlib/CategoryTheory/Conj.lean", "def_pos": [82, 9], "def_end_pos": [82, 19]}]], "state_before": "k G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\nV W : FdRep k G\ni : V \u2245 W\ng : G\n\u22a2 \u2191(\u03c1 W) g = \u2191(LinearEquiv.conj (isoToLinearEquiv i)) (\u2191(\u03c1 V) g)", "state_after": "k G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\nV W : FdRep k G\ni : V \u2245 W\ng : G\n\u22a2 \u2191(\u03c1 W) g =\n    ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).inv \u226b\n      \u2191(\u03c1 V) g \u226b ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).hom"}, {"tactic": "rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]", "annotated_tactic": ["rw [<a>Iso.eq_inv_comp</a> ((<a>Action.forget</a> (<a>FGModuleCat</a> k) (<a>MonCat.of</a> G)).<a>mapIso</a> i)]", [{"full_name": "CategoryTheory.Iso.eq_inv_comp", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [218, 9], "def_end_pos": [218, 20]}, {"full_name": "Action.forget", "def_path": "Mathlib/RepresentationTheory/Action.lean", "def_pos": [301, 5], "def_end_pos": [301, 11]}, {"full_name": "FGModuleCat", "def_path": "Mathlib/Algebra/Category/FGModuleCat/Basic.lean", "def_pos": [49, 5], "def_end_pos": [49, 16]}, {"full_name": "MonCat.of", "def_path": "Mathlib/Algebra/Category/MonCat/Basic.lean", "def_pos": [109, 5], "def_end_pos": [109, 7]}, {"full_name": "CategoryTheory.Functor.mapIso", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [589, 5], "def_end_pos": [589, 11]}]], "state_before": "k G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\nV W : FdRep k G\ni : V \u2245 W\ng : G\n\u22a2 \u2191(\u03c1 W) g =\n    ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).inv \u226b\n      \u2191(\u03c1 V) g \u226b ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).hom", "state_after": "k G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\nV W : FdRep k G\ni : V \u2245 W\ng : G\n\u22a2 ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).hom \u226b \u2191(\u03c1 W) g =\n    \u2191(\u03c1 V) g \u226b ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).hom"}, {"tactic": "exact (i.hom.comm g).symm", "annotated_tactic": ["exact (i.hom.comm g).<a>symm</a>", [{"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "k G : Type u\ninst\u271d\u00b9 : Field k\ninst\u271d : Monoid G\nV W : FdRep k G\ni : V \u2245 W\ng : G\n\u22a2 ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).hom \u226b \u2191(\u03c1 W) g =\n    \u2191(\u03c1 V) g \u226b ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).hom", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/AlgebraicGeometry/Pullbacks.lean", "full_name": "AlgebraicGeometry.Scheme.Pullback.lift_comp_\u03b9", "start": [442, 1], "end": [466, 43], "traced_tactics": [{"tactic": "rw [\u2190 pullback.condition_assoc, Category.assoc, p_comm]", "annotated_tactic": ["rw [\u2190 <a>pullback.condition_assoc</a>, <a>Category.assoc</a>, <a>p_comm</a>]", [{"full_name": "CategoryTheory.Limits.pullback.condition_assoc", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [1208, 3], "def_end_pos": [1208, 10]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [159, 3], "def_end_pos": [159, 8]}, {"full_name": "AlgebraicGeometry.Scheme.Pullback.p_comm", "def_path": "Mathlib/AlgebraicGeometry/Pullbacks.lean", "def_pos": [293, 9], "def_end_pos": [293, 15]}]], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g", "state_after": "no goals"}, {"tactic": "apply ((gluing \ud835\udcb0 f g).openCover.pullbackCover pullback.fst).hom_ext", "annotated_tactic": ["apply ((<a>gluing</a> \ud835\udcb0 f g).openCover.pullbackCover <a>pullback.fst</a>).<a>hom_ext</a>", [{"full_name": "AlgebraicGeometry.Scheme.Pullback.gluing", "def_path": "Mathlib/AlgebraicGeometry/Pullbacks.lean", "def_pos": [249, 5], "def_end_pos": [249, 11]}, {"full_name": "CategoryTheory.Limits.pullback.fst", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [1113, 8], "def_end_pos": [1113, 20]}, {"full_name": "AlgebraicGeometry.Scheme.OpenCover.hom_ext", "def_path": "Mathlib/AlgebraicGeometry/Gluing.lean", "def_pos": [475, 9], "def_end_pos": [475, 16]}]], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n        (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n      GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n    pullback.fst", "state_after": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 \u2200 (x : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J),\n    OpenCover.map (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst) x \u226b\n        pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n            (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n          GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n      OpenCover.map (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst) x \u226b pullback.fst"}, {"tactic": "intro j", "annotated_tactic": ["intro j", []], "state_before": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\n\u22a2 \u2200 (x : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J),\n    OpenCover.map (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst) x \u226b\n        pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n            (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n          GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n      OpenCover.map (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst) x \u226b pullback.fst", "state_after": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 OpenCover.map (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst) j \u226b\n      pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n          (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n        GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n    OpenCover.map (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst) j \u226b pullback.fst"}, {"tactic": "dsimp only [OpenCover.pullbackCover]", "annotated_tactic": ["dsimp only [<a>OpenCover.pullbackCover</a>]", [{"full_name": "AlgebraicGeometry.Scheme.OpenCover.pullbackCover", "def_path": "Mathlib/AlgebraicGeometry/OpenImmersion.lean", "def_pos": [734, 5], "def_end_pos": [734, 35]}]], "state_before": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 OpenCover.map (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst) j \u226b\n      pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n          (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n        GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n    OpenCover.map (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst) j \u226b pullback.fst", "state_after": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b\n      pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n          (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n        GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n    pullback.fst \u226b pullback.fst"}, {"tactic": "trans pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i \u226b (gluing \ud835\udcb0 f g).\u03b9 _", "annotated_tactic": ["trans <a>pullbackFst\u03b9ToV</a> \ud835\udcb0 f g i j \u226b <a>fV</a> \ud835\udcb0 f g j i \u226b (<a>gluing</a> \ud835\udcb0 f g).<a>\u03b9</a> _", [{"full_name": "AlgebraicGeometry.Scheme.Pullback.pullbackFst\u03b9ToV", "def_path": "Mathlib/AlgebraicGeometry/Pullbacks.lean", "def_pos": [406, 5], "def_end_pos": [406, 20]}, {"full_name": "AlgebraicGeometry.Scheme.Pullback.fV", "def_path": "Mathlib/AlgebraicGeometry/Pullbacks.lean", "def_pos": [117, 8], "def_end_pos": [117, 10]}, {"full_name": "AlgebraicGeometry.Scheme.Pullback.gluing", "def_path": "Mathlib/AlgebraicGeometry/Pullbacks.lean", "def_pos": [249, 5], "def_end_pos": [249, 11]}, {"full_name": "AlgebraicGeometry.Scheme.GlueData.\u03b9", "def_path": "Mathlib/AlgebraicGeometry/Gluing.lean", "def_pos": [159, 8], "def_end_pos": [159, 9]}]], "state_before": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b\n      pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n          (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n        GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n    pullback.fst \u226b pullback.fst", "state_after": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b\n      pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n          (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n        GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n    pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j\n\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j = pullback.fst \u226b pullback.fst"}, {"tactic": "rw [\u2190 show _ = fV \ud835\udcb0 f g j i \u226b _ from (gluing \ud835\udcb0 f g).glue_condition j i]", "annotated_tactic": ["rw [\u2190 show _ = <a>fV</a> \ud835\udcb0 f g j i \u226b _ from (<a>gluing</a> \ud835\udcb0 f g).<a>glue_condition</a> j i]", [{"full_name": "AlgebraicGeometry.Scheme.Pullback.fV", "def_path": "Mathlib/AlgebraicGeometry/Pullbacks.lean", "def_pos": [117, 8], "def_end_pos": [117, 10]}, {"full_name": "AlgebraicGeometry.Scheme.Pullback.gluing", "def_path": "Mathlib/AlgebraicGeometry/Pullbacks.lean", "def_pos": [249, 5], "def_end_pos": [249, 11]}, {"full_name": "AlgebraicGeometry.Scheme.GlueData.glue_condition", "def_path": "Mathlib/AlgebraicGeometry/Gluing.lean", "def_pos": [185, 9], "def_end_pos": [185, 23]}]], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b\n      pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n          (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n        GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n    pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j", "state_after": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b\n      pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n          (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n        GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n    pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b\n      GlueData.t (gluing \ud835\udcb0 f g).toGlueData j i \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i"}, {"tactic": "simp_rw [\u2190 Category.assoc]", "annotated_tactic": ["simp_rw [\u2190 <a>Category.assoc</a>]", [{"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [159, 3], "def_end_pos": [159, 8]}]], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b\n      pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n          (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n        GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n    pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b\n      GlueData.t (gluing \ud835\udcb0 f g).toGlueData j i \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData i j \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) i", "state_after": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 (pullback.fst \u226b\n        pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n          (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g)) \u226b\n      GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n    ((pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b GlueData.t (gluing \ud835\udcb0 f g).toGlueData j i) \u226b\n        GlueData.f (gluing \ud835\udcb0 f g).toGlueData i j) \u226b\n      GlueData.\u03b9 (gluing \ud835\udcb0 f g) i"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 (pullback.fst \u226b\n        pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n          (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g)) \u226b\n      GlueData.\u03b9 (gluing \ud835\udcb0 f g) i =\n    ((pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b GlueData.t (gluing \ud835\udcb0 f g).toGlueData j i) \u226b\n        GlueData.f (gluing \ud835\udcb0 f g).toGlueData i j) \u226b\n      GlueData.\u03b9 (gluing \ud835\udcb0 f g) i", "state_after": "case e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b\n      pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n        (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) =\n    (pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b GlueData.t (gluing \ud835\udcb0 f g).toGlueData j i) \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData i j"}, {"tactic": "rw [gluing_f, gluing_t]", "annotated_tactic": ["rw [<a>gluing_f</a>, <a>gluing_t</a>]", [{"full_name": "AlgebraicGeometry.Scheme.Pullback.gluing_f", "def_path": "Mathlib/AlgebraicGeometry/Pullbacks.lean", "def_pos": [248, 3], "def_end_pos": [248, 8]}, {"full_name": "AlgebraicGeometry.Scheme.Pullback.gluing_t", "def_path": "Mathlib/AlgebraicGeometry/Pullbacks.lean", "def_pos": [248, 3], "def_end_pos": [248, 8]}]], "state_before": "case e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b\n      pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n        (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) =\n    (pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b GlueData.t (gluing \ud835\udcb0 f g).toGlueData j i) \u226b GlueData.f (gluing \ud835\udcb0 f g).toGlueData i j", "state_after": "case e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b\n      pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n        (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) =\n    (pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b t \ud835\udcb0 f g j i) \u226b pullback.fst"}, {"tactic": "apply pullback.hom_ext <;> simp_rw [Category.assoc]", "annotated_tactic": ["apply <a>pullback.hom_ext</a> <;> simp_rw [<a>Category.assoc</a>]", [{"full_name": "CategoryTheory.Limits.pullback.hom_ext", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [1269, 9], "def_end_pos": [1269, 25]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [159, 3], "def_end_pos": [159, 8]}]], "state_before": "case e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b\n      pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n        (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) =\n    (pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b t \ud835\udcb0 f g j i) \u226b pullback.fst", "state_after": "case e_a.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b\n      pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n          (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n        pullback.fst =\n    pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b t \ud835\udcb0 f g j i \u226b pullback.fst \u226b pullback.fst\n\ncase e_a.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b\n      pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n          (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n        pullback.snd =\n    pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b t \ud835\udcb0 f g j i \u226b pullback.fst \u226b pullback.snd"}, {"tactic": "rw [t_fst_fst, pullback.lift_fst, pullbackFst\u03b9ToV_snd]", "annotated_tactic": ["rw [<a>t_fst_fst</a>, <a>pullback.lift_fst</a>, <a>pullbackFst\u03b9ToV_snd</a>]", [{"full_name": "AlgebraicGeometry.Scheme.Pullback.t_fst_fst", "def_path": "Mathlib/AlgebraicGeometry/Pullbacks.lean", "def_pos": [69, 9], "def_end_pos": [69, 18]}, {"full_name": "CategoryTheory.Limits.pullback.lift_fst", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [1168, 9], "def_end_pos": [1168, 26]}, {"full_name": "AlgebraicGeometry.Scheme.Pullback.pullbackFst\u03b9ToV_snd", "def_path": "Mathlib/AlgebraicGeometry/Pullbacks.lean", "def_pos": [426, 9], "def_end_pos": [426, 28]}]], "state_before": "case e_a.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b\n      pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n          (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n        pullback.fst =\n    pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b t \ud835\udcb0 f g j i \u226b pullback.fst \u226b pullback.fst", "state_after": "case e_a.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b pullback.snd = pullback.fst \u226b pullback.snd"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case e_a.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b pullback.snd = pullback.fst \u226b pullback.snd", "state_after": "no goals"}, {"tactic": "rw [t_fst_snd, pullback.lift_snd, pullbackFst\u03b9ToV_fst_assoc, pullback.condition_assoc]", "annotated_tactic": ["rw [<a>t_fst_snd</a>, <a>pullback.lift_snd</a>, <a>pullbackFst\u03b9ToV_fst_assoc</a>, <a>pullback.condition_assoc</a>]", [{"full_name": "AlgebraicGeometry.Scheme.Pullback.t_fst_snd", "def_path": "Mathlib/AlgebraicGeometry/Pullbacks.lean", "def_pos": [82, 9], "def_end_pos": [82, 18]}, {"full_name": "CategoryTheory.Limits.pullback.lift_snd", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [1175, 9], "def_end_pos": [1175, 26]}, {"full_name": "AlgebraicGeometry.Scheme.Pullback.pullbackFst\u03b9ToV_fst_assoc", "def_path": "Mathlib/AlgebraicGeometry/Pullbacks.lean", "def_pos": [413, 9], "def_end_pos": [413, 16]}, {"full_name": "CategoryTheory.Limits.pullback.condition_assoc", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [1208, 3], "def_end_pos": [1208, 10]}]], "state_before": "case e_a.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.fst \u226b\n      pullback.lift pullback.snd (pullback.fst \u226b p2 \ud835\udcb0 f g)\n          (_ : pullback.snd \u226b OpenCover.map \ud835\udcb0 i \u226b f = (pullback.fst \u226b p2 \ud835\udcb0 f g) \u226b g) \u226b\n        pullback.snd =\n    pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b t \ud835\udcb0 f g j i \u226b pullback.fst \u226b pullback.snd", "state_after": "case e_a.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.snd \u226b OpenCover.map (GlueData.openCover (gluing \ud835\udcb0 f g)) j \u226b p2 \ud835\udcb0 f g = pullback.snd \u226b pullback.snd"}, {"tactic": "erw [Multicoequalizer.\u03c0_desc]", "annotated_tactic": ["erw [<a>Multicoequalizer.\u03c0_desc</a>]", [{"full_name": "CategoryTheory.Limits.Multicoequalizer.\u03c0_desc", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean", "def_pos": [876, 9], "def_end_pos": [876, 15]}]], "state_before": "case e_a.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.snd \u226b OpenCover.map (GlueData.openCover (gluing \ud835\udcb0 f g)) j \u226b p2 \ud835\udcb0 f g = pullback.snd \u226b pullback.snd", "state_after": "case e_a.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.snd \u226b pullback.snd = pullback.snd \u226b pullback.snd"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case e_a.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.snd \u226b pullback.snd = pullback.snd \u226b pullback.snd", "state_after": "no goals"}, {"tactic": "rw [pullback.condition, \u2190 Category.assoc]", "annotated_tactic": ["rw [<a>pullback.condition</a>, \u2190 <a>Category.assoc</a>]", [{"full_name": "CategoryTheory.Limits.pullback.condition", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [1209, 9], "def_end_pos": [1209, 27]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [159, 3], "def_end_pos": [159, 8]}]], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j = pullback.fst \u226b pullback.fst", "state_after": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 (pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i) \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j =\n    pullback.snd \u226b OpenCover.map (GlueData.openCover (gluing \ud835\udcb0 f g)) j"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 (pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i) \u226b GlueData.\u03b9 (gluing \ud835\udcb0 f g) j =\n    pullback.snd \u226b OpenCover.map (GlueData.openCover (gluing \ud835\udcb0 f g)) j", "state_after": "case e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i = pullback.snd"}, {"tactic": "apply pullback.hom_ext", "annotated_tactic": ["apply <a>pullback.hom_ext</a>", [{"full_name": "CategoryTheory.Limits.pullback.hom_ext", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [1269, 9], "def_end_pos": [1269, 25]}]], "state_before": "case e_a\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i = pullback.snd", "state_after": "case e_a.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 (pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i) \u226b pullback.fst = pullback.snd \u226b pullback.fst\n\ncase e_a.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 (pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i) \u226b pullback.snd = pullback.snd \u226b pullback.snd"}, {"tactic": "simp only [pullbackFst\u03b9ToV_fst]", "annotated_tactic": ["simp only [<a>pullbackFst\u03b9ToV_fst</a>]", [{"full_name": "AlgebraicGeometry.Scheme.Pullback.pullbackFst\u03b9ToV_fst", "def_path": "Mathlib/AlgebraicGeometry/Pullbacks.lean", "def_pos": [414, 9], "def_end_pos": [414, 28]}]], "state_before": "case e_a.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 (pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i) \u226b pullback.fst = pullback.snd \u226b pullback.fst", "state_after": "case e_a.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.snd \u226b pullback.fst = pullback.snd \u226b pullback.fst"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case e_a.h\u2080\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.snd \u226b pullback.fst = pullback.snd \u226b pullback.fst", "state_after": "no goals"}, {"tactic": "simp only [pullbackFst\u03b9ToV_fst]", "annotated_tactic": ["simp only [<a>pullbackFst\u03b9ToV_fst</a>]", [{"full_name": "AlgebraicGeometry.Scheme.Pullback.pullbackFst\u03b9ToV_fst", "def_path": "Mathlib/AlgebraicGeometry/Pullbacks.lean", "def_pos": [414, 9], "def_end_pos": [414, 28]}]], "state_before": "case e_a.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 (pullbackFst\u03b9ToV \ud835\udcb0 f g i j \u226b fV \ud835\udcb0 f g j i) \u226b pullback.snd = pullback.snd \u226b pullback.snd", "state_after": "case e_a.h\u2081\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : OpenCover X\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (OpenCover.map \ud835\udcb0 i \u226b f) g\ns : PullbackCone f g\ni : \ud835\udcb0.J\nj : (OpenCover.pullbackCover (GlueData.openCover (gluing \ud835\udcb0 f g)) pullback.fst).J\n\u22a2 pullback.snd \u226b pullback.snd = pullback.snd \u226b pullback.snd"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/BilinearMap.lean", "full_name": "LinearMap.map_smul\u2082", "start": [173, 1], "end": [174, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Logic/Relation.lean", "full_name": "Symmetric.swap_eq", "start": [88, 1], "end": [89, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Category/TopCat/Opens.lean", "full_name": "TopologicalSpace.Opens.openEmbedding_obj_top", "start": [340, 1], "end": [343, 47], "traced_tactics": [{"tactic": "ext1", "annotated_tactic": ["ext1", []], "state_before": "X : TopCat\nU : Opens \u2191X\n\u22a2 (IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion U))).obj \u22a4 = U", "state_after": "case h\nX : TopCat\nU : Opens \u2191X\n\u22a2 \u2191((IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion U))).obj \u22a4) = \u2191U"}, {"tactic": "exact Set.image_univ.trans Subtype.range_coe", "annotated_tactic": ["exact Set.image_univ.trans <a>Subtype.range_coe</a>", [{"full_name": "Subtype.range_coe", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 18]}]], "state_before": "case h\nX : TopCat\nU : Opens \u2191X\n\u22a2 \u2191((IsOpenMap.functor (_ : IsOpenMap \u2191(inclusion U))).obj \u22a4) = \u2191U", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/FieldTheory/IsAlgClosed/Spectrum.lean", "full_name": "spectrum.pow_image_subset", "start": [133, 1], "end": [134, 92], "traced_tactics": [{"tactic": "simpa only [eval_pow, eval_X, aeval_X_pow] using subset_polynomial_aeval a (X ^ n : \ud835\udd5c[X])", "annotated_tactic": ["simpa only [<a>eval_pow</a>, <a>eval_X</a>, <a>aeval_X_pow</a>] using <a>subset_polynomial_aeval</a> a (<a>X</a> ^ n : \ud835\udd5c[X])", [{"full_name": "Polynomial.eval_pow", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [1093, 9], "def_end_pos": [1093, 17]}, {"full_name": "Polynomial.eval_X", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "Polynomial.aeval_X_pow", "def_path": "Mathlib/Data/Polynomial/AlgebraMap.lean", "def_pos": [214, 9], "def_end_pos": [214, 20]}, {"full_name": "spectrum.subset_polynomial_aeval", "def_path": "Mathlib/FieldTheory/IsAlgClosed/Spectrum.lean", "def_pos": [85, 9], "def_end_pos": [85, 32]}, {"full_name": "Polynomial.X", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [566, 5], "def_end_pos": [566, 6]}]], "state_before": "\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\nn : \u2115\n\u22a2 (fun x => x ^ n) '' \u03c3 a \u2286 \u03c3 (a ^ n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/MetricSeparated.lean", "full_name": "IsMetricSeparated.union_right", "start": [95, 1], "end": [97, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/ModelTheory/Substructures.lean", "full_name": "FirstOrder.Language.Substructure.le_comap_of_map_le", "start": [479, 1], "end": [480, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "full_name": "Nat.mul_lt_mul'", "start": [534, 11], "end": [535, 92], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/GroupCompletion.lean", "full_name": "AddMonoidHom.extension_coe", "start": [251, 1], "end": [253, 92], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/IsROrC/Basic.lean", "full_name": "IsROrC.algebraMap_eq_ofReal", "start": [113, 1], "end": [114, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "full_name": "List.foldr_nil", "start": [236, 9], "end": [236, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Symmetrized.lean", "full_name": "SymAlg.mul_def", "start": [200, 1], "end": [201, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean", "full_name": "NonUnitalSubsemiring.mem_sSup_of_directedOn", "start": [844, 1], "end": [847, 95], "traced_tactics": [{"tactic": "haveI : Nonempty S := Sne.to_subtype", "annotated_tactic": ["haveI : <a>Nonempty</a> S := Sne.to_subtype", [{"full_name": "Nonempty", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [686, 17], "def_end_pos": [686, 25]}]], "state_before": "R : Type u\nS\u271d : Type v\nT : Type w\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring S\u271d\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring T\nF : Type u_1\ninst\u271d : NonUnitalRingHomClass F R S\u271d\nS : Set (NonUnitalSubsemiring R)\nSne : Set.Nonempty S\nhS : DirectedOn (fun x x_1 => x \u2264 x_1) S\nx : R\n\u22a2 x \u2208 sSup S \u2194 \u2203 s, s \u2208 S \u2227 x \u2208 s", "state_after": "R : Type u\nS\u271d : Type v\nT : Type w\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring S\u271d\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring T\nF : Type u_1\ninst\u271d : NonUnitalRingHomClass F R S\u271d\nS : Set (NonUnitalSubsemiring R)\nSne : Set.Nonempty S\nhS : DirectedOn (fun x x_1 => x \u2264 x_1) S\nx : R\nthis : Nonempty \u2191S\n\u22a2 x \u2208 sSup S \u2194 \u2203 s, s \u2208 S \u2227 x \u2208 s"}, {"tactic": "simp only [sSup_eq_iSup', mem_iSup_of_directed hS.directed_val, Subtype.exists, exists_prop]", "annotated_tactic": ["simp only [<a>sSup_eq_iSup'</a>, <a>mem_iSup_of_directed</a> hS.directed_val, <a>Subtype.exists</a>, <a>exists_prop</a>]", [{"full_name": "sSup_eq_iSup'", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [689, 9], "def_end_pos": [689, 22]}, {"full_name": "NonUnitalSubsemiring.mem_iSup_of_directed", "def_path": "Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean", "def_pos": [826, 9], "def_end_pos": [826, 29]}, {"full_name": "Subtype.exists", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [54, 19], "def_end_pos": [54, 27]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}]], "state_before": "R : Type u\nS\u271d : Type v\nT : Type w\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring R\nM : Subsemigroup R\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring S\u271d\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring T\nF : Type u_1\ninst\u271d : NonUnitalRingHomClass F R S\u271d\nS : Set (NonUnitalSubsemiring R)\nSne : Set.Nonempty S\nhS : DirectedOn (fun x x_1 => x \u2264 x_1) S\nx : R\nthis : Nonempty \u2191S\n\u22a2 x \u2208 sSup S \u2194 \u2203 s, s \u2208 S \u2227 x \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Logic/Relation.lean", "full_name": "Relation.ReflGen.mono", "start": [257, 1], "end": [259, 40], "traced_tactics": [{"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d b c d : \u03b1\np : \u03b1 \u2192 \u03b1 \u2192 Prop\nhp : \u2200 (a b : \u03b1), r a b \u2192 p a b\na : \u03b1\n\u22a2 ReflGen p a a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Lattice.lean", "full_name": "inf_right_comm", "start": [532, 1], "end": [533, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Normed/Group/Pointwise.lean", "full_name": "inv_ball", "start": [91, 1], "end": [91, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Interval.lean", "full_name": "Interval.coe_dual", "start": [500, 1], "end": [503, 50], "traced_tactics": [{"tactic": "cases s with\n| none => rfl\n| some s\u2080 => exact NonemptyInterval.coe_dual s\u2080", "annotated_tactic": ["cases s with\n  | <a>none</a> => rfl\n  | <a>some</a> s\u2080 => exact <a>NonemptyInterval.coe_dual</a> s\u2080", [{"full_name": "Option.none", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2141, 5], "def_end_pos": [2141, 9]}, {"full_name": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}, {"full_name": "NonemptyInterval.coe_dual", "def_path": "Mathlib/Order/Interval.lean", "def_pos": [279, 9], "def_end_pos": [279, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PartialOrder \u03b2\ns\u271d t : Interval \u03b1\na b : \u03b1\ns : Interval \u03b1\n\u22a2 \u2191(\u2191dual s) = \u2191ofDual \u207b\u00b9' \u2191s", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PartialOrder \u03b2\ns t : Interval \u03b1\na b : \u03b1\n\u22a2 \u2191(\u2191dual none) = \u2191ofDual \u207b\u00b9' \u2191none", "state_after": "no goals"}, {"tactic": "exact NonemptyInterval.coe_dual s\u2080", "annotated_tactic": ["exact <a>NonemptyInterval.coe_dual</a> s\u2080", [{"full_name": "NonemptyInterval.coe_dual", "def_path": "Mathlib/Order/Interval.lean", "def_pos": [279, 9], "def_end_pos": [279, 17]}]], "state_before": "case some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\n\u03ba : \u03b9 \u2192 Sort u_6\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PartialOrder \u03b2\ns t : Interval \u03b1\na b : \u03b1\ns\u2080 : NonemptyInterval \u03b1\n\u22a2 \u2191(\u2191dual (some s\u2080)) = \u2191ofDual \u207b\u00b9' \u2191(some s\u2080)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Order/LeftRightLim.lean", "full_name": "Antitone.le_leftLim", "start": [284, 1], "end": [285, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Sigma/Basic.lean", "full_name": "Prod.fst_comp_toSigma", "start": [185, 1], "end": [186, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.enumOrd_def_nonempty", "start": [2209, 1], "end": [2211, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/Dioph.lean", "full_name": "Dioph.div_dioph", "start": [699, 1], "end": [718, 92], "traced_tactics": [{"tactic": "refine Iff.trans ?_ eq_comm", "annotated_tactic": ["refine <a>Iff.trans</a> ?_ <a>eq_comm</a>", [{"full_name": "Iff.trans", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [666, 9], "def_end_pos": [666, 18]}, {"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "\u03b1 \u03b2 : Type\nn : \u2115\nf g : (\u03b1 \u2192 \u2115) \u2192 \u2115\ndf : DiophFn f\ndg : DiophFn g\nthis : Dioph fun v => v &2 = 0 \u2227 v &0 = 0 \u2228 v &0 * v &2 \u2264 v &1 \u2227 v &1 < (v &0 + 1) * v &2\nz x y : \u2115\n\u22a2 y = 0 \u2227 z = 0 \u2228 z * y \u2264 x \u2227 x < (z + 1) * y \u2194 x / y = z", "state_after": "\u03b1 \u03b2 : Type\nn : \u2115\nf g : (\u03b1 \u2192 \u2115) \u2192 \u2115\ndf : DiophFn f\ndg : DiophFn g\nthis : Dioph fun v => v &2 = 0 \u2227 v &0 = 0 \u2228 v &0 * v &2 \u2264 v &1 \u2227 v &1 < (v &0 + 1) * v &2\nz x y : \u2115\n\u22a2 y = 0 \u2227 z = 0 \u2228 z * y \u2264 x \u2227 x < (z + 1) * y \u2194 z = x / y"}, {"tactic": "exact y.eq_zero_or_pos.elim\n  (fun y0 => by\n    rw [y0, Nat.div_zero]\n    exact \u27e8fun o => (o.resolve_right fun \u27e8_, h2\u27e9 => Nat.not_lt_zero _ h2).right,\n      fun z0 => Or.inl \u27e8rfl, z0\u27e9\u27e9)\n  fun ypos =>\n    Iff.trans \u27e8fun o => o.resolve_left fun \u27e8h1, _\u27e9 => Nat.ne_of_gt ypos h1, Or.inr\u27e9\n      (le_antisymm_iff.trans <| and_congr (Nat.le_div_iff_mul_le ypos) <|\n        Iff.trans \u27e8lt_succ_of_le, le_of_lt_succ\u27e9 (div_lt_iff_lt_mul ypos)).symm", "annotated_tactic": ["exact y.eq_zero_or_pos.elim\n              (fun y0 => by\n                rw [y0, <a>Nat.div_zero</a>]\n                exact \u27e8fun o => (o.resolve_right fun \u27e8_, h2\u27e9 => <a>Nat.not_lt_zero</a> _ h2).<a>right</a>,\n                  fun z0 => <a>Or.inl</a> \u27e8<a>rfl</a>, z0\u27e9\u27e9)\n              fun ypos =>\n                <a>Iff.trans</a> \u27e8fun o => o.resolve_left fun \u27e8h1, _\u27e9 => <a>Nat.ne_of_gt</a> ypos h1, <a>Or.inr</a>\u27e9\n                  (le_antisymm_iff.trans <| <a>and_congr</a> (<a>Nat.le_div_iff_mul_le</a> ypos) <|\n                    <a>Iff.trans</a> \u27e8<a>lt_succ_of_le</a>, <a>le_of_lt_succ</a>\u27e9 (<a>div_lt_iff_lt_mul</a> ypos)).<a>symm</a>", [{"full_name": "Nat.div_zero", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [559, 27], "def_end_pos": [559, 35]}, {"full_name": "Nat.not_lt_zero", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1575, 9], "def_end_pos": [1575, 24]}, {"full_name": "And.right", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [507, 3], "def_end_pos": [507, 8]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Iff.trans", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [666, 9], "def_end_pos": [666, 18]}, {"full_name": "Nat.ne_of_gt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [136, 9], "def_end_pos": [136, 17]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "and_congr", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [156, 9], "def_end_pos": [156, 18]}, {"full_name": "Nat.le_div_iff_mul_le", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [565, 9], "def_end_pos": [565, 26]}, {"full_name": "Iff.trans", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [666, 9], "def_end_pos": [666, 18]}, {"full_name": "Nat.lt_succ_of_le", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 22]}, {"full_name": "Nat.le_of_lt_succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1631, 9], "def_end_pos": [1631, 26]}, {"full_name": "Nat.div_lt_iff_lt_mul", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [595, 9], "def_end_pos": [595, 26]}, {"full_name": "Iff.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [671, 9], "def_end_pos": [671, 17]}]], "state_before": "\u03b1 \u03b2 : Type\nn : \u2115\nf g : (\u03b1 \u2192 \u2115) \u2192 \u2115\ndf : DiophFn f\ndg : DiophFn g\nthis : Dioph fun v => v &2 = 0 \u2227 v &0 = 0 \u2228 v &0 * v &2 \u2264 v &1 \u2227 v &1 < (v &0 + 1) * v &2\nz x y : \u2115\n\u22a2 y = 0 \u2227 z = 0 \u2228 z * y \u2264 x \u2227 x < (z + 1) * y \u2194 z = x / y", "state_after": "no goals"}, {"tactic": "rw [y0, Nat.div_zero]", "annotated_tactic": ["rw [y0, <a>Nat.div_zero</a>]", [{"full_name": "Nat.div_zero", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [559, 27], "def_end_pos": [559, 35]}]], "state_before": "\u03b1 \u03b2 : Type\nn : \u2115\nf g : (\u03b1 \u2192 \u2115) \u2192 \u2115\ndf : DiophFn f\ndg : DiophFn g\nthis : Dioph fun v => v &2 = 0 \u2227 v &0 = 0 \u2228 v &0 * v &2 \u2264 v &1 \u2227 v &1 < (v &0 + 1) * v &2\nz x y : \u2115\ny0 : y = 0\n\u22a2 y = 0 \u2227 z = 0 \u2228 z * y \u2264 x \u2227 x < (z + 1) * y \u2194 z = x / y", "state_after": "\u03b1 \u03b2 : Type\nn : \u2115\nf g : (\u03b1 \u2192 \u2115) \u2192 \u2115\ndf : DiophFn f\ndg : DiophFn g\nthis : Dioph fun v => v &2 = 0 \u2227 v &0 = 0 \u2228 v &0 * v &2 \u2264 v &1 \u2227 v &1 < (v &0 + 1) * v &2\nz x y : \u2115\ny0 : y = 0\n\u22a2 0 = 0 \u2227 z = 0 \u2228 z * 0 \u2264 x \u2227 x < (z + 1) * 0 \u2194 z = 0"}, {"tactic": "exact \u27e8fun o => (o.resolve_right fun \u27e8_, h2\u27e9 => Nat.not_lt_zero _ h2).right,\n  fun z0 => Or.inl \u27e8rfl, z0\u27e9\u27e9", "annotated_tactic": ["exact \u27e8fun o => (o.resolve_right fun \u27e8_, h2\u27e9 => <a>Nat.not_lt_zero</a> _ h2).<a>right</a>,\n                  fun z0 => <a>Or.inl</a> \u27e8<a>rfl</a>, z0\u27e9\u27e9", [{"full_name": "Nat.not_lt_zero", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1575, 9], "def_end_pos": [1575, 24]}, {"full_name": "And.right", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [507, 3], "def_end_pos": [507, 8]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03b1 \u03b2 : Type\nn : \u2115\nf g : (\u03b1 \u2192 \u2115) \u2192 \u2115\ndf : DiophFn f\ndg : DiophFn g\nthis : Dioph fun v => v &2 = 0 \u2227 v &0 = 0 \u2228 v &0 * v &2 \u2264 v &1 \u2227 v &1 < (v &0 + 1) * v &2\nz x y : \u2115\ny0 : y = 0\n\u22a2 0 = 0 \u2227 z = 0 \u2228 z * 0 \u2264 x \u2227 x < (z + 1) * 0 \u2194 z = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/NonUnitalSubring/Basic.lean", "full_name": "NonUnitalSubring.mem_closure_iff", "start": [712, 1], "end": [737, 76], "traced_tactics": [{"tactic": "rw [zero_mul q]", "annotated_tactic": ["rw [<a>zero_mul</a> q]", [{"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}]], "state_before": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf : R \u2192\u2099+* S\ns : Set R\nx\u271d : R\nh : x\u271d \u2208 closure s\nx y : R\nhx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nq : R\nhq : q \u2208 \u2191(Subsemigroup.closure s)\n\u22a2 0 * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)", "state_after": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf : R \u2192\u2099+* S\ns : Set R\nx\u271d : R\nh : x\u271d \u2208 closure s\nx y : R\nhx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nq : R\nhq : q \u2208 \u2191(Subsemigroup.closure s)\n\u22a2 0 \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)"}, {"tactic": "apply AddSubgroup.zero_mem _", "annotated_tactic": ["apply <a>AddSubgroup.zero_mem</a> _", [{"full_name": "AddSubgroup.zero_mem", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [565, 3], "def_end_pos": [565, 14]}]], "state_before": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf : R \u2192\u2099+* S\ns : Set R\nx\u271d : R\nh : x\u271d \u2208 closure s\nx y : R\nhx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nq : R\nhq : q \u2208 \u2191(Subsemigroup.closure s)\n\u22a2 0 \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)", "state_after": "no goals"}, {"tactic": "rw [add_mul p\u2081 p\u2082 q]", "annotated_tactic": ["rw [<a>add_mul</a> p\u2081 p\u2082 q]", [{"full_name": "add_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [91, 7], "def_end_pos": [91, 14]}]], "state_before": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf : R \u2192\u2099+* S\ns : Set R\nx\u271d : R\nh : x\u271d \u2208 closure s\nx y : R\nhx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nq : R\nhq : q \u2208 \u2191(Subsemigroup.closure s)\np\u2081 p\u2082 : R\nihp\u2081 : p\u2081 * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nihp\u2082 : p\u2082 * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\n\u22a2 (p\u2081 + p\u2082) * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)", "state_after": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf : R \u2192\u2099+* S\ns : Set R\nx\u271d : R\nh : x\u271d \u2208 closure s\nx y : R\nhx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nq : R\nhq : q \u2208 \u2191(Subsemigroup.closure s)\np\u2081 p\u2082 : R\nihp\u2081 : p\u2081 * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nihp\u2082 : p\u2082 * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\n\u22a2 p\u2081 * q + p\u2082 * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)"}, {"tactic": "apply AddSubgroup.add_mem _ ihp\u2081 ihp\u2082", "annotated_tactic": ["apply <a>AddSubgroup.add_mem</a> _ ihp\u2081 ihp\u2082", [{"full_name": "AddSubgroup.add_mem", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [572, 3], "def_end_pos": [572, 14]}]], "state_before": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf : R \u2192\u2099+* S\ns : Set R\nx\u271d : R\nh : x\u271d \u2208 closure s\nx y : R\nhx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nq : R\nhq : q \u2208 \u2191(Subsemigroup.closure s)\np\u2081 p\u2082 : R\nihp\u2081 : p\u2081 * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nihp\u2082 : p\u2082 * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\n\u22a2 p\u2081 * q + p\u2082 * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)", "state_after": "no goals"}, {"tactic": "have f : -x * q = -(x * q) := by simp", "annotated_tactic": ["have f : -x * q = -(x * q) := by simp", []], "state_before": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf : R \u2192\u2099+* S\ns : Set R\nx\u271d\u00b9 : R\nh : x\u271d\u00b9 \u2208 closure s\nx\u271d y : R\nhx\u271d : x\u271d \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nq : R\nhq : q \u2208 \u2191(Subsemigroup.closure s)\nx : R\nhx : x * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\n\u22a2 -x * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)", "state_after": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf\u271d : R \u2192\u2099+* S\ns : Set R\nx\u271d\u00b9 : R\nh : x\u271d\u00b9 \u2208 closure s\nx\u271d y : R\nhx\u271d : x\u271d \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nq : R\nhq : q \u2208 \u2191(Subsemigroup.closure s)\nx : R\nhx : x * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nf : -x * q = -(x * q)\n\u22a2 -x * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)"}, {"tactic": "rw [f]", "annotated_tactic": ["rw [f]", []], "state_before": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf\u271d : R \u2192\u2099+* S\ns : Set R\nx\u271d\u00b9 : R\nh : x\u271d\u00b9 \u2208 closure s\nx\u271d y : R\nhx\u271d : x\u271d \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nq : R\nhq : q \u2208 \u2191(Subsemigroup.closure s)\nx : R\nhx : x * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nf : -x * q = -(x * q)\n\u22a2 -x * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)", "state_after": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf\u271d : R \u2192\u2099+* S\ns : Set R\nx\u271d\u00b9 : R\nh : x\u271d\u00b9 \u2208 closure s\nx\u271d y : R\nhx\u271d : x\u271d \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nq : R\nhq : q \u2208 \u2191(Subsemigroup.closure s)\nx : R\nhx : x * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nf : -x * q = -(x * q)\n\u22a2 -(x * q) \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)"}, {"tactic": "apply AddSubgroup.neg_mem _ hx", "annotated_tactic": ["apply <a>AddSubgroup.neg_mem</a> _ hx", [{"full_name": "AddSubgroup.neg_mem", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [579, 3], "def_end_pos": [579, 14]}]], "state_before": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf\u271d : R \u2192\u2099+* S\ns : Set R\nx\u271d\u00b9 : R\nh : x\u271d\u00b9 \u2208 closure s\nx\u271d y : R\nhx\u271d : x\u271d \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nq : R\nhq : q \u2208 \u2191(Subsemigroup.closure s)\nx : R\nhx : x * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nf : -x * q = -(x * q)\n\u22a2 -(x * q) \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf : R \u2192\u2099+* S\ns : Set R\nx\u271d\u00b9 : R\nh : x\u271d\u00b9 \u2208 closure s\nx\u271d y : R\nhx\u271d : x\u271d \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nq : R\nhq : q \u2208 \u2191(Subsemigroup.closure s)\nx : R\nhx : x * q \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\n\u22a2 -x * q = -(x * q)", "state_after": "no goals"}, {"tactic": "rw [mul_zero x]", "annotated_tactic": ["rw [<a>mul_zero</a> x]", [{"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf : R \u2192\u2099+* S\ns : Set R\nx\u271d : R\nh : x\u271d \u2208 closure s\nx y : R\nhx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\n\u22a2 x * 0 \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)", "state_after": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf : R \u2192\u2099+* S\ns : Set R\nx\u271d : R\nh : x\u271d \u2208 closure s\nx y : R\nhx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\n\u22a2 0 \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)"}, {"tactic": "apply AddSubgroup.zero_mem _", "annotated_tactic": ["apply <a>AddSubgroup.zero_mem</a> _", [{"full_name": "AddSubgroup.zero_mem", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [565, 3], "def_end_pos": [565, 14]}]], "state_before": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf : R \u2192\u2099+* S\ns : Set R\nx\u271d : R\nh : x\u271d \u2208 closure s\nx y : R\nhx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\n\u22a2 0 \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)", "state_after": "no goals"}, {"tactic": "rw [mul_add x q\u2081 q\u2082]", "annotated_tactic": ["rw [<a>mul_add</a> x q\u2081 q\u2082]", [{"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}]], "state_before": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf : R \u2192\u2099+* S\ns : Set R\nx\u271d : R\nh : x\u271d \u2208 closure s\nx y : R\nhx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nq\u2081 q\u2082 : R\nihq\u2081 : x * q\u2081 \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nihq\u2082 : x * q\u2082 \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\n\u22a2 x * (q\u2081 + q\u2082) \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)", "state_after": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf : R \u2192\u2099+* S\ns : Set R\nx\u271d : R\nh : x\u271d \u2208 closure s\nx y : R\nhx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nq\u2081 q\u2082 : R\nihq\u2081 : x * q\u2081 \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nihq\u2082 : x * q\u2082 \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\n\u22a2 x * q\u2081 + x * q\u2082 \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)"}, {"tactic": "apply AddSubgroup.add_mem _ ihq\u2081 ihq\u2082", "annotated_tactic": ["apply <a>AddSubgroup.add_mem</a> _ ihq\u2081 ihq\u2082", [{"full_name": "AddSubgroup.add_mem", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [572, 3], "def_end_pos": [572, 14]}]], "state_before": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf : R \u2192\u2099+* S\ns : Set R\nx\u271d : R\nh : x\u271d \u2208 closure s\nx y : R\nhx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nq\u2081 q\u2082 : R\nihq\u2081 : x * q\u2081 \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nihq\u2082 : x * q\u2082 \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\n\u22a2 x * q\u2081 + x * q\u2082 \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)", "state_after": "no goals"}, {"tactic": "have f : x * -z = -(x * z) := by simp", "annotated_tactic": ["have f : x * -z = -(x * z) := by simp", []], "state_before": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf : R \u2192\u2099+* S\ns : Set R\nx\u271d : R\nh : x\u271d \u2208 closure s\nx y : R\nhx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nz : R\nhz : x * z \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\n\u22a2 x * -z \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)", "state_after": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf\u271d : R \u2192\u2099+* S\ns : Set R\nx\u271d : R\nh : x\u271d \u2208 closure s\nx y : R\nhx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nz : R\nhz : x * z \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nf : x * -z = -(x * z)\n\u22a2 x * -z \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)"}, {"tactic": "rw [f]", "annotated_tactic": ["rw [f]", []], "state_before": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf\u271d : R \u2192\u2099+* S\ns : Set R\nx\u271d : R\nh : x\u271d \u2208 closure s\nx y : R\nhx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nz : R\nhz : x * z \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nf : x * -z = -(x * z)\n\u22a2 x * -z \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)", "state_after": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf\u271d : R \u2192\u2099+* S\ns : Set R\nx\u271d : R\nh : x\u271d \u2208 closure s\nx y : R\nhx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nz : R\nhz : x * z \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nf : x * -z = -(x * z)\n\u22a2 -(x * z) \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)"}, {"tactic": "apply AddSubgroup.neg_mem _ hz", "annotated_tactic": ["apply <a>AddSubgroup.neg_mem</a> _ hz", [{"full_name": "AddSubgroup.neg_mem", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [579, 3], "def_end_pos": [579, 14]}]], "state_before": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf\u271d : R \u2192\u2099+* S\ns : Set R\nx\u271d : R\nh : x\u271d \u2208 closure s\nx y : R\nhx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nz : R\nhz : x * z \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nf : x * -z = -(x * z)\n\u22a2 -(x * z) \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "F : Type w\nR : Type u\nS : Type v\nT : Type u_1\ninst\u271d\u00b3 : NonUnitalNonAssocRing R\ninst\u271d\u00b2 : NonUnitalNonAssocRing S\ninst\u271d\u00b9 : NonUnitalNonAssocRing T\ninst\u271d : NonUnitalRingHomClass F R S\ng : S \u2192\u2099+* T\nf : R \u2192\u2099+* S\ns : Set R\nx\u271d : R\nh : x\u271d \u2208 closure s\nx y : R\nhx : x \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nhy : y \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\nz : R\nhz : x * z \u2208 AddSubgroup.closure \u2191(Subsemigroup.closure s)\n\u22a2 x * -z = -(x * z)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Equicontinuity.lean", "full_name": "Metric.equicontinuousAt_iff_pair", "start": [58, 11], "end": [69, 52], "traced_tactics": [{"tactic": "rw [equicontinuousAt_iff_pair]", "annotated_tactic": ["rw [<a>equicontinuousAt_iff_pair</a>]", [{"full_name": "equicontinuousAt_iff_pair", "def_path": "Mathlib/Topology/UniformSpace/Equicontinuity.lean", "def_pos": [183, 9], "def_end_pos": [183, 34]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\n\u22a2 EquicontinuousAt F x\u2080 \u2194\n    \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\n\u22a2 (\u2200 (U : Set (\u03b1 \u00d7 \u03b1)), U \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 V, V \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U) \u2194\n    \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5"}, {"tactic": "constructor <;> intro H", "annotated_tactic": ["constructor <;> intro H", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\n\u22a2 (\u2200 (U : Set (\u03b1 \u00d7 \u03b1)), U \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 V, V \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U) \u2194\n    \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\nH : \u2200 (U : Set (\u03b1 \u00d7 \u03b1)), U \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 V, V \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5\n\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5\n\u22a2 \u2200 (U : Set (\u03b1 \u00d7 \u03b1)), U \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 V, V \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U"}, {"tactic": "intro \u03b5 h\u03b5", "annotated_tactic": ["intro \u03b5 h\u03b5", []], "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\nH : \u2200 (U : Set (\u03b1 \u00d7 \u03b1)), U \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 V, V \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\nH : \u2200 (U : Set (\u03b1 \u00d7 \u03b1)), U \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 V, V \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5"}, {"tactic": "exact H _ (dist_mem_uniformity h\u03b5)", "annotated_tactic": ["exact H _ (<a>dist_mem_uniformity</a> h\u03b5)", [{"full_name": "Metric.dist_mem_uniformity", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [826, 9], "def_end_pos": [826, 28]}]], "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\nH : \u2200 (U : Set (\u03b1 \u00d7 \u03b1)), U \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 V, V \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5", "state_after": "no goals"}, {"tactic": "intro U hU", "annotated_tactic": ["intro U hU", []], "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5\n\u22a2 \u2200 (U : Set (\u03b1 \u00d7 \u03b1)), U \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 V, V \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U", "state_after": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\n\u22a2 \u2203 V, V \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U"}, {"tactic": "rcases mem_uniformity_dist.mp hU with \u27e8\u03b5, h\u03b5, h\u03b5U\u27e9", "annotated_tactic": ["rcases mem_uniformity_dist.mp hU with \u27e8\u03b5, h\u03b5, h\u03b5U\u27e9", []], "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\n\u22a2 \u2203 V, V \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U", "state_after": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5U : \u2200 {a b : \u03b1}, dist a b < \u03b5 \u2192 (a, b) \u2208 U\n\u22a2 \u2203 V, V \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U"}, {"tactic": "refine' Exists.imp (fun V => And.imp_right fun h => _) (H _ h\u03b5)", "annotated_tactic": ["refine' <a>Exists.imp</a> (fun V => <a>And.imp_right</a> fun h => _) (H _ h\u03b5)", [{"full_name": "Exists.imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [360, 9], "def_end_pos": [360, 19]}, {"full_name": "And.imp_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [154, 9], "def_end_pos": [154, 22]}]], "state_before": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5U : \u2200 {a b : \u03b1}, dist a b < \u03b5 \u2192 (a, b) \u2208 U\n\u22a2 \u2203 V, V \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U", "state_after": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5U : \u2200 {a b : \u03b1}, dist a b < \u03b5 \u2192 (a, b) \u2208 U\nV : Set \u03b2\nh : \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (x' : \u03b2), x' \u2208 V \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5\n\u22a2 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U"}, {"tactic": "exact fun x hx x' hx' i => h\u03b5U (h _ hx _ hx' i)", "annotated_tactic": ["exact fun x hx x' hx' i => h\u03b5U (h _ hx _ hx' i)", []], "state_before": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5U : \u2200 {a b : \u03b1}, dist a b < \u03b5 \u2192 (a, b) \u2208 U\nV : Set \u03b2\nh : \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (x' : \u03b2), x' \u2208 V \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5\n\u22a2 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Basic.lean", "full_name": "IsMin.Iio_eq", "start": [706, 1], "end": [707, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/LocalExtr.lean", "full_name": "IsLocalMinOn.min", "start": [511, 8], "end": [513, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Matrix/Basic.lean", "full_name": "Matrix.transpose_smul", "start": [2013, 1], "end": [2015, 6], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o \u2192 Type u_5\nn' : o \u2192 Type u_6\nR\u271d : Type u_7\nS : Type u_8\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type u_9\nR : Type u_10\ninst\u271d : SMul R \u03b1\nc : R\nM : Matrix m n \u03b1\n\u22a2 (c \u2022 M)\u1d40 = c \u2022 M\u1d40", "state_after": "case a.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o \u2192 Type u_5\nn' : o \u2192 Type u_6\nR\u271d : Type u_7\nS : Type u_8\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type u_9\nR : Type u_10\ninst\u271d : SMul R \u03b1\nc : R\nM : Matrix m n \u03b1\ni\u271d : n\nx\u271d : m\n\u22a2 (c \u2022 M)\u1d40 i\u271d x\u271d = (c \u2022 M\u1d40) i\u271d x\u271d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case a.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o \u2192 Type u_5\nn' : o \u2192 Type u_6\nR\u271d : Type u_7\nS : Type u_8\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type u_9\nR : Type u_10\ninst\u271d : SMul R \u03b1\nc : R\nM : Matrix m n \u03b1\ni\u271d : n\nx\u271d : m\n\u22a2 (c \u2022 M)\u1d40 i\u271d x\u271d = (c \u2022 M\u1d40) i\u271d x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Complex/CauchyIntegral.lean", "full_name": "DifferentiableOn.analyticOn", "start": [604, 1], "end": [605, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean", "full_name": "TopCat.Presheaf.isSheaf_iff_isSheafEqualizerProducts", "start": [535, 1], "end": [539, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/Sort.lean", "full_name": "List.eq_of_perm_of_sorted", "start": [80, 1], "end": [96, 42], "traced_tactics": [{"tactic": "induction' s\u2081 with a l\u2081 h\u2081 s\u2081 IH generalizing l\u2082", "annotated_tactic": ["induction' s\u2081 with a l\u2081 h\u2081 s\u2081 IH generalizing l\u2082", []], "state_before": "\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\ns\u2081 : Sorted r l\u2081\ns\u2082 : Sorted r l\u2082\n\u22a2 l\u2081 = l\u2082", "state_after": "case nil\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082\u271d : List \u03b1\np\u271d : l\u2081 ~ l\u2082\u271d\ns\u2082\u271d : Sorted r l\u2082\u271d\nl\u2082 : List \u03b1\np : [] ~ l\u2082\ns\u2082 : Sorted r l\u2082\n\u22a2 [] = l\u2082\n\ncase cons\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081\u271d l\u2082\u271d : List \u03b1\np\u271d : l\u2081\u271d ~ l\u2082\u271d\ns\u2082\u271d : Sorted r l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 r a a'\ns\u2081 : Pairwise r l\u2081\nIH : \u2200 {l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 l\u2081 = l\u2082\nl\u2082 : List \u03b1\np : a :: l\u2081 ~ l\u2082\ns\u2082 : Sorted r l\u2082\n\u22a2 a :: l\u2081 = l\u2082"}, {"tactic": "exact p.nil_eq", "annotated_tactic": ["exact p.nil_eq", []], "state_before": "case nil\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082\u271d : List \u03b1\np\u271d : l\u2081 ~ l\u2082\u271d\ns\u2082\u271d : Sorted r l\u2082\u271d\nl\u2082 : List \u03b1\np : [] ~ l\u2082\ns\u2082 : Sorted r l\u2082\n\u22a2 [] = l\u2082", "state_after": "no goals"}, {"tactic": "have : a \u2208 l\u2082 := p.subset (mem_cons_self _ _)", "annotated_tactic": ["have : a \u2208 l\u2082 := p.subset (<a>mem_cons_self</a> _ _)", [{"full_name": "List.mem_cons_self", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [66, 9], "def_end_pos": [66, 22]}]], "state_before": "case cons\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081\u271d l\u2082\u271d : List \u03b1\np\u271d : l\u2081\u271d ~ l\u2082\u271d\ns\u2082\u271d : Sorted r l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 r a a'\ns\u2081 : Pairwise r l\u2081\nIH : \u2200 {l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 l\u2081 = l\u2082\nl\u2082 : List \u03b1\np : a :: l\u2081 ~ l\u2082\ns\u2082 : Sorted r l\u2082\n\u22a2 a :: l\u2081 = l\u2082", "state_after": "case cons\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081\u271d l\u2082\u271d : List \u03b1\np\u271d : l\u2081\u271d ~ l\u2082\u271d\ns\u2082\u271d : Sorted r l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 r a a'\ns\u2081 : Pairwise r l\u2081\nIH : \u2200 {l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 l\u2081 = l\u2082\nl\u2082 : List \u03b1\np : a :: l\u2081 ~ l\u2082\ns\u2082 : Sorted r l\u2082\nthis : a \u2208 l\u2082\n\u22a2 a :: l\u2081 = l\u2082"}, {"tactic": "rcases mem_split this with \u27e8u\u2082, v\u2082, rfl\u27e9", "annotated_tactic": ["rcases <a>mem_split</a> this with \u27e8u\u2082, v\u2082, rfl\u27e9", [{"full_name": "List.mem_split", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 18]}]], "state_before": "case cons\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081\u271d l\u2082\u271d : List \u03b1\np\u271d : l\u2081\u271d ~ l\u2082\u271d\ns\u2082\u271d : Sorted r l\u2082\u271d\na : \u03b1\nl\u2081 : List \u03b1\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 r a a'\ns\u2081 : Pairwise r l\u2081\nIH : \u2200 {l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 l\u2081 = l\u2082\nl\u2082 : List \u03b1\np : a :: l\u2081 ~ l\u2082\ns\u2082 : Sorted r l\u2082\nthis : a \u2208 l\u2082\n\u22a2 a :: l\u2081 = l\u2082", "state_after": "case cons.intro.intro\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081\u271d l\u2082 : List \u03b1\np\u271d : l\u2081\u271d ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nl\u2081 : List \u03b1\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 r a a'\ns\u2081 : Pairwise r l\u2081\nIH : \u2200 {l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 l\u2081 = l\u2082\nu\u2082 v\u2082 : List \u03b1\np : a :: l\u2081 ~ u\u2082 ++ a :: v\u2082\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis : a \u2208 u\u2082 ++ a :: v\u2082\n\u22a2 a :: l\u2081 = u\u2082 ++ a :: v\u2082"}, {"tactic": "have p' := (perm_cons a).1 (p.trans perm_middle)", "annotated_tactic": ["have p' := (<a>perm_cons</a> a).1 (p.trans <a>perm_middle</a>)", [{"full_name": "List.perm_cons", "def_path": "Mathlib/Data/List/Perm.lean", "def_pos": [643, 9], "def_end_pos": [643, 18]}, {"full_name": "List.perm_middle", "def_path": "Mathlib/Data/List/Perm.lean", "def_pos": [130, 9], "def_end_pos": [130, 20]}]], "state_before": "case cons.intro.intro\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081\u271d l\u2082 : List \u03b1\np\u271d : l\u2081\u271d ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nl\u2081 : List \u03b1\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 r a a'\ns\u2081 : Pairwise r l\u2081\nIH : \u2200 {l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 l\u2081 = l\u2082\nu\u2082 v\u2082 : List \u03b1\np : a :: l\u2081 ~ u\u2082 ++ a :: v\u2082\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis : a \u2208 u\u2082 ++ a :: v\u2082\n\u22a2 a :: l\u2081 = u\u2082 ++ a :: v\u2082", "state_after": "case cons.intro.intro\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081\u271d l\u2082 : List \u03b1\np\u271d : l\u2081\u271d ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nl\u2081 : List \u03b1\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 r a a'\ns\u2081 : Pairwise r l\u2081\nIH : \u2200 {l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 l\u2081 = l\u2082\nu\u2082 v\u2082 : List \u03b1\np : a :: l\u2081 ~ u\u2082 ++ a :: v\u2082\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis : a \u2208 u\u2082 ++ a :: v\u2082\np' : l\u2081 ~ u\u2082 ++ v\u2082\n\u22a2 a :: l\u2081 = u\u2082 ++ a :: v\u2082"}, {"tactic": "obtain rfl := IH p' (s\u2082.sublist <| by simp)", "annotated_tactic": ["obtain rfl := IH p' (s\u2082.sublist <| by simp)", []], "state_before": "case cons.intro.intro\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081\u271d l\u2082 : List \u03b1\np\u271d : l\u2081\u271d ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nl\u2081 : List \u03b1\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 r a a'\ns\u2081 : Pairwise r l\u2081\nIH : \u2200 {l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 l\u2081 = l\u2082\nu\u2082 v\u2082 : List \u03b1\np : a :: l\u2081 ~ u\u2082 ++ a :: v\u2082\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis : a \u2208 u\u2082 ++ a :: v\u2082\np' : l\u2081 ~ u\u2082 ++ v\u2082\n\u22a2 a :: l\u2081 = u\u2082 ++ a :: v\u2082", "state_after": "case cons.intro.intro\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\n\u22a2 a :: (u\u2082 ++ v\u2082) = u\u2082 ++ a :: v\u2082"}, {"tactic": "change a :: u\u2082 ++ v\u2082 = u\u2082 ++ ([a] ++ v\u2082)", "annotated_tactic": ["change a :: u\u2082 ++ v\u2082 = u\u2082 ++ ([a] ++ v\u2082)", []], "state_before": "case cons.intro.intro\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\n\u22a2 a :: (u\u2082 ++ v\u2082) = u\u2082 ++ a :: v\u2082", "state_after": "case cons.intro.intro\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\n\u22a2 a :: u\u2082 ++ v\u2082 = u\u2082 ++ ([a] ++ v\u2082)"}, {"tactic": "rw [\u2190 append_assoc]", "annotated_tactic": ["rw [\u2190 <a>append_assoc</a>]", [{"full_name": "List.append_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 21]}]], "state_before": "case cons.intro.intro\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\n\u22a2 a :: u\u2082 ++ v\u2082 = u\u2082 ++ ([a] ++ v\u2082)", "state_after": "case cons.intro.intro\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\n\u22a2 a :: u\u2082 ++ v\u2082 = u\u2082 ++ [a] ++ v\u2082"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case cons.intro.intro\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\n\u22a2 a :: u\u2082 ++ v\u2082 = u\u2082 ++ [a] ++ v\u2082", "state_after": "case cons.intro.intro.e_a\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\n\u22a2 a :: u\u2082 = u\u2082 ++ [a]"}, {"tactic": "have : \u2200 (x : \u03b1) (_ : x \u2208 u\u2082), x = a := fun x m =>\n  antisymm ((pairwise_append.1 s\u2082).2.2 _ m a (mem_cons_self _ _)) (h\u2081 _ (by simp [m]))", "annotated_tactic": ["have : \u2200 (x : \u03b1) (_ : x \u2208 u\u2082), x = a := fun x m =>\n      <a>antisymm</a> ((<a>pairwise_append</a>.1 s\u2082).2.2 _ m a (<a>mem_cons_self</a> _ _)) (h\u2081 _ (by simp [m]))", [{"full_name": "antisymm", "def_path": "Mathlib/Init/Algebra/Classes.lean", "def_pos": [316, 9], "def_end_pos": [316, 17]}, {"full_name": "List.pairwise_append", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1457, 9], "def_end_pos": [1457, 24]}, {"full_name": "List.mem_cons_self", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [66, 9], "def_end_pos": [66, 22]}]], "state_before": "case cons.intro.intro.e_a\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\n\u22a2 a :: u\u2082 = u\u2082 ++ [a]", "state_after": "case cons.intro.intro.e_a\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis\u271d : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\nthis : \u2200 (x : \u03b1), x \u2208 u\u2082 \u2192 x = a\n\u22a2 a :: u\u2082 = u\u2082 ++ [a]"}, {"tactic": "rw [(@eq_replicate _ a (length u\u2082 + 1) (a :: u\u2082)).2,\n      (@eq_replicate _ a (length u\u2082 + 1) (u\u2082 ++ [a])).2] <;>\n    constructor <;>\n  simp [iff_true_intro this, or_comm]", "annotated_tactic": ["rw [(@<a>eq_replicate</a> _ a (<a>length</a> u\u2082 + 1) (a :: u\u2082)).2,\n          (@<a>eq_replicate</a> _ a (<a>length</a> u\u2082 + 1) (u\u2082 ++ [a])).2] <;>\n        constructor <;>\n      simp [<a>iff_true_intro</a> this, <a>or_comm</a>]", [{"full_name": "List.eq_replicate", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [338, 9], "def_end_pos": [338, 21]}, {"full_name": "List.length", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2232, 5], "def_end_pos": [2232, 16]}, {"full_name": "List.eq_replicate", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [338, 9], "def_end_pos": [338, 21]}, {"full_name": "List.length", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2232, 5], "def_end_pos": [2232, 16]}, {"full_name": "iff_true_intro", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [71, 9], "def_end_pos": [71, 23]}, {"full_name": "or_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [263, 9], "def_end_pos": [263, 16]}]], "state_before": "case cons.intro.intro.e_a\n\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis\u271d : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\nthis : \u2200 (x : \u03b1), x \u2208 u\u2082 \u2192 x = a\n\u22a2 a :: u\u2082 = u\u2082 ++ [a]", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081\u271d l\u2082 : List \u03b1\np\u271d : l\u2081\u271d ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nl\u2081 : List \u03b1\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 l\u2081 \u2192 r a a'\ns\u2081 : Pairwise r l\u2081\nIH : \u2200 {l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 l\u2081 = l\u2082\nu\u2082 v\u2082 : List \u03b1\np : a :: l\u2081 ~ u\u2082 ++ a :: v\u2082\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis : a \u2208 u\u2082 ++ a :: v\u2082\np' : l\u2081 ~ u\u2082 ++ v\u2082\n\u22a2 u\u2082 ++ v\u2082 <+ u\u2082 ++ a :: v\u2082", "state_after": "no goals"}, {"tactic": "simp [m]", "annotated_tactic": ["simp [m]", []], "state_before": "\u03b1 : Type uu\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : IsAntisymm \u03b1 r\nl\u2081 l\u2082 : List \u03b1\np\u271d : l\u2081 ~ l\u2082\ns\u2082\u271d : Sorted r l\u2082\na : \u03b1\nu\u2082 v\u2082 : List \u03b1\ns\u2082 : Sorted r (u\u2082 ++ a :: v\u2082)\nthis : a \u2208 u\u2082 ++ a :: v\u2082\nh\u2081 : \u2200 (a' : \u03b1), a' \u2208 u\u2082 ++ v\u2082 \u2192 r a a'\ns\u2081 : Pairwise r (u\u2082 ++ v\u2082)\nIH : \u2200 {l\u2082 : List \u03b1}, u\u2082 ++ v\u2082 ~ l\u2082 \u2192 Sorted r l\u2082 \u2192 u\u2082 ++ v\u2082 = l\u2082\np : a :: (u\u2082 ++ v\u2082) ~ u\u2082 ++ a :: v\u2082\np' : u\u2082 ++ v\u2082 ~ u\u2082 ++ v\u2082\nx : \u03b1\nm : x \u2208 u\u2082\n\u22a2 x \u2208 u\u2082 ++ v\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Real/ENNReal.lean", "full_name": "ENNReal.toReal_mono'", "start": [2047, 1], "end": [2050, 35], "traced_tactics": [{"tactic": "rcases eq_or_ne a \u221e with rfl | ha", "annotated_tactic": ["rcases <a>eq_or_ne</a> a \u221e with rfl | ha", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a \u2264 b\nht : b = \u22a4 \u2192 a = \u22a4\n\u22a2 ENNReal.toReal a \u2264 ENNReal.toReal b", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : \u22a4 \u2264 b\nht : b = \u22a4 \u2192 \u22a4 = \u22a4\n\u22a2 ENNReal.toReal \u22a4 \u2264 ENNReal.toReal b\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a \u2264 b\nht : b = \u22a4 \u2192 a = \u22a4\nha : a \u2260 \u22a4\n\u22a2 ENNReal.toReal a \u2264 ENNReal.toReal b"}, {"tactic": "exact toReal_nonneg", "annotated_tactic": ["exact <a>toReal_nonneg</a>", [{"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nb c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : \u22a4 \u2264 b\nht : b = \u22a4 \u2192 \u22a4 = \u22a4\n\u22a2 ENNReal.toReal \u22a4 \u2264 ENNReal.toReal b", "state_after": "no goals"}, {"tactic": "exact toReal_mono (mt ht ha) h", "annotated_tactic": ["exact <a>toReal_mono</a> (<a>mt</a> ht ha) h", [{"full_name": "ENNReal.toReal_mono", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2042, 9], "def_end_pos": [2042, 20]}, {"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nh : a \u2264 b\nht : b = \u22a4 \u2192 a = \u22a4\nha : a \u2260 \u22a4\n\u22a2 ENNReal.toReal a \u2264 ENNReal.toReal b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/ContinuousOn.lean", "full_name": "frontier_inter_open_inter", "start": [1305, 1], "end": [1309, 37], "traced_tactics": [{"tactic": "simp only [\u2190 Subtype.preimage_coe_eq_preimage_coe_iff,\n  ht.isOpenMap_subtype_val.preimage_frontier_eq_frontier_preimage continuous_subtype_val,\n  Subtype.preimage_coe_inter_self]", "annotated_tactic": ["simp only [\u2190 <a>Subtype.preimage_coe_eq_preimage_coe_iff</a>,\n    ht.isOpenMap_subtype_val.preimage_frontier_eq_frontier_preimage <a>continuous_subtype_val</a>,\n    <a>Subtype.preimage_coe_inter_self</a>]", [{"full_name": "Subtype.preimage_coe_eq_preimage_coe_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1456, 9], "def_end_pos": [1456, 41]}, {"full_name": "continuous_subtype_val", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1000, 9], "def_end_pos": [1000, 31]}, {"full_name": "Subtype.preimage_coe_inter_self", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1463, 9], "def_end_pos": [1463, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\ns t : Set \u03b1\nht : IsOpen t\n\u22a2 frontier (s \u2229 t) \u2229 t = frontier s \u2229 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.lift_down", "start": [1117, 1], "end": [1125, 53], "traced_tactics": [{"tactic": "rw [\u2190 lift_id #\u03b2, \u2190 lift_umax, \u2190 lift_umax.{u, v}, lift_mk_le.{v}]", "annotated_tactic": ["rw [\u2190 <a>lift_id</a> #\u03b2, \u2190 <a>lift_umax</a>, \u2190 <a>lift_umax</a>.{u, v}, <a>lift_mk_le</a>.{v}]", [{"full_name": "Cardinal.lift_id", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [224, 9], "def_end_pos": [224, 16]}, {"full_name": "Cardinal.lift_umax", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [202, 9], "def_end_pos": [202, 18]}, {"full_name": "Cardinal.lift_umax", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [202, 9], "def_end_pos": [202, 18]}, {"full_name": "Cardinal.lift_mk_le", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [310, 9], "def_end_pos": [310, 19]}]], "state_before": "\u03b1\u271d \u03b2\u271d : Type u\na : Cardinal.{u}\nb : Cardinal.{max u v}\n\u03b1 : Type u\n\u03b2 : Type (max u v)\n\u22a2 #\u03b2 \u2264 lift.{v, u} #\u03b1 \u2192 \u2203 a', lift.{v, u} a' = #\u03b2", "state_after": "\u03b1\u271d \u03b2\u271d : Type u\na : Cardinal.{u}\nb : Cardinal.{max u v}\n\u03b1 : Type u\n\u03b2 : Type (max u v)\n\u22a2 Nonempty (\u03b2 \u21aa \u03b1) \u2192 \u2203 a', lift.{max u v, u} a' = lift.{max u v, max u v} #\u03b2"}, {"tactic": "exact fun \u27e8f\u27e9 =>\n  \u27e8#(Set.range f),\n    Eq.symm <| lift_mk_eq.{_, _, v}.2\n      \u27e8Function.Embedding.equivOfSurjective (Embedding.codRestrict _ f Set.mem_range_self)\n          fun \u27e8a, \u27e8b, e\u27e9\u27e9 => \u27e8b, Subtype.eq e\u27e9\u27e9\u27e9", "annotated_tactic": ["exact fun \u27e8f\u27e9 =>\n      \u27e8#(<a>Set.range</a> f),\n        <a>Eq.symm</a> <| <a>lift_mk_eq</a>.{_, _, v}.2\n          \u27e8<a>Function.Embedding.equivOfSurjective</a> (<a>Embedding.codRestrict</a> _ f <a>Set.mem_range_self</a>)\n              fun \u27e8a, \u27e8b, e\u27e9\u27e9 => \u27e8b, <a>Subtype.eq</a> e\u27e9\u27e9\u27e9", [{"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "Cardinal.lift_mk_eq", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [324, 9], "def_end_pos": [324, 19]}, {"full_name": "Function.Embedding.equivOfSurjective", "def_path": "Mathlib/Logic/Embedding/Basic.lean", "def_pos": [176, 29], "def_end_pos": [176, 46]}, {"full_name": "Function.Embedding.codRestrict", "def_path": "Mathlib/Logic/Embedding/Set.lean", "def_pos": [63, 5], "def_end_pos": [63, 16]}, {"full_name": "Set.mem_range_self", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [680, 9], "def_end_pos": [680, 23]}, {"full_name": "Subtype.eq", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [960, 19], "def_end_pos": [960, 21]}]], "state_before": "\u03b1\u271d \u03b2\u271d : Type u\na : Cardinal.{u}\nb : Cardinal.{max u v}\n\u03b1 : Type u\n\u03b2 : Type (max u v)\n\u22a2 Nonempty (\u03b2 \u21aa \u03b1) \u2192 \u2203 a', lift.{max u v, u} a' = lift.{max u v, max u v} #\u03b2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Antilipschitz.lean", "full_name": "AntilipschitzWith.comp", "start": [128, 1], "end": [133, 53], "traced_tactics": [{"tactic": "rw [ENNReal.coe_mul, mul_assoc]", "annotated_tactic": ["rw [<a>ENNReal.coe_mul</a>, <a>mul_assoc</a>]", [{"full_name": "ENNReal.coe_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [390, 9], "def_end_pos": [390, 16]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nKg : \u211d\u22650\ng : \u03b2 \u2192 \u03b3\nhg : AntilipschitzWith Kg g\nKf : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nhf : AntilipschitzWith Kf f\nx y : \u03b1\n\u22a2 \u2191Kf * (\u2191Kg * edist (g (f x)) (g (f y))) = \u2191(Kf * Kg) * edist ((g \u2218 f) x) ((g \u2218 f) y)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nKg : \u211d\u22650\ng : \u03b2 \u2192 \u03b3\nhg : AntilipschitzWith Kg g\nKf : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nhf : AntilipschitzWith Kf f\nx y : \u03b1\n\u22a2 \u2191Kf * (\u2191Kg * edist (g (f x)) (g (f y))) = \u2191Kf * (\u2191Kg * edist ((g \u2218 f) x) ((g \u2218 f) y))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b2\ninst\u271d : PseudoEMetricSpace \u03b3\nK : \u211d\u22650\nf\u271d : \u03b1 \u2192 \u03b2\nKg : \u211d\u22650\ng : \u03b2 \u2192 \u03b3\nhg : AntilipschitzWith Kg g\nKf : \u211d\u22650\nf : \u03b1 \u2192 \u03b2\nhf : AntilipschitzWith Kf f\nx y : \u03b1\n\u22a2 \u2191Kf * (\u2191Kg * edist (g (f x)) (g (f y))) = \u2191Kf * (\u2191Kg * edist ((g \u2218 f) x) ((g \u2218 f) y))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Option/Lemmas.lean", "full_name": "Option.to_list_none", "start": [224, 9], "end": [224, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Perm/Basic.lean", "full_name": "Equiv.Perm.zpow_apply_comm", "start": [118, 1], "end": [120, 69], "traced_tactics": [{"tactic": "rw [\u2190 Equiv.Perm.mul_apply, \u2190 Equiv.Perm.mul_apply, zpow_mul_comm]", "annotated_tactic": ["rw [\u2190 <a>Equiv.Perm.mul_apply</a>, \u2190 <a>Equiv.Perm.mul_apply</a>, <a>zpow_mul_comm</a>]", [{"full_name": "Equiv.Perm.mul_apply", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [67, 9], "def_end_pos": [67, 18]}, {"full_name": "Equiv.Perm.mul_apply", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [67, 9], "def_end_pos": [67, 18]}, {"full_name": "zpow_mul_comm", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [254, 9], "def_end_pos": [254, 22]}]], "state_before": "\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\n\u03c3 : Perm \u03b1\nm n : \u2124\nx : \u03b1\n\u22a2 \u2191(\u03c3 ^ m) (\u2191(\u03c3 ^ n) x) = \u2191(\u03c3 ^ n) (\u2191(\u03c3 ^ m) x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "full_name": "Polynomial.nextCoeff_X_sub_C", "start": [1561, 1], 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\u211d E\nx y : E\ns : Set E\nh : [x-[\u211d]y] \u2286 s\nA : Continuous fun t => (1 - t) \u2022 x + t \u2022 y\n\u22a2 (fun t => (1 - t) \u2022 x + t \u2022 y) '' unitInterval = [x-[\u211d]y]", "state_after": "no goals"}, {"tactic": "continuity", "annotated_tactic": ["continuity", []], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE\u271d : Type u_3\ninst\u271d\u2079 : AddCommGroup E\u271d\ninst\u271d\u2078 : Module \u211d E\u271d\ninst\u271d\u2077 : TopologicalSpace E\u271d\ninst\u271d\u2076 : TopologicalAddGroup E\u271d\ninst\u271d\u2075 : ContinuousSMul \u211d E\u271d\nE : Type u_4\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\nx y : E\ns : Set E\nh : [x-[\u211d]y] \u2286 s\n\u22a2 Continuous fun t => (1 - t) \u2022 x + t \u2022 y", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE\u271d : Type u_3\ninst\u271d\u2079 : AddCommGroup E\u271d\ninst\u271d\u2078 : Module \u211d E\u271d\ninst\u271d\u2077 : TopologicalSpace E\u271d\ninst\u271d\u2076 : TopologicalAddGroup E\u271d\ninst\u271d\u2075 : ContinuousSMul \u211d E\u271d\nE : Type u_4\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\nx y : E\ns : Set E\nh : [x-[\u211d]y] \u2286 s\nA : Continuous fun t => (1 - t) \u2022 x + t \u2022 y\n\u22a2 (1 - 0) \u2022 x + 0 \u2022 y = x", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE\u271d : Type u_3\ninst\u271d\u2079 : AddCommGroup E\u271d\ninst\u271d\u2078 : Module \u211d E\u271d\ninst\u271d\u2077 : TopologicalSpace E\u271d\ninst\u271d\u2076 : TopologicalAddGroup E\u271d\ninst\u271d\u2075 : ContinuousSMul \u211d E\u271d\nE : Type u_4\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\nx y : E\ns : Set E\nh : [x-[\u211d]y] \u2286 s\nA : Continuous fun t => (1 - t) \u2022 x + t \u2022 y\n\u22a2 (1 - 1) \u2022 x + 1 \u2022 y = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/RingDivision.lean", "full_name": "Polynomial.Monic.comp", "start": [859, 1], "end": [860, 88], "traced_tactics": [{"tactic": "rw [Monic.def, leadingCoeff_comp h, Monic.def.1 hp, Monic.def.1 hq, one_pow, one_mul]", "annotated_tactic": ["rw [<a>Monic.def</a>, <a>leadingCoeff_comp</a> h, <a>Monic.def</a>.1 hp, <a>Monic.def</a>.1 hq, <a>one_pow</a>, <a>one_mul</a>]", [{"full_name": "Polynomial.Monic.def", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [84, 9], "def_end_pos": [84, 18]}, {"full_name": "Polynomial.leadingCoeff_comp", "def_path": "Mathlib/Data/Polynomial/Degree/Lemmas.lean", "def_pos": [393, 9], "def_end_pos": [393, 26]}, {"full_name": "Polynomial.Monic.def", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [84, 9], "def_end_pos": [84, 18]}, {"full_name": "Polynomial.Monic.def", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [84, 9], "def_end_pos": [84, 18]}, {"full_name": "one_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [90, 9], "def_end_pos": [90, 16]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np q : R[X]\nhp : Monic p\nhq : Monic q\nh : natDegree q \u2260 0\n\u22a2 Monic (Polynomial.comp p q)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean", "full_name": "InnerProductGeometry.norm_div_sin_angle_sub_of_inner_eq_zero", "start": [338, 1], "end": [342, 80], "traced_tactics": [{"tactic": "rw [\u2190 neg_eq_zero, \u2190 inner_neg_right] at h", "annotated_tactic": ["rw [\u2190 <a>neg_eq_zero</a>, \u2190 <a>inner_neg_right</a>] at h", [{"full_name": "neg_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [422, 3], "def_end_pos": [422, 14]}, {"full_name": "inner_neg_right", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [635, 9], "def_end_pos": [635, 24]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : inner x y = 0\nh0 : x = 0 \u2228 y \u2260 0\n\u22a2 \u2016y\u2016 / Real.sin (angle x (x - y)) = \u2016x - y\u2016", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : inner x (-y) = 0\nh0 : x = 0 \u2228 y \u2260 0\n\u22a2 \u2016y\u2016 / Real.sin (angle x (x - y)) = \u2016x - y\u2016"}, {"tactic": "rw [\u2190 neg_ne_zero] at h0", "annotated_tactic": ["rw [\u2190 <a>neg_ne_zero</a>] at h0", [{"full_name": "neg_ne_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [434, 3], "def_end_pos": [434, 14]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : inner x (-y) = 0\nh0 : x = 0 \u2228 y \u2260 0\n\u22a2 \u2016y\u2016 / Real.sin (angle x (x - y)) = \u2016x - y\u2016", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : inner x (-y) = 0\nh0 : x = 0 \u2228 -y \u2260 0\n\u22a2 \u2016y\u2016 / Real.sin (angle x (x - y)) = \u2016x - y\u2016"}, {"tactic": "rw [sub_eq_add_neg, \u2190 norm_neg, norm_div_sin_angle_add_of_inner_eq_zero h h0]", "annotated_tactic": ["rw [<a>sub_eq_add_neg</a>, \u2190 <a>norm_neg</a>, <a>norm_div_sin_angle_add_of_inner_eq_zero</a> h h0]", [{"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "norm_neg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [437, 30], "def_end_pos": [437, 38]}, {"full_name": "InnerProductGeometry.norm_div_sin_angle_add_of_inner_eq_zero", "def_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean", "def_pos": [211, 9], "def_end_pos": [211, 48]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : inner x (-y) = 0\nh0 : x = 0 \u2228 -y \u2260 0\n\u22a2 \u2016y\u2016 / Real.sin (angle x (x - y)) = \u2016x - y\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Manifold/BumpFunction.lean", "full_name": "SmoothBumpFunction.nhds_basis_tsupport", "start": [289, 1], "end": [297, 35], "traced_tactics": [{"tactic": "have :\n  (\ud835\udcdd c).HasBasis (fun _ : SmoothBumpFunction I c => True) fun f =>\n    (extChartAt I c).symm '' (closedBall (extChartAt I c c) f.rOut \u2229 range I) := by\n  rw [\u2190 map_extChartAt_symm_nhdsWithin_range I c]\n  exact nhdsWithin_range_basis.map _", "annotated_tactic": ["have :\n    (\ud835\udcdd c).<a>HasBasis</a> (fun _ : <a>SmoothBumpFunction</a> I c => <a>True</a>) fun f =>\n      (<a>extChartAt</a> I c).<a>symm</a> '' (<a>closedBall</a> (<a>extChartAt</a> I c c) f.rOut \u2229 <a>range</a> I) := by\n    rw [\u2190 <a>map_extChartAt_symm_nhdsWithin_range</a> I c]\n    exact nhdsWithin_range_basis.map _", [{"full_name": "Filter.HasBasis", "def_path": "Mathlib/Order/Filter/Bases.lean", "def_pos": [231, 11], "def_end_pos": [231, 19]}, {"full_name": "SmoothBumpFunction", "def_path": "Mathlib/Geometry/Manifold/BumpFunction.lean", "def_pos": [61, 11], "def_end_pos": [61, 29]}, {"full_name": "True", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [176, 11], "def_end_pos": [176, 15]}, {"full_name": "extChartAt", "def_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "def_pos": [1242, 5], "def_end_pos": [1242, 15]}, {"full_name": "LocalEquiv.symm", "def_path": "Mathlib/Logic/Equiv/LocalEquiv.lean", "def_pos": [151, 15], "def_end_pos": [151, 19]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [474, 5], "def_end_pos": [474, 15]}, {"full_name": "extChartAt", "def_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "def_pos": [1242, 5], "def_end_pos": [1242, 15]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}, {"full_name": "map_extChartAt_symm_nhdsWithin_range", "def_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "def_pos": [1422, 9], "def_end_pos": [1422, 45]}]], "state_before": "E : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ninst\u271d : T2Space M\n\u22a2 HasBasis (\ud835\udcdd c) (fun x => True) fun f => tsupport \u2191f", "state_after": "E : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ninst\u271d : T2Space M\nthis :\n  HasBasis (\ud835\udcdd c) (fun x => True) fun f =>\n    \u2191(LocalEquiv.symm (extChartAt I c)) '' (closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I)\n\u22a2 HasBasis (\ud835\udcdd c) (fun x => True) fun f => tsupport \u2191f"}, {"tactic": "refine' this.to_hasBasis' (fun f _ => \u27e8f, trivial, f.tsupport_subset_symm_image_closedBall\u27e9)\n  fun f _ => f.tsupport_mem_nhds", "annotated_tactic": ["refine' this.to_hasBasis' (fun f _ => \u27e8f, <a>trivial</a>, f.tsupport_subset_symm_image_closedBall\u27e9)\n    fun f _ => f.tsupport_mem_nhds", [{"full_name": "trivial", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [514, 31], "def_end_pos": [514, 38]}]], "state_before": "E : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ninst\u271d : T2Space M\nthis :\n  HasBasis (\ud835\udcdd c) (fun x => True) fun f =>\n    \u2191(LocalEquiv.symm (extChartAt I c)) '' (closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I)\n\u22a2 HasBasis (\ud835\udcdd c) (fun x => True) fun f => tsupport \u2191f", "state_after": "no goals"}, {"tactic": "rw [\u2190 map_extChartAt_symm_nhdsWithin_range I c]", "annotated_tactic": ["rw [\u2190 <a>map_extChartAt_symm_nhdsWithin_range</a> I c]", [{"full_name": "map_extChartAt_symm_nhdsWithin_range", "def_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "def_pos": [1422, 9], "def_end_pos": [1422, 45]}]], "state_before": "E : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ninst\u271d : T2Space M\n\u22a2 HasBasis (\ud835\udcdd c) (fun x => True) fun f =>\n    \u2191(LocalEquiv.symm (extChartAt I c)) '' (closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I)", "state_after": "E : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ninst\u271d : T2Space M\n\u22a2 HasBasis (map (\u2191(LocalEquiv.symm (extChartAt I c))) (\ud835\udcdd[range \u2191I] \u2191(extChartAt I c) c)) (fun x => True) fun f =>\n    \u2191(LocalEquiv.symm (extChartAt I c)) '' (closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I)"}, {"tactic": "exact nhdsWithin_range_basis.map _", "annotated_tactic": ["exact nhdsWithin_range_basis.map _", []], "state_before": "E : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u2074 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : ChartedSpace H M\ninst\u271d\u00b9 : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\ninst\u271d : T2Space M\n\u22a2 HasBasis (map (\u2191(LocalEquiv.symm (extChartAt I c))) (\ud835\udcdd[range \u2191I] \u2191(extChartAt I c) c)) (fun x => True) fun f =>\n    \u2191(LocalEquiv.symm (extChartAt I c)) '' (closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/Colex.lean", "full_name": "Colex.sum_two_pow_lt_iff_lt", "start": [391, 1], "end": [414, 19], "traced_tactics": [{"tactic": "refine'\n  \u27e8fun h => (lt_trichotomy A B).resolve_right fun h\u2081 => h\u2081.elim _ (not_lt_of_gt h \u2218 z _ _), z A B\u27e9", "annotated_tactic": ["refine'\n    \u27e8fun h => (<a>lt_trichotomy</a> A B).<a>resolve_right</a> fun h\u2081 => h\u2081.elim _ (<a>not_lt_of_gt</a> h \u2218 z _ _), z A B\u27e9", [{"full_name": "Colex.lt_trichotomy", "def_path": "Mathlib/Combinatorics/Colex.lean", "def_pos": [168, 9], "def_end_pos": [168, 22]}, {"full_name": "Or.resolve_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [272, 9], "def_end_pos": [272, 25]}, {"full_name": "not_lt_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [307, 9], "def_end_pos": [307, 21]}]], "state_before": "\u03b1 : Type u_1\nA B : Finset \u2115\nz : \u2200 (A B : Finset \u2115), toColex A < toColex B \u2192 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i\n\u22a2 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i \u2194 toColex A < toColex B", "state_after": "\u03b1 : Type u_1\nA B : Finset \u2115\nz : \u2200 (A B : Finset \u2115), toColex A < toColex B \u2192 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i\nh : \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i\nh\u2081 : A = B \u2228 B < A\n\u22a2 A = B \u2192 False"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "\u03b1 : Type u_1\nA B : Finset \u2115\nz : \u2200 (A B : Finset \u2115), toColex A < toColex B \u2192 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i\nh : \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i\nh\u2081 : A = B \u2228 B < A\n\u22a2 A = B \u2192 False", "state_after": "\u03b1 : Type u_1\nA : Finset \u2115\nz : \u2200 (A B : Finset \u2115), toColex A < toColex B \u2192 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i\nh : \u2211 i in A, 2 ^ i < \u2211 i in A, 2 ^ i\nh\u2081 : A = A \u2228 A < A\n\u22a2 False"}, {"tactic": "apply irrefl _ h", "annotated_tactic": ["apply <a>irrefl</a> _ h", [{"full_name": "irrefl", "def_path": "Mathlib/Init/Algebra/Classes.lean", "def_pos": [300, 9], "def_end_pos": [300, 15]}]], "state_before": "\u03b1 : Type u_1\nA : Finset \u2115\nz : \u2200 (A B : Finset \u2115), toColex A < toColex B \u2192 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i\nh : \u2211 i in A, 2 ^ i < \u2211 i in A, 2 ^ i\nh\u2081 : A = A \u2228 A < A\n\u22a2 False", "state_after": "no goals"}, {"tactic": "intro A B", "annotated_tactic": ["intro A B", []], "state_before": "\u03b1 : Type u_1\nA B : Finset \u2115\n\u22a2 \u2200 (A B : Finset \u2115), toColex A < toColex B \u2192 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i", "state_after": "\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\n\u22a2 toColex A < toColex B \u2192 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i"}, {"tactic": "rw [\u2190 sdiff_lt_sdiff_iff_lt, Colex.lt_def]", "annotated_tactic": ["rw [\u2190 <a>sdiff_lt_sdiff_iff_lt</a>, <a>Colex.lt_def</a>]", [{"full_name": "Colex.sdiff_lt_sdiff_iff_lt", "def_path": "Mathlib/Combinatorics/Colex.lean", "def_pos": [308, 9], "def_end_pos": [308, 30]}, {"full_name": "Colex.lt_def", "def_path": "Mathlib/Combinatorics/Colex.lean", "def_pos": [91, 9], "def_end_pos": [91, 21]}]], "state_before": "\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\n\u22a2 toColex A < toColex B \u2192 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i", "state_after": "\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\n\u22a2 (\u2203 k, (\u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)) \u2227 \u00ack \u2208 A \\ B \u2227 k \u2208 B \\ A) \u2192 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i"}, {"tactic": "rintro \u27e8k, z, kA, kB\u27e9", "annotated_tactic": ["rintro \u27e8k, z, kA, kB\u27e9", []], "state_before": "\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\n\u22a2 (\u2203 k, (\u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)) \u2227 \u00ack \u2208 A \\ B \u2227 k \u2208 B \\ A) \u2192 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\n\u22a2 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i"}, {"tactic": "rw [\u2190 sdiff_union_inter A B]", "annotated_tactic": ["rw [\u2190 <a>sdiff_union_inter</a> A B]", [{"full_name": "Finset.sdiff_union_inter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2183, 9], "def_end_pos": [2183, 26]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\n\u22a2 \u2211 i in A, 2 ^ i < \u2211 i in B, 2 ^ i", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\n\u22a2 \u2211 i in A \\ B \u222a A \u2229 B, 2 ^ i < \u2211 i in B, 2 ^ i"}, {"tactic": "conv_rhs => rw [\u2190 sdiff_union_inter B A]", "annotated_tactic": ["conv_rhs => rw [\u2190 <a>sdiff_union_inter</a> B A]", [{"full_name": "Finset.sdiff_union_inter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2183, 9], "def_end_pos": [2183, 26]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\n\u22a2 \u2211 i in A \\ B \u222a A \u2229 B, 2 ^ i < \u2211 i in B, 2 ^ i", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\n\u22a2 \u2211 i in A \\ B \u222a A \u2229 B, 2 ^ i < \u2211 i in B \\ A \u222a B \u2229 A, 2 ^ i"}, {"tactic": "rw [sum_union (disjoint_sdiff_inter _ _), sum_union (disjoint_sdiff_inter _ _), inter_comm,\n  add_lt_add_iff_right]", "annotated_tactic": ["rw [<a>sum_union</a> (<a>disjoint_sdiff_inter</a> _ _), <a>sum_union</a> (<a>disjoint_sdiff_inter</a> _ _), <a>inter_comm</a>,\n      <a>add_lt_add_iff_right</a>]", [{"full_name": "Finset.sum_union", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [408, 3], "def_end_pos": [408, 14]}, {"full_name": "Finset.disjoint_sdiff_inter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2399, 9], "def_end_pos": [2399, 29]}, {"full_name": "Finset.sum_union", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [408, 3], "def_end_pos": [408, 14]}, {"full_name": "Finset.disjoint_sdiff_inter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2399, 9], "def_end_pos": [2399, 29]}, {"full_name": "Finset.inter_comm", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1642, 9], "def_end_pos": [1642, 19]}, {"full_name": "add_lt_add_iff_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [112, 3], "def_end_pos": [112, 14]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\n\u22a2 \u2211 i in A \\ B \u222a A \u2229 B, 2 ^ i < \u2211 i in B \\ A \u222a B \u2229 A, 2 ^ i", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\n\u22a2 \u2211 x in A \\ B, 2 ^ x < \u2211 x in B \\ A, 2 ^ x"}, {"tactic": "apply lt_of_lt_of_le (@Nat.sum_two_pow_lt k (A \\ B) _)", "annotated_tactic": ["apply <a>lt_of_lt_of_le</a> (@<a>Nat.sum_two_pow_lt</a> k (A \\ B) _)", [{"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}, {"full_name": "Nat.sum_two_pow_lt", "def_path": "Mathlib/Combinatorics/Colex.lean", "def_pos": [102, 9], "def_end_pos": [102, 27]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\n\u22a2 \u2211 x in A \\ B, 2 ^ x < \u2211 x in B \\ A, 2 ^ x", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\n\u22a2 2 ^ k \u2264 \u2211 x in B \\ A, 2 ^ x\n\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\n\u22a2 \u2200 {x : \u2115}, x \u2208 A \\ B \u2192 x < k"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\n\u22a2 \u2200 {x : \u2115}, x \u2208 A \\ B \u2192 x < k", "state_after": "\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\nx : \u2115\nhx : x \u2208 A \\ B\n\u22a2 x < k"}, {"tactic": "apply lt_of_le_of_ne (le_of_not_lt _)", "annotated_tactic": ["apply <a>lt_of_le_of_ne</a> (<a>le_of_not_lt</a> _)", [{"full_name": "lt_of_le_of_ne", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [196, 9], "def_end_pos": [196, 23]}, {"full_name": "le_of_not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [321, 9], "def_end_pos": [321, 21]}]], "state_before": "\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\nx : \u2115\nhx : x \u2208 A \\ B\n\u22a2 x < k", "state_after": "\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\nx : \u2115\nhx : x \u2208 A \\ B\n\u22a2 x \u2260 k\n\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\nx : \u2115\nhx : x \u2208 A \\ B\n\u22a2 \u00ack < x"}, {"tactic": "intro kx", "annotated_tactic": ["intro kx", []], "state_before": "\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\nx : \u2115\nhx : x \u2208 A \\ B\n\u22a2 \u00ack < x", "state_after": "\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\nx : \u2115\nhx : x \u2208 A \\ B\nkx : k < x\n\u22a2 False"}, {"tactic": "have := (z kx).1 hx", "annotated_tactic": ["have := (z kx).1 hx", []], "state_before": "\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\nx : \u2115\nhx : x \u2208 A \\ B\nkx : k < x\n\u22a2 False", "state_after": "\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\nx : \u2115\nhx : x \u2208 A \\ B\nkx : k < x\nthis : x \u2208 B \\ A\n\u22a2 False"}, {"tactic": "rw [mem_sdiff] at this hx", "annotated_tactic": ["rw [<a>mem_sdiff</a>] at this hx", [{"full_name": "Finset.mem_sdiff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2077, 9], "def_end_pos": [2077, 18]}]], "state_before": "\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\nx : \u2115\nhx : x \u2208 A \\ B\nkx : k < x\nthis : x \u2208 B \\ A\n\u22a2 False", "state_after": "\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\nx : \u2115\nhx : x \u2208 A \u2227 \u00acx \u2208 B\nkx : k < x\nthis : x \u2208 B \u2227 \u00acx \u2208 A\n\u22a2 False"}, {"tactic": "exact hx.2 this.1", "annotated_tactic": ["exact hx.2 this.1", []], "state_before": "\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\nx : \u2115\nhx : x \u2208 A \u2227 \u00acx \u2208 B\nkx : k < x\nthis : x \u2208 B \u2227 \u00acx \u2208 A\n\u22a2 False", "state_after": "no goals"}, {"tactic": "apply single_le_sum (fun _ _ => Nat.zero_le _) kB", "annotated_tactic": ["apply <a>single_le_sum</a> (fun _ _ => <a>Nat.zero_le</a> _) kB", [{"full_name": "Finset.single_le_sum", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [202, 15], "def_end_pos": [202, 28]}, {"full_name": "Nat.zero_le", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 20]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\n\u22a2 2 ^ k \u2264 \u2211 x in B \\ A, 2 ^ x", "state_after": "no goals"}, {"tactic": "apply ne_of_mem_of_not_mem hx kA", "annotated_tactic": ["apply <a>ne_of_mem_of_not_mem</a> hx kA", [{"full_name": "ne_of_mem_of_not_mem", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [719, 9], "def_end_pos": [719, 29]}]], "state_before": "\u03b1 : Type u_1\nA\u271d B\u271d A B : Finset \u2115\nk : \u2115\nz : \u2200 {x : \u2115}, k < x \u2192 (x \u2208 A \\ B \u2194 x \u2208 B \\ A)\nkA : \u00ack \u2208 A \\ B\nkB : k \u2208 B \\ A\nx : \u2115\nhx : x \u2208 A \\ B\n\u22a2 x \u2260 k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "full_name": "MvPowerSeries.map_X", "start": [640, 1], "end": [640, 73], "traced_tactics": [{"tactic": "simp [MvPowerSeries.X]", "annotated_tactic": ["simp [<a>MvPowerSeries.X</a>]", [{"full_name": "MvPowerSeries.X", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [405, 5], "def_end_pos": [405, 6]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\nS : Type u_3\nT : Type u_4\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Semiring T\nf : R \u2192+* S\ng : S \u2192+* T\ns : \u03c3\n\u22a2 \u2191(map \u03c3 f) (X s) = X s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Matrix/Block.lean", "full_name": "Matrix.blockDiagonal_apply_ne", "start": [386, 1], "end": [388, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Dimension.lean", "full_name": "Basis.card_le_card_of_submodule", "start": [825, 1], "end": [827, 90], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.dom_bool\u2082", "start": [708, 1], "end": [710, 20], "traced_tactics": [{"tactic": "cases a <;> rfl", "annotated_tactic": ["cases a <;> rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03c3 : Type u_5\ninst\u271d\u2074 : Primcodable \u03b1\ninst\u271d\u00b3 : Primcodable \u03b2\ninst\u271d\u00b2 : Primcodable \u03b3\ninst\u271d\u00b9 : Primcodable \u03b4\ninst\u271d : Primcodable \u03c3\nf : Bool \u2192 Bool \u2192 \u03b1\nx\u271d : Bool \u00d7 Bool\na b : Bool\n\u22a2 (bif (a, b).1 then f true (a, b).2 else f false (a, b).2) = f (a, b).1 (a, b).2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Sigma.lean", "full_name": "Set.mem_sigma_iff", "start": [59, 1], "end": [60, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "full_name": "Zsqrtd.sqLe_cancel", "start": [444, 1], "end": [454, 84], "traced_tactics": [{"tactic": "apply le_of_not_gt", "annotated_tactic": ["apply <a>le_of_not_gt</a>", [{"full_name": "le_of_not_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [328, 9], "def_end_pos": [328, 21]}]], "state_before": "d\u271d : \u2124\nc d x y z w : \u2115\nzw : SqLe y d x c\nh : SqLe (x + z) c (y + w) d\n\u22a2 SqLe z c w d", "state_after": "case a\nd\u271d : \u2124\nc d x y z w : \u2115\nzw : SqLe y d x c\nh : SqLe (x + z) c (y + w) d\n\u22a2 \u00acc * z * z > d * w * w"}, {"tactic": "intro l", "annotated_tactic": ["intro l", []], "state_before": "case a\nd\u271d : \u2124\nc d x y z w : \u2115\nzw : SqLe y d x c\nh : SqLe (x + z) c (y + w) d\n\u22a2 \u00acc * z * z > d * w * w", "state_after": "case a\nd\u271d : \u2124\nc d x y z w : \u2115\nzw : SqLe y d x c\nh : SqLe (x + z) c (y + w) d\nl : c * z * z > d * w * w\n\u22a2 False"}, {"tactic": "refine' not_le_of_gt _ h", "annotated_tactic": ["refine' <a>not_le_of_gt</a> _ h", [{"full_name": "not_le_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [148, 9], "def_end_pos": [148, 21]}]], "state_before": "case a\nd\u271d : \u2124\nc d x y z w : \u2115\nzw : SqLe y d x c\nh : SqLe (x + z) c (y + w) d\nl : c * z * z > d * w * w\n\u22a2 False", "state_after": "case a\nd\u271d : \u2124\nc d x y z w : \u2115\nzw : SqLe y d x c\nh : SqLe (x + z) c (y + w) d\nl : c * z * z > d * w * w\n\u22a2 c * (x + z) * (x + z) > d * (y + w) * (y + w)"}, {"tactic": "simp only [SqLe, mul_add, mul_comm, mul_left_comm, add_assoc, gt_iff_lt]", "annotated_tactic": ["simp only [<a>SqLe</a>, <a>mul_add</a>, <a>mul_comm</a>, <a>mul_left_comm</a>, <a>add_assoc</a>, <a>gt_iff_lt</a>]", [{"full_name": "Zsqrtd.SqLe", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [422, 5], "def_end_pos": [422, 9]}, {"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "mul_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [96, 9], "def_end_pos": [96, 22]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "gt_iff_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 18]}]], "state_before": "case a\nd\u271d : \u2124\nc d x y z w : \u2115\nzw : SqLe y d x c\nh : SqLe (x + z) c (y + w) d\nl : c * z * z > d * w * w\n\u22a2 c * (x + z) * (x + z) > d * (y + w) * (y + w)", "state_after": "case a\nd\u271d : \u2124\nc d x y z w : \u2115\nzw : SqLe y d x c\nh : SqLe (x + z) c (y + w) d\nl : c * z * z > d * w * w\n\u22a2 d * (y * y) + (d * (y * w) + (d * (y * w) + d * (w * w))) < c * (x * x) + (c * (x * z) + (c * (x * z) + c * (z * z)))"}, {"tactic": "have hm := sqLe_add_mixed zw (le_of_lt l)", "annotated_tactic": ["have hm := <a>sqLe_add_mixed</a> zw (<a>le_of_lt</a> l)", [{"full_name": "Zsqrtd.sqLe_add_mixed", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [431, 9], "def_end_pos": [431, 23]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case a\nd\u271d : \u2124\nc d x y z w : \u2115\nzw : SqLe y d x c\nh : SqLe (x + z) c (y + w) d\nl : c * z * z > d * w * w\n\u22a2 d * (y * y) + (d * (y * w) + (d * (y * w) + d * (w * w))) < c * (x * x) + (c * (x * z) + (c * (x * z) + c * (z * z)))", "state_after": "case a\nd\u271d : \u2124\nc d x y z w : \u2115\nzw : SqLe y d x c\nh : SqLe (x + z) c (y + w) d\nl : c * z * z > d * w * w\nhm : d * (y * w) \u2264 c * (x * z)\n\u22a2 d * (y * y) + (d * (y * w) + (d * (y * w) + d * (w * w))) < c * (x * x) + (c * (x * z) + (c * (x * z) + c * (z * z)))"}, {"tactic": "simp only [SqLe, mul_assoc, gt_iff_lt] at l zw", "annotated_tactic": ["simp only [<a>SqLe</a>, <a>mul_assoc</a>, <a>gt_iff_lt</a>] at l zw", [{"full_name": "Zsqrtd.SqLe", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [422, 5], "def_end_pos": [422, 9]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "gt_iff_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 18]}]], "state_before": "case a\nd\u271d : \u2124\nc d x y z w : \u2115\nzw : SqLe y d x c\nh : SqLe (x + z) c (y + w) d\nl : c * z * z > d * w * w\nhm : d * (y * w) \u2264 c * (x * z)\n\u22a2 d * (y * y) + (d * (y * w) + (d * (y * w) + d * (w * w))) < c * (x * x) + (c * (x * z) + (c * (x * z) + c * (z * z)))", "state_after": "case a\nd\u271d : \u2124\nc d x y z w : \u2115\nh : SqLe (x + z) c (y + w) d\nhm : d * (y * w) \u2264 c * (x * z)\nl : d * (w * w) < c * (z * z)\nzw : d * (y * y) \u2264 c * (x * x)\n\u22a2 d * (y * y) + (d * (y * w) + (d * (y * w) + d * (w * w))) < c * (x * x) + (c * (x * z) + (c * (x * z) + c * (z * z)))"}, {"tactic": "exact\n  lt_of_le_of_lt (add_le_add_right zw _)\n    (add_lt_add_left (add_lt_add_of_le_of_lt hm (add_lt_add_of_le_of_lt hm l)) _)", "annotated_tactic": ["exact\n    <a>lt_of_le_of_lt</a> (<a>add_le_add_right</a> zw _)\n      (<a>add_lt_add_left</a> (<a>add_lt_add_of_le_of_lt</a> hm (<a>add_lt_add_of_le_of_lt</a> hm l)) _)", [{"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}, {"full_name": "add_le_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [66, 15], "def_end_pos": [66, 31]}, {"full_name": "add_lt_add_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [120, 15], "def_end_pos": [120, 30]}, {"full_name": "add_lt_add_of_le_of_lt", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [170, 3], "def_end_pos": [170, 14]}, {"full_name": "add_lt_add_of_le_of_lt", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [170, 3], "def_end_pos": [170, 14]}]], "state_before": "case a\nd\u271d : \u2124\nc d x y z w : \u2115\nh : SqLe (x + z) c (y + w) d\nhm : d * (y * w) \u2264 c * (x * z)\nl : d * (w * w) < c * (z * z)\nzw : d * (y * y) \u2264 c * (x * x)\n\u22a2 d * (y * y) + (d * (y * w) + (d * (y * w) + d * (w * w))) < c * (x * x) + (c * (x * z) + (c * (x * z) + c * (z * z)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/Basic.lean", "full_name": "Vector.mapAccumr_cons", "start": [812, 1], "end": [817, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Logic/Equiv/LocalEquiv.lean", "full_name": "LocalEquiv.trans_refl_restr", "start": [758, 1], "end": [759, 72], "traced_tactics": [{"tactic": "simp [trans_source]", "annotated_tactic": ["simp [<a>trans_source</a>]", [{"full_name": "LocalEquiv.trans_source", "def_path": "Mathlib/Logic/Equiv/LocalEquiv.lean", "def_pos": [707, 9], "def_end_pos": [707, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ne : LocalEquiv \u03b1 \u03b2\ne' : LocalEquiv \u03b2 \u03b3\ns : Set \u03b2\n\u22a2 (LocalEquiv.trans e (LocalEquiv.restr (LocalEquiv.refl \u03b2) s)).source = (LocalEquiv.restr e (\u2191e \u207b\u00b9' s)).source", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Group/Commute/Basic.lean", "full_name": "Commute.div_mul_div_comm", "start": [38, 11], "end": [40, 83], "traced_tactics": [{"tactic": "simp_rw [div_eq_mul_inv, mul_inv_rev, hbd.inv_inv.symm.eq, hbc.mul_mul_mul_comm]", "annotated_tactic": ["simp_rw [<a>div_eq_mul_inv</a>, <a>mul_inv_rev</a>, hbd.inv_inv.symm.eq, hbc.mul_mul_mul_comm]", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}, {"full_name": "mul_inv_rev", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1050, 9], "def_end_pos": [1050, 20]}]], "state_before": "G : Type u_1\ninst\u271d : DivisionMonoid G\na b c d : G\nhbd : Commute b d\nhbc : Commute b\u207b\u00b9 c\n\u22a2 a / b * (c / d) = a * c / (b * d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/HashMap/WF.lean", "full_name": "Std.HashMap.Imp.Buckets.WF.mk'", "start": [40, 1], "end": [44, 81], "traced_tactics": [{"tactic": "refine \u27e8fun _ h => ?_, fun i h => ?_\u27e9", "annotated_tactic": ["refine \u27e8fun _ h => ?_, fun i h => ?_\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nh : 0 < n\n\u22a2 WF (Buckets.mk n)", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : Nat\ninst\u271d\u00b3 : BEq \u03b1\ninst\u271d\u00b2 : Hashable \u03b1\nh\u271d : 0 < n\ninst\u271d\u00b9 : LawfulHashable \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nx\u271d : AssocList \u03b1 \u03b2\nh : x\u271d \u2208 (Buckets.mk n).val.data\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList x\u271d)\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nh\u271d : 0 < n\ni : Nat\nh : i < Array.size (Buckets.mk n).val\n\u22a2 AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size (Buckets.mk n).val) = i)\n    (Buckets.mk n).val[i]"}, {"tactic": "simp [Buckets.mk, empty', mkArray, List.mem_replicate] at h", "annotated_tactic": ["simp [<a>Buckets.mk</a>, <a>empty'</a>, <a>mkArray</a>, <a>List.mem_replicate</a>] at h", [{"full_name": "Std.HashMap.Imp.Buckets.mk", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [28, 5], "def_end_pos": [28, 7]}, {"full_name": "Std.HashMap.Imp.empty'", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [83, 15], "def_end_pos": [83, 21]}, {"full_name": "Array.mkArray", "def_path": "lake-packages/lean4/src/lean/Init/Data/Array/Basic.lean", "def_pos": [20, 5], "def_end_pos": [20, 12]}, {"full_name": "List.mem_replicate", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [325, 9], "def_end_pos": [325, 22]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : Nat\ninst\u271d\u00b3 : BEq \u03b1\ninst\u271d\u00b2 : Hashable \u03b1\nh\u271d : 0 < n\ninst\u271d\u00b9 : LawfulHashable \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nx\u271d : AssocList \u03b1 \u03b2\nh : x\u271d \u2208 (Buckets.mk n).val.data\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList x\u271d)", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : Nat\ninst\u271d\u00b3 : BEq \u03b1\ninst\u271d\u00b2 : Hashable \u03b1\nh\u271d : 0 < n\ninst\u271d\u00b9 : LawfulHashable \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nx\u271d : AssocList \u03b1 \u03b2\nh : \u00acn = 0 \u2227 x\u271d = AssocList.nil\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList x\u271d)"}, {"tactic": "simp [h, List.Pairwise.nil]", "annotated_tactic": ["simp [h, <a>List.Pairwise.nil</a>]", [{"full_name": "List.Pairwise.nil", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1120, 5], "def_end_pos": [1120, 8]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : Nat\ninst\u271d\u00b3 : BEq \u03b1\ninst\u271d\u00b2 : Hashable \u03b1\nh\u271d : 0 < n\ninst\u271d\u00b9 : LawfulHashable \u03b1\ninst\u271d : PartialEquivBEq \u03b1\nx\u271d : AssocList \u03b1 \u03b2\nh : \u00acn = 0 \u2227 x\u271d = AssocList.nil\n\u22a2 List.Pairwise (fun a b => \u00ac(a.fst == b.fst) = true) (AssocList.toList x\u271d)", "state_after": "no goals"}, {"tactic": "simp [Buckets.mk, empty', mkArray, Array.getElem_eq_data_get, AssocList.All]", "annotated_tactic": ["simp [<a>Buckets.mk</a>, <a>empty'</a>, <a>mkArray</a>, <a>Array.getElem_eq_data_get</a>, <a>AssocList.All</a>]", [{"full_name": "Std.HashMap.Imp.Buckets.mk", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [28, 5], "def_end_pos": [28, 7]}, {"full_name": "Std.HashMap.Imp.empty'", "def_path": "lake-packages/std/Std/Data/HashMap/Basic.lean", "def_pos": [83, 15], "def_end_pos": [83, 21]}, {"full_name": "Array.mkArray", "def_path": "lake-packages/lean4/src/lean/Init/Data/Array/Basic.lean", "def_pos": [20, 5], "def_end_pos": [20, 12]}, {"full_name": "Array.getElem_eq_data_get", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [28, 9], "def_end_pos": [28, 28]}, {"full_name": "Std.AssocList.All", "def_path": "lake-packages/std/Std/Data/AssocList.lean", "def_pos": [145, 5], "def_end_pos": [145, 8]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : Nat\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : Hashable \u03b1\nh\u271d : 0 < n\ni : Nat\nh : i < Array.size (Buckets.mk n).val\n\u22a2 AssocList.All (fun k x => USize.toNat (UInt64.toUSize (hash k) % Array.size (Buckets.mk n).val) = i)\n    (Buckets.mk n).val[i]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean", "full_name": "EuclideanGeometry.collinear_iff_eq_or_eq_or_angle_eq_zero_or_angle_eq_pi", "start": [416, 1], "end": [439, 29], "traced_tactics": [{"tactic": "refine' \u27e8fun h => _, fun h => _\u27e9", "annotated_tactic": ["refine' \u27e8fun h => _, fun h => _\u27e9", []], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083} \u2194 p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0", "state_after": "case refine'_1\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Collinear \u211d {p\u2081, p\u2082, p\u2083}\n\u22a2 p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n\ncase refine'_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083}"}, {"tactic": "replace h := h.wbtw_or_wbtw_or_wbtw", "annotated_tactic": ["replace h := h.wbtw_or_wbtw_or_wbtw", []], "state_before": "case refine'_1\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Collinear \u211d {p\u2081, p\u2082, p\u2083}\n\u22a2 p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0", "state_after": "case refine'_1\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081 \u2228 Wbtw \u211d p\u2083 p\u2081 p\u2082\n\u22a2 p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0"}, {"tactic": "by_cases h\u2081\u2082 : p\u2081 = p\u2082", "annotated_tactic": ["by_cases h\u2081\u2082 : p\u2081 = p\u2082", []], "state_before": "case refine'_1\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081 \u2228 Wbtw \u211d p\u2083 p\u2081 p\u2082\n\u22a2 p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0", "state_after": "case pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081 \u2228 Wbtw \u211d p\u2083 p\u2081 p\u2082\nh\u2081\u2082 : p\u2081 = p\u2082\n\u22a2 p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081 \u2228 Wbtw \u211d p\u2083 p\u2081 p\u2082\nh\u2081\u2082 : \u00acp\u2081 = p\u2082\n\u22a2 p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0"}, {"tactic": "by_cases h\u2083\u2082 : p\u2083 = p\u2082", "annotated_tactic": ["by_cases h\u2083\u2082 : p\u2083 = p\u2082", []], "state_before": "case neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081 \u2228 Wbtw \u211d p\u2083 p\u2081 p\u2082\nh\u2081\u2082 : \u00acp\u2081 = p\u2082\n\u22a2 p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0", "state_after": "case pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081 \u2228 Wbtw \u211d p\u2083 p\u2081 p\u2082\nh\u2081\u2082 : \u00acp\u2081 = p\u2082\nh\u2083\u2082 : p\u2083 = p\u2082\n\u22a2 p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081 \u2228 Wbtw \u211d p\u2083 p\u2081 p\u2082\nh\u2081\u2082 : \u00acp\u2081 = p\u2082\nh\u2083\u2082 : \u00acp\u2083 = p\u2082\n\u22a2 p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0"}, {"tactic": "rw [or_iff_right h\u2081\u2082, or_iff_right h\u2083\u2082]", "annotated_tactic": ["rw [<a>or_iff_right</a> h\u2081\u2082, <a>or_iff_right</a> h\u2083\u2082]", [{"full_name": "or_iff_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [303, 9], "def_end_pos": [303, 21]}, {"full_name": "or_iff_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [303, 9], "def_end_pos": [303, 21]}]], "state_before": "case neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081 \u2228 Wbtw \u211d p\u2083 p\u2081 p\u2082\nh\u2081\u2082 : \u00acp\u2081 = p\u2082\nh\u2083\u2082 : \u00acp\u2083 = p\u2082\n\u22a2 p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0", "state_after": "case neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081 \u2228 Wbtw \u211d p\u2083 p\u2081 p\u2082\nh\u2081\u2082 : \u00acp\u2081 = p\u2082\nh\u2083\u2082 : \u00acp\u2083 = p\u2082\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0"}, {"tactic": "rcases h with (h | h | h)", "annotated_tactic": ["rcases h with (h | h | h)", []], "state_before": "case neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081 \u2228 Wbtw \u211d p\u2083 p\u2081 p\u2082\nh\u2081\u2082 : \u00acp\u2081 = p\u2082\nh\u2083\u2082 : \u00acp\u2083 = p\u2082\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0", "state_after": "case neg.inl\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh\u2081\u2082 : \u00acp\u2081 = p\u2082\nh\u2083\u2082 : \u00acp\u2083 = p\u2082\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n\ncase neg.inr.inl\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh\u2081\u2082 : \u00acp\u2081 = p\u2082\nh\u2083\u2082 : \u00acp\u2083 = p\u2082\nh : Wbtw \u211d p\u2082 p\u2083 p\u2081\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n\ncase neg.inr.inr\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh\u2081\u2082 : \u00acp\u2081 = p\u2082\nh\u2083\u2082 : \u00acp\u2083 = p\u2082\nh : Wbtw \u211d p\u2083 p\u2081 p\u2082\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0"}, {"tactic": "exact Or.inl h\u2081\u2082", "annotated_tactic": ["exact <a>Or.inl</a> h\u2081\u2082", [{"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}]], "state_before": "case pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081 \u2228 Wbtw \u211d p\u2083 p\u2081 p\u2082\nh\u2081\u2082 : p\u2081 = p\u2082\n\u22a2 p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0", "state_after": "no goals"}, {"tactic": "exact Or.inr (Or.inl h\u2083\u2082)", "annotated_tactic": ["exact <a>Or.inr</a> (<a>Or.inl</a> h\u2083\u2082)", [{"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}]], "state_before": "case pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081 \u2228 Wbtw \u211d p\u2083 p\u2081 p\u2082\nh\u2081\u2082 : \u00acp\u2081 = p\u2082\nh\u2083\u2082 : p\u2083 = p\u2082\n\u22a2 p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0", "state_after": "no goals"}, {"tactic": "exact Or.inr (angle_eq_pi_iff_sbtw.2 \u27e8h, Ne.symm h\u2081\u2082, Ne.symm h\u2083\u2082\u27e9)", "annotated_tactic": ["exact <a>Or.inr</a> (<a>angle_eq_pi_iff_sbtw</a>.2 \u27e8h, <a>Ne.symm</a> h\u2081\u2082, <a>Ne.symm</a> h\u2083\u2082\u27e9)", [{"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "EuclideanGeometry.angle_eq_pi_iff_sbtw", "def_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean", "def_pos": [307, 9], "def_end_pos": [307, 29]}, {"full_name": "Ne.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [575, 9], "def_end_pos": [575, 16]}, {"full_name": "Ne.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [575, 9], "def_end_pos": [575, 16]}]], "state_before": "case neg.inl\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh\u2081\u2082 : \u00acp\u2081 = p\u2082\nh\u2083\u2082 : \u00acp\u2083 = p\u2082\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0", "state_after": "no goals"}, {"tactic": "exact Or.inl (h.angle\u2083\u2081\u2082_eq_zero_of_ne h\u2083\u2082)", "annotated_tactic": ["exact <a>Or.inl</a> (h.angle\u2083\u2081\u2082_eq_zero_of_ne h\u2083\u2082)", [{"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}]], "state_before": "case neg.inr.inl\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh\u2081\u2082 : \u00acp\u2081 = p\u2082\nh\u2083\u2082 : \u00acp\u2083 = p\u2082\nh : Wbtw \u211d p\u2082 p\u2083 p\u2081\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0", "state_after": "no goals"}, {"tactic": "exact Or.inl (h.angle\u2082\u2083\u2081_eq_zero_of_ne h\u2081\u2082)", "annotated_tactic": ["exact <a>Or.inl</a> (h.angle\u2082\u2083\u2081_eq_zero_of_ne h\u2081\u2082)", [{"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}]], "state_before": "case neg.inr.inr\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh\u2081\u2082 : \u00acp\u2081 = p\u2082\nh\u2083\u2082 : \u00acp\u2083 = p\u2082\nh : Wbtw \u211d p\u2083 p\u2081 p\u2082\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0", "state_after": "no goals"}, {"tactic": "rcases h with (rfl | rfl | h | h)", "annotated_tactic": ["rcases h with (rfl | rfl | h | h)", []], "state_before": "case refine'_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : p\u2081 = p\u2082 \u2228 p\u2083 = p\u2082 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083}", "state_after": "case refine'_2.inl\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2083 : P\n\u22a2 Collinear \u211d {p\u2081, p\u2081, p\u2083}\n\ncase refine'_2.inr.inl\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2083 : P\n\u22a2 Collinear \u211d {p\u2081, p\u2083, p\u2083}\n\ncase refine'_2.inr.inr.inl\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : \u2220 p\u2081 p\u2082 p\u2083 = 0\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083}\n\ncase refine'_2.inr.inr.inr\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083}"}, {"tactic": "simpa using collinear_pair \u211d p\u2081 p\u2083", "annotated_tactic": ["simpa using <a>collinear_pair</a> \u211d p\u2081 p\u2083", [{"full_name": "collinear_pair", "def_path": "Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean", "def_pos": [433, 9], "def_end_pos": [433, 23]}]], "state_before": "case refine'_2.inl\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2083 : P\n\u22a2 Collinear \u211d {p\u2081, p\u2081, p\u2083}", "state_after": "no goals"}, {"tactic": "simpa using collinear_pair \u211d p\u2081 p\u2083", "annotated_tactic": ["simpa using <a>collinear_pair</a> \u211d p\u2081 p\u2083", [{"full_name": "collinear_pair", "def_path": "Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean", "def_pos": [433, 9], "def_end_pos": [433, 23]}]], "state_before": "case refine'_2.inr.inl\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2083 : P\n\u22a2 Collinear \u211d {p\u2081, p\u2083, p\u2083}", "state_after": "no goals"}, {"tactic": "rw [angle_eq_zero_iff_ne_and_wbtw] at h", "annotated_tactic": ["rw [<a>angle_eq_zero_iff_ne_and_wbtw</a>] at h", [{"full_name": "EuclideanGeometry.angle_eq_zero_iff_ne_and_wbtw", "def_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean", "def_pos": [385, 9], "def_end_pos": [385, 38]}]], "state_before": "case refine'_2.inr.inr.inl\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : \u2220 p\u2081 p\u2082 p\u2083 = 0\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083}", "state_after": "case refine'_2.inr.inr.inl\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : p\u2081 \u2260 p\u2082 \u2227 Wbtw \u211d p\u2082 p\u2081 p\u2083 \u2228 p\u2083 \u2260 p\u2082 \u2227 Wbtw \u211d p\u2082 p\u2083 p\u2081\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083}"}, {"tactic": "rcases h with (\u27e8-, h\u27e9 | \u27e8-, h\u27e9)", "annotated_tactic": ["rcases h with (\u27e8-, h\u27e9 | \u27e8-, h\u27e9)", []], "state_before": "case refine'_2.inr.inr.inl\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : p\u2081 \u2260 p\u2082 \u2227 Wbtw \u211d p\u2082 p\u2081 p\u2083 \u2228 p\u2083 \u2260 p\u2082 \u2227 Wbtw \u211d p\u2082 p\u2083 p\u2081\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083}", "state_after": "case refine'_2.inr.inr.inl.inl.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2082 p\u2081 p\u2083\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083}\n\ncase refine'_2.inr.inr.inl.inr.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2082 p\u2083 p\u2081\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083}"}, {"tactic": "rw [Set.insert_comm]", "annotated_tactic": ["rw [<a>Set.insert_comm</a>]", [{"full_name": "Set.insert_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1198, 9], "def_end_pos": [1198, 20]}]], "state_before": "case refine'_2.inr.inr.inl.inl.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2082 p\u2081 p\u2083\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083}", "state_after": "case refine'_2.inr.inr.inl.inl.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2082 p\u2081 p\u2083\n\u22a2 Collinear \u211d {p\u2082, p\u2081, p\u2083}"}, {"tactic": "exact h.collinear", "annotated_tactic": ["exact h.collinear", []], "state_before": "case refine'_2.inr.inr.inl.inl.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2082 p\u2081 p\u2083\n\u22a2 Collinear \u211d {p\u2082, p\u2081, p\u2083}", "state_after": "no goals"}, {"tactic": "rw [Set.insert_comm, Set.pair_comm]", "annotated_tactic": ["rw [<a>Set.insert_comm</a>, <a>Set.pair_comm</a>]", [{"full_name": "Set.insert_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1198, 9], "def_end_pos": [1198, 20]}, {"full_name": "Set.pair_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1402, 9], "def_end_pos": [1402, 18]}]], "state_before": "case refine'_2.inr.inr.inl.inr.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2082 p\u2083 p\u2081\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083}", "state_after": "case refine'_2.inr.inr.inl.inr.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2082 p\u2083 p\u2081\n\u22a2 Collinear \u211d {p\u2082, p\u2083, p\u2081}"}, {"tactic": "exact h.collinear", "annotated_tactic": ["exact h.collinear", []], "state_before": "case refine'_2.inr.inr.inl.inr.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2082 p\u2083 p\u2081\n\u22a2 Collinear \u211d {p\u2082, p\u2083, p\u2081}", "state_after": "no goals"}, {"tactic": "rw [angle_eq_pi_iff_sbtw] at h", "annotated_tactic": ["rw [<a>angle_eq_pi_iff_sbtw</a>] at h", [{"full_name": "EuclideanGeometry.angle_eq_pi_iff_sbtw", "def_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean", "def_pos": [307, 9], "def_end_pos": [307, 29]}]], "state_before": "case refine'_2.inr.inr.inr\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : \u2220 p\u2081 p\u2082 p\u2083 = \u03c0\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083}", "state_after": "case refine'_2.inr.inr.inr\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Sbtw \u211d p\u2081 p\u2082 p\u2083\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083}"}, {"tactic": "exact h.wbtw.collinear", "annotated_tactic": ["exact h.wbtw.collinear", []], "state_before": "case refine'_2.inr.inr.inr\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np\u2081 p\u2082 p\u2083 : P\nh : Sbtw \u211d p\u2081 p\u2082 p\u2083\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Ideal.lean", "full_name": "Order.Cofinal.le_above", "start": [541, 1], "end": [542, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "inv_mem_nhds_one", "start": [642, 1], "end": [643, 31], "traced_tactics": [{"tactic": "rwa [\u2190 nhds_one_symm'] at hS", "annotated_tactic": ["rwa [\u2190 <a>nhds_one_symm'</a>] at hS", [{"full_name": "nhds_one_symm'", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [636, 9], "def_end_pos": [636, 23]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nG : Type w\nH : Type x\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : TopologicalSpace \u03b1\nf : \u03b1 \u2192 G\ns : Set \u03b1\nx : \u03b1\nS : Set G\nhS : S \u2208 \ud835\udcdd 1\n\u22a2 S\u207b\u00b9 \u2208 \ud835\udcdd 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Subsemigroup/Operations.lean", "full_name": "Subsemigroup.map_strictMono_of_injective", "start": [437, 1], "end": [438, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/ContDiff.lean", "full_name": "HasFTaylorSeriesUpToOn.prod", "start": [503, 1], "end": [513, 82], "traced_tactics": [{"tactic": "set L := fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun _ : Fin m => E) F G", "annotated_tactic": ["set L := fun m => <a>ContinuousMultilinearMap.prodL</a> \ud835\udd5c (fun _ : <a>Fin</a> m => E) F G", [{"full_name": "ContinuousMultilinearMap.prodL", "def_path": "Mathlib/Analysis/NormedSpace/Multilinear.lean", "def_pos": [550, 5], "def_end_pos": [550, 10]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\n\u22a2 HasFTaylorSeriesUpToOn n (fun y => (f y, g y)) (fun y k => ContinuousMultilinearMap.prod (p y k) (q y k)) s", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G\n\u22a2 HasFTaylorSeriesUpToOn n (fun y => (f y, g y)) (fun y k => ContinuousMultilinearMap.prod (p y k) (q y k)) s"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G\n\u22a2 HasFTaylorSeriesUpToOn n (fun y => (f y, g y)) (fun y k => ContinuousMultilinearMap.prod (p y k) (q y k)) s", "state_after": "case zero_eq\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (ContinuousMultilinearMap.prod (p x 0) (q x 0)) = (f x, g x)\n\ncase fderivWithin\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G\n\u22a2 \u2200 (m : \u2115),\n    \u2191m < n \u2192\n      \u2200 (x : E),\n        x \u2208 s \u2192\n          HasFDerivWithinAt (fun x => ContinuousMultilinearMap.prod (p x m) (q x m))\n            (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.prod (p x (Nat.succ m)) (q x (Nat.succ m)))) s\n            x\n\ncase cont\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => ContinuousMultilinearMap.prod (p x m) (q x m)) s"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case zero_eq\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 ContinuousMultilinearMap.uncurry0 (ContinuousMultilinearMap.prod (p x 0) (q x 0)) = (f x, g x)", "state_after": "case zero_eq\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G\nx : E\nhx : x \u2208 s\n\u22a2 ContinuousMultilinearMap.uncurry0 (ContinuousMultilinearMap.prod (p x 0) (q x 0)) = (f x, g x)"}, {"tactic": "rw [\u2190 hf.zero_eq x hx, \u2190 hg.zero_eq x hx]", "annotated_tactic": ["rw [\u2190 hf.zero_eq x hx, \u2190 hg.zero_eq x hx]", []], "state_before": "case zero_eq\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G\nx : E\nhx : x \u2208 s\n\u22a2 ContinuousMultilinearMap.uncurry0 (ContinuousMultilinearMap.prod (p x 0) (q x 0)) = (f x, g x)", "state_after": "case zero_eq\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G\nx : E\nhx : x \u2208 s\n\u22a2 ContinuousMultilinearMap.uncurry0 (ContinuousMultilinearMap.prod (p x 0) (q x 0)) =\n    (ContinuousMultilinearMap.uncurry0 (p x 0), ContinuousMultilinearMap.uncurry0 (q x 0))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case zero_eq\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G\nx : E\nhx : x \u2208 s\n\u22a2 ContinuousMultilinearMap.uncurry0 (ContinuousMultilinearMap.prod (p x 0) (q x 0)) =\n    (ContinuousMultilinearMap.uncurry0 (p x 0), ContinuousMultilinearMap.uncurry0 (q x 0))", "state_after": "no goals"}, {"tactic": "intro m hm x hx", "annotated_tactic": ["intro m hm x hx", []], "state_before": "case fderivWithin\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G\n\u22a2 \u2200 (m : \u2115),\n    \u2191m < n \u2192\n      \u2200 (x : E),\n        x \u2208 s \u2192\n          HasFDerivWithinAt (fun x => ContinuousMultilinearMap.prod (p x m) (q x m))\n            (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.prod (p x (Nat.succ m)) (q x (Nat.succ m)))) s\n            x", "state_after": "case fderivWithin\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\n\u22a2 HasFDerivWithinAt (fun x => ContinuousMultilinearMap.prod (p x m) (q x m))\n    (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.prod (p x (Nat.succ m)) (q x (Nat.succ m)))) s x"}, {"tactic": "convert (L m).hasFDerivAt.comp_hasFDerivWithinAt x\n    ((hf.fderivWithin m hm x hx).prod (hg.fderivWithin m hm x hx))", "annotated_tactic": ["convert (L m).hasFDerivAt.comp_hasFDerivWithinAt x\n        ((hf.fderivWithin m hm x hx).<a>prod</a> (hg.fderivWithin m hm x hx))", [{"full_name": "HasFDerivWithinAt.prod", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Prod.lean", "def_pos": [73, 16], "def_end_pos": [73, 38]}]], "state_before": "case fderivWithin\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\n\u22a2 HasFDerivWithinAt (fun x => ContinuousMultilinearMap.prod (p x m) (q x m))\n    (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.prod (p x (Nat.succ m)) (q x (Nat.succ m)))) s x", "state_after": "no goals"}, {"tactic": "intro m hm", "annotated_tactic": ["intro m hm", []], "state_before": "case cont\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => ContinuousMultilinearMap.prod (p x m) (q x m)) s", "state_after": "case cont\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 ContinuousOn (fun x => ContinuousMultilinearMap.prod (p x m) (q x m)) s"}, {"tactic": "exact (L m).continuous.comp_continuousOn ((hf.rst.imnt m hm).prod (hg.cont m hm))", "annotated_tactic": ["exact (L m).continuous.comp_continuousOn ((hf.rst.imnt m hm).<a>prod</a> (hg.cont m hm))", [{"full_name": "ContinuousOn.prod", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [1113, 9], "def_end_pos": [1113, 26]}]], "state_before": "case cont\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng\u271d : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nhf : HasFTaylorSeriesUpToOn n f p s\ng : E \u2192 G\nq : E \u2192 FormalMultilinearSeries \ud835\udd5c E G\nhg : HasFTaylorSeriesUpToOn n g q s\nL : (m : \u2115) \u2192\n  ContinuousMultilinearMap \ud835\udd5c (fun x => E) F \u00d7 ContinuousMultilinearMap \ud835\udd5c (fun x => E) G \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun x => E) (F \u00d7 G) :=\n  fun m => ContinuousMultilinearMap.prodL \ud835\udd5c (fun x => E) F G\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 ContinuousOn (fun x => ContinuousMultilinearMap.prod (p x m) (q x m)) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "full_name": "csSup_Iic", "start": [774, 1], "end": [775, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "full_name": "Monoid.coe_one", "start": [1318, 1], "end": [1318, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.injective_valMinAbs", "start": [980, 1], "end": [981, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/MvPolynomial/Symmetric.lean", "full_name": "MvPolynomial.support_esymm''", "start": [221, 1], "end": [243, 62], "traced_tactics": [{"tactic": "rw [esymm_eq_sum_monomial]", "annotated_tactic": ["rw [<a>esymm_eq_sum_monomial</a>]", [{"full_name": "MvPolynomial.esymm_eq_sum_monomial", "def_path": "Mathlib/RingTheory/MvPolynomial/Symmetric.lean", "def_pos": [189, 9], "def_end_pos": [189, 30]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\n\u22a2 support (esymm \u03c3 R n) = Finset.biUnion (powersetCard n univ) fun t => (fun\u2080 | \u2211 i in t, fun\u2080 | i => 1 => 1).support", "state_after": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\n\u22a2 support (\u2211 t in powersetCard n univ, \u2191(monomial (\u2211 i in t, fun\u2080 | i => 1)) 1) =\n    Finset.biUnion (powersetCard n univ) fun t => (fun\u2080 | \u2211 i in t, fun\u2080 | i => 1 => 1).support"}, {"tactic": "simp only [\u2190 single_eq_monomial]", "annotated_tactic": ["simp only [\u2190 <a>single_eq_monomial</a>]", [{"full_name": "MvPolynomial.single_eq_monomial", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [171, 9], "def_end_pos": [171, 27]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\n\u22a2 support (\u2211 t in powersetCard n univ, \u2191(monomial (\u2211 i in t, fun\u2080 | i => 1)) 1) =\n    Finset.biUnion (powersetCard n univ) fun t => (fun\u2080 | \u2211 i in t, fun\u2080 | i => 1 => 1).support", "state_after": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\n\u22a2 support (\u2211 x in powersetCard n univ, fun\u2080 | \u2211 i in x, fun\u2080 | i => 1 => 1) =\n    Finset.biUnion (powersetCard n univ) fun t => (fun\u2080 | \u2211 i in t, fun\u2080 | i => 1 => 1).support"}, {"tactic": "refine' Finsupp.support_sum_eq_biUnion (powersetCard n (univ : Finset \u03c3)) _", "annotated_tactic": ["refine' <a>Finsupp.support_sum_eq_biUnion</a> (<a>powersetCard</a> n (<a>univ</a> : <a>Finset</a> \u03c3)) _", [{"full_name": "Finsupp.support_sum_eq_biUnion", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [571, 9], "def_end_pos": [571, 31]}, {"full_name": "Finset.powersetCard", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [197, 5], "def_end_pos": [197, 17]}, {"full_name": "Finset.univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [67, 5], "def_end_pos": [67, 9]}, {"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [138, 11], "def_end_pos": [138, 17]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\n\u22a2 support (\u2211 x in powersetCard n univ, fun\u2080 | \u2211 i in x, fun\u2080 | i => 1 => 1) =\n    Finset.biUnion (powersetCard n univ) fun t => (fun\u2080 | \u2211 i in t, fun\u2080 | i => 1 => 1).support", "state_after": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\n\u22a2 \u2200 (i\u2081 i\u2082 : Finset \u03c3),\n    i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | \u2211 i in i\u2081, fun\u2080 | i => 1 => 1).support (fun\u2080 | \u2211 i in i\u2082, fun\u2080 | i => 1 => 1).support"}, {"tactic": "intro s t hst", "annotated_tactic": ["intro s t hst", []], "state_before": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\n\u22a2 \u2200 (i\u2081 i\u2082 : Finset \u03c3),\n    i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | \u2211 i in i\u2081, fun\u2080 | i => 1 => 1).support (fun\u2080 | \u2211 i in i\u2082, fun\u2080 | i => 1 => 1).support", "state_after": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\n\u22a2 Disjoint (fun\u2080 | \u2211 i in s, fun\u2080 | i => 1 => 1).support (fun\u2080 | \u2211 i in t, fun\u2080 | i => 1 => 1).support"}, {"tactic": "rw [Finset.disjoint_left, Finsupp.support_single_ne_zero _ one_ne_zero]", "annotated_tactic": ["rw [<a>Finset.disjoint_left</a>, <a>Finsupp.support_single_ne_zero</a> _ <a>one_ne_zero</a>]", [{"full_name": "Finset.disjoint_left", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [939, 9], "def_end_pos": [939, 22]}, {"full_name": "Finsupp.support_single_ne_zero", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [358, 9], "def_end_pos": [358, 31]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\n\u22a2 Disjoint (fun\u2080 | \u2211 i in s, fun\u2080 | i => 1 => 1).support (fun\u2080 | \u2211 i in t, fun\u2080 | i => 1 => 1).support", "state_after": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\n\u22a2 \u2200 \u2983a : \u03c3 \u2192\u2080 \u2115\u2984, a \u2208 {\u2211 i in s, fun\u2080 | i => 1} \u2192 \u00aca \u2208 (fun\u2080 | \u2211 i in t, fun\u2080 | i => 1 => 1).support"}, {"tactic": "rw [Finsupp.support_single_ne_zero _ one_ne_zero]", "annotated_tactic": ["rw [<a>Finsupp.support_single_ne_zero</a> _ <a>one_ne_zero</a>]", [{"full_name": "Finsupp.support_single_ne_zero", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [358, 9], "def_end_pos": [358, 31]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\n\u22a2 \u2200 \u2983a : \u03c3 \u2192\u2080 \u2115\u2984, a \u2208 {\u2211 i in s, fun\u2080 | i => 1} \u2192 \u00aca \u2208 (fun\u2080 | \u2211 i in t, fun\u2080 | i => 1 => 1).support", "state_after": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\n\u22a2 \u2200 \u2983a : \u03c3 \u2192\u2080 \u2115\u2984, a \u2208 {\u2211 i in s, fun\u2080 | i => 1} \u2192 \u00aca \u2208 {\u2211 i in t, fun\u2080 | i => 1}"}, {"tactic": "simp only [one_ne_zero, mem_singleton, Finsupp.mem_support_iff]", "annotated_tactic": ["simp only [<a>one_ne_zero</a>, <a>mem_singleton</a>, <a>Finsupp.mem_support_iff</a>]", [{"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}, {"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 22]}, {"full_name": "Finsupp.mem_support_iff", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [186, 9], "def_end_pos": [186, 24]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\n\u22a2 \u2200 \u2983a : \u03c3 \u2192\u2080 \u2115\u2984, a \u2208 {\u2211 i in s, fun\u2080 | i => 1} \u2192 \u00aca \u2208 {\u2211 i in t, fun\u2080 | i => 1}", "state_after": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\n\u22a2 \u2200 \u2983a : \u03c3 \u2192\u2080 \u2115\u2984, (a = \u2211 i in s, fun\u2080 | i => 1) \u2192 \u00aca = \u2211 i in t, fun\u2080 | i => 1"}, {"tactic": "rintro a h rfl", "annotated_tactic": ["rintro a h rfl", []], "state_before": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\n\u22a2 \u2200 \u2983a : \u03c3 \u2192\u2080 \u2115\u2984, (a = \u2211 i in s, fun\u2080 | i => 1) \u2192 \u00aca = \u2211 i in t, fun\u2080 | i => 1", "state_after": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\n\u22a2 False"}, {"tactic": "have := congr_arg Finsupp.support h", "annotated_tactic": ["have := <a>congr_arg</a> <a>Finsupp.support</a> h", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "Finsupp.support", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [96, 3], "def_end_pos": [96, 10]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\n\u22a2 False", "state_after": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 False"}, {"tactic": "rw [Finsupp.support_sum_eq_biUnion, Finsupp.support_sum_eq_biUnion] at this", "annotated_tactic": ["rw [<a>Finsupp.support_sum_eq_biUnion</a>, <a>Finsupp.support_sum_eq_biUnion</a>] at this", [{"full_name": "Finsupp.support_sum_eq_biUnion", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [571, 9], "def_end_pos": [571, 31]}, {"full_name": "Finsupp.support_sum_eq_biUnion", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [571, 9], "def_end_pos": [571, 31]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 False", "state_after": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = Finset.biUnion s fun i => (fun\u2080 | i => 1).support\n\u22a2 False\n\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | i\u2081 => 1).support (fun\u2080 | i\u2082 => 1).support\n\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | i\u2081 => 1).support (fun\u2080 | i\u2082 => 1).support"}, {"tactic": "have hsingle : \u2200 s : Finset \u03c3, \u2200 x : \u03c3, x \u2208 s \u2192 (Finsupp.single x 1).support = {x} := by\n  intros _ x _\n  rw [Finsupp.support_single_ne_zero x one_ne_zero]", "annotated_tactic": ["have hsingle : \u2200 s : <a>Finset</a> \u03c3, \u2200 x : \u03c3, x \u2208 s \u2192 (<a>Finsupp.single</a> x 1).<a>support</a> = {x} := by\n    intros _ x _\n    rw [<a>Finsupp.support_single_ne_zero</a> x <a>one_ne_zero</a>]", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [138, 11], "def_end_pos": [138, 17]}, {"full_name": "Finsupp.single", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [289, 5], "def_end_pos": [289, 11]}, {"full_name": "Finsupp.support", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [96, 3], "def_end_pos": [96, 10]}, {"full_name": "Finsupp.support_single_ne_zero", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [358, 9], "def_end_pos": [358, 31]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = Finset.biUnion s fun i => (fun\u2080 | i => 1).support\n\u22a2 False\n\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | i\u2081 => 1).support (fun\u2080 | i\u2082 => 1).support\n\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | i\u2081 => 1).support (fun\u2080 | i\u2082 => 1).support", "state_after": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = Finset.biUnion s fun i => (fun\u2080 | i => 1).support\nhsingle : \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (fun\u2080 | x => 1).support = {x}\n\u22a2 False\n\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | i\u2081 => 1).support (fun\u2080 | i\u2082 => 1).support\n\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | i\u2081 => 1).support (fun\u2080 | i\u2082 => 1).support"}, {"tactic": "have hs := biUnion_congr (of_eq_true (eq_self s)) (hsingle s)", "annotated_tactic": ["have hs := <a>biUnion_congr</a> (<a>of_eq_true</a> (<a>eq_self</a> s)) (hsingle s)", [{"full_name": "Finset.biUnion_congr", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3630, 9], "def_end_pos": [3630, 22]}, {"full_name": "of_eq_true", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [12, 9], "def_end_pos": [12, 19]}, {"full_name": "eq_self", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [28, 17], "def_end_pos": [28, 24]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = Finset.biUnion s fun i => (fun\u2080 | i => 1).support\nhsingle : \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (fun\u2080 | x => 1).support = {x}\n\u22a2 False\n\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | i\u2081 => 1).support (fun\u2080 | i\u2082 => 1).support\n\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | i\u2081 => 1).support (fun\u2080 | i\u2082 => 1).support", "state_after": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = Finset.biUnion s fun i => (fun\u2080 | i => 1).support\nhsingle : \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (fun\u2080 | x => 1).support = {x}\nhs : (Finset.biUnion s fun a => (fun\u2080 | a => 1).support) = Finset.biUnion s fun a => {a}\n\u22a2 False\n\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | i\u2081 => 1).support (fun\u2080 | i\u2082 => 1).support\n\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | i\u2081 => 1).support (fun\u2080 | i\u2082 => 1).support"}, {"tactic": "have ht := biUnion_congr (of_eq_true (eq_self t)) (hsingle t)", "annotated_tactic": ["have ht := <a>biUnion_congr</a> (<a>of_eq_true</a> (<a>eq_self</a> t)) (hsingle t)", [{"full_name": "Finset.biUnion_congr", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3630, 9], "def_end_pos": [3630, 22]}, {"full_name": "of_eq_true", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [12, 9], "def_end_pos": [12, 19]}, {"full_name": "eq_self", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [28, 17], "def_end_pos": [28, 24]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = Finset.biUnion s fun i => (fun\u2080 | i => 1).support\nhsingle : \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (fun\u2080 | x => 1).support = {x}\nhs : (Finset.biUnion s fun a => (fun\u2080 | a => 1).support) = Finset.biUnion s fun a => {a}\n\u22a2 False\n\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | i\u2081 => 1).support (fun\u2080 | i\u2082 => 1).support\n\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | i\u2081 => 1).support (fun\u2080 | i\u2082 => 1).support", "state_after": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = Finset.biUnion s fun i => (fun\u2080 | i => 1).support\nhsingle : \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (fun\u2080 | x => 1).support = {x}\nhs : (Finset.biUnion s fun a => (fun\u2080 | a => 1).support) = Finset.biUnion s fun a => {a}\nht : (Finset.biUnion t fun a => (fun\u2080 | a => 1).support) = Finset.biUnion t fun a => {a}\n\u22a2 False\n\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | i\u2081 => 1).support (fun\u2080 | i\u2082 => 1).support\n\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | i\u2081 => 1).support (fun\u2080 | i\u2082 => 1).support"}, {"tactic": "rw [hs, ht] at this", "annotated_tactic": ["rw [hs, ht] at this", []], "state_before": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = Finset.biUnion s fun i => (fun\u2080 | i => 1).support\nhsingle : \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (fun\u2080 | x => 1).support = {x}\nhs : (Finset.biUnion s fun a => (fun\u2080 | a => 1).support) = Finset.biUnion s fun a => {a}\nht : (Finset.biUnion t fun a => (fun\u2080 | a => 1).support) = Finset.biUnion t fun a => {a}\n\u22a2 False\n\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | i\u2081 => 1).support (fun\u2080 | i\u2082 => 1).support\n\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | i\u2081 => 1).support (fun\u2080 | i\u2082 => 1).support", "state_after": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun a => {a}) = Finset.biUnion s fun a => {a}\nhsingle : \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (fun\u2080 | x => 1).support = {x}\nhs : (Finset.biUnion s fun a => (fun\u2080 | a => 1).support) = Finset.biUnion s fun a => {a}\nht : (Finset.biUnion t fun a => (fun\u2080 | a => 1).support) = Finset.biUnion t fun a => {a}\n\u22a2 False\n\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | i\u2081 => 1).support (fun\u2080 | i\u2082 => 1).support\n\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | i\u2081 => 1).support (fun\u2080 | i\u2082 => 1).support"}, {"tactic": "all_goals intro x y; simp [Finsupp.support_single_disjoint]", "annotated_tactic": ["all_goals intro x y; simp [<a>Finsupp.support_single_disjoint</a>]", [{"full_name": "Finsupp.support_single_disjoint", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [432, 9], "def_end_pos": [432, 32]}]], "state_before": "case h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | i\u2081 => 1).support (fun\u2080 | i\u2082 => 1).support\n\ncase h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | i\u2081 => 1).support (fun\u2080 | i\u2082 => 1).support", "state_after": "no goals"}, {"tactic": "intros _ x _", "annotated_tactic": ["intros _ x _", []], "state_before": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = Finset.biUnion s fun i => (fun\u2080 | i => 1).support\n\u22a2 \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (fun\u2080 | x => 1).support = {x}", "state_after": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = Finset.biUnion s fun i => (fun\u2080 | i => 1).support\ns\u271d : Finset \u03c3\nx : \u03c3\na\u271d : x \u2208 s\u271d\n\u22a2 (fun\u2080 | x => 1).support = {x}"}, {"tactic": "rw [Finsupp.support_single_ne_zero x one_ne_zero]", "annotated_tactic": ["rw [<a>Finsupp.support_single_ne_zero</a> x <a>one_ne_zero</a>]", [{"full_name": "Finsupp.support_single_ne_zero", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [358, 9], "def_end_pos": [358, 31]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = Finset.biUnion s fun i => (fun\u2080 | i => 1).support\ns\u271d : Finset \u03c3\nx : \u03c3\na\u271d : x \u2208 s\u271d\n\u22a2 (fun\u2080 | x => 1).support = {x}", "state_after": "no goals"}, {"tactic": "simp only [biUnion_singleton_eq_self] at this", "annotated_tactic": ["simp only [<a>biUnion_singleton_eq_self</a>] at this", [{"full_name": "Finset.biUnion_singleton_eq_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3706, 9], "def_end_pos": [3706, 34]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\nthis : (Finset.biUnion t fun a => {a}) = Finset.biUnion s fun a => {a}\nhsingle : \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (fun\u2080 | x => 1).support = {x}\nhs : (Finset.biUnion s fun a => (fun\u2080 | a => 1).support) = Finset.biUnion s fun a => {a}\nht : (Finset.biUnion t fun a => (fun\u2080 | a => 1).support) = Finset.biUnion t fun a => {a}\n\u22a2 False", "state_after": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\nhsingle : \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (fun\u2080 | x => 1).support = {x}\nhs : (Finset.biUnion s fun a => (fun\u2080 | a => 1).support) = Finset.biUnion s fun a => {a}\nht : (Finset.biUnion t fun a => (fun\u2080 | a => 1).support) = Finset.biUnion t fun a => {a}\nthis : t = s\n\u22a2 False"}, {"tactic": "exact absurd this hst.symm", "annotated_tactic": ["exact <a>absurd</a> this hst.symm", [{"full_name": "absurd", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [233, 21], "def_end_pos": [233, 27]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis\u271d\u00b9 : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nthis\u271d : (Finset.biUnion t fun i => (fun\u2080 | i => 1).support) = (\u2211 i in s, fun\u2080 | i => 1).support\nhsingle : \u2200 (s : Finset \u03c3) (x : \u03c3), x \u2208 s \u2192 (fun\u2080 | x => 1).support = {x}\nhs : (Finset.biUnion s fun a => (fun\u2080 | a => 1).support) = Finset.biUnion s fun a => {a}\nht : (Finset.biUnion t fun a => (fun\u2080 | a => 1).support) = Finset.biUnion t fun a => {a}\nthis : t = s\n\u22a2 False", "state_after": "no goals"}, {"tactic": "intro x y", "annotated_tactic": ["intro x y", []], "state_before": "case h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\n\u22a2 \u2200 (i\u2081 i\u2082 : \u03c3), i\u2081 \u2260 i\u2082 \u2192 Disjoint (fun\u2080 | i\u2081 => 1).support (fun\u2080 | i\u2082 => 1).support", "state_after": "case h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nx y : \u03c3\n\u22a2 x \u2260 y \u2192 Disjoint (fun\u2080 | x => 1).support (fun\u2080 | y => 1).support"}, {"tactic": "simp [Finsupp.support_single_disjoint]", "annotated_tactic": ["simp [<a>Finsupp.support_single_disjoint</a>]", [{"full_name": "Finsupp.support_single_disjoint", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [432, 9], "def_end_pos": [432, 32]}]], "state_before": "case h\n\u03c3 : Type u_1\nR : Type u_2\n\u03c4 : Type u_3\nS : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : CommSemiring S\ninst\u271d\u00b3 : Fintype \u03c3\ninst\u271d\u00b2 : Fintype \u03c4\nn : \u2115\ninst\u271d\u00b9 : DecidableEq \u03c3\ninst\u271d : Nontrivial R\ns t : Finset \u03c3\nhst : s \u2260 t\nh : (\u2211 i in t, fun\u2080 | i => 1) = \u2211 i in s, fun\u2080 | i => 1\nthis : (\u2211 i in t, fun\u2080 | i => 1).support = (\u2211 i in s, fun\u2080 | i => 1).support\nx y : \u03c3\n\u22a2 x \u2260 y \u2192 Disjoint (fun\u2080 | x => 1).support (fun\u2080 | y => 1).support", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Image.lean", "full_name": "Finset.image_singleton", "start": [516, 1], "end": [517, 99], "traced_tactics": [{"tactic": "simpa only [mem_image, exists_prop, mem_singleton, exists_eq_left] using eq_comm", "annotated_tactic": ["simpa only [<a>mem_image</a>, <a>exists_prop</a>, <a>mem_singleton</a>, <a>exists_eq_left</a>] using <a>eq_comm</a>", [{"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 22]}, {"full_name": "exists_eq_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [459, 17], "def_end_pos": [459, 31]}, {"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\na\u271d : \u03b1\nb c : \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nx : \u03b2\n\u22a2 x \u2208 image f {a} \u2194 x \u2208 {f a}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.integral_nonneg_of_ae", "start": [1360, 1], "end": [1361, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "measurable_mem", "start": [1141, 9], "end": [1141, 98], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/DedekindDomain/AdicValuation.lean", "full_name": "IsDedekindDomain.HeightOneSpectrum.IntValuation.le_max_iff_min_le", "start": [178, 1], "end": [183, 18], "traced_tactics": [{"tactic": "rw [le_max_iff, ofAdd_le, ofAdd_le, neg_le_neg_iff, neg_le_neg_iff, Int.ofNat_le, Int.ofNat_le,\n  \u2190 min_le_iff]", "annotated_tactic": ["rw [<a>le_max_iff</a>, <a>ofAdd_le</a>, <a>ofAdd_le</a>, <a>neg_le_neg_iff</a>, <a>neg_le_neg_iff</a>, <a>Int.ofNat_le</a>, <a>Int.ofNat_le</a>,\n    \u2190 <a>min_le_iff</a>]", [{"full_name": "le_max_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [38, 9], "def_end_pos": [38, 19]}, {"full_name": "Multiplicative.ofAdd_le", "def_path": "Mathlib/Algebra/Order/Monoid/TypeTags.lean", "def_pos": [149, 9], "def_end_pos": [149, 17]}, {"full_name": "Multiplicative.ofAdd_le", "def_path": "Mathlib/Algebra/Order/Monoid/TypeTags.lean", "def_pos": [149, 9], "def_end_pos": [149, 17]}, {"full_name": "neg_le_neg_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [342, 3], "def_end_pos": [342, 14]}, {"full_name": "neg_le_neg_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [342, 3], "def_end_pos": [342, 14]}, {"full_name": "Int.ofNat_le", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [574, 28], "def_end_pos": [574, 36]}, {"full_name": "Int.ofNat_le", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [574, 28], "def_end_pos": [574, 36]}, {"full_name": "min_le_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [43, 9], "def_end_pos": [43, 19]}]], "state_before": "R : Type u_1\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\na b c : \u2115\n\u22a2 \u2191ofAdd (-\u2191c) \u2264 max (\u2191ofAdd (-\u2191a)) (\u2191ofAdd (-\u2191b)) \u2194 min a b \u2264 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/PythagoreanTriples.lean", "full_name": "coprime_sq_sub_sq_sum_of_odd_odd", "start": [402, 9], "end": [432, 9], "traced_tactics": [{"tactic": "cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hm) with m0 hm2", "annotated_tactic": ["cases' <a>exists_eq_mul_left_of_dvd</a> (<a>Int.dvd_sub_of_emod_eq</a> hm) with m0 hm2", [{"full_name": "exists_eq_mul_left_of_dvd", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [154, 9], "def_end_pos": [154, 34]}, {"full_name": "Int.dvd_sub_of_emod_eq", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [695, 9], "def_end_pos": [695, 27]}]], "state_before": "m n : \u2124\nh : Int.gcd m n = 1\nhm : m % 2 = 1\nhn : n % 2 = 1\n\u22a2 2 \u2223 m ^ 2 + n ^ 2 \u2227\n    2 \u2223 m ^ 2 - n ^ 2 \u2227 (m ^ 2 - n ^ 2) / 2 % 2 = 0 \u2227 Int.gcd ((m ^ 2 - n ^ 2) / 2) ((m ^ 2 + n ^ 2) / 2) = 1", "state_after": "case intro\nm n : \u2124\nh : Int.gcd m n = 1\nhm : m % 2 = 1\nhn : n % 2 = 1\nm0 : \u2124\nhm2 : m - 1 = m0 * 2\n\u22a2 2 \u2223 m ^ 2 + n ^ 2 \u2227\n    2 \u2223 m ^ 2 - n ^ 2 \u2227 (m ^ 2 - n ^ 2) / 2 % 2 = 0 \u2227 Int.gcd ((m ^ 2 - n ^ 2) / 2) ((m ^ 2 + n ^ 2) / 2) = 1"}, {"tactic": "cases' exists_eq_mul_left_of_dvd (Int.dvd_sub_of_emod_eq hn) with n0 hn2", "annotated_tactic": ["cases' <a>exists_eq_mul_left_of_dvd</a> (<a>Int.dvd_sub_of_emod_eq</a> hn) with n0 hn2", [{"full_name": "exists_eq_mul_left_of_dvd", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [154, 9], "def_end_pos": [154, 34]}, {"full_name": "Int.dvd_sub_of_emod_eq", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [695, 9], "def_end_pos": [695, 27]}]], "state_before": "case intro\nm n : \u2124\nh : Int.gcd m n = 1\nhm : m % 2 = 1\nhn : n % 2 = 1\nm0 : \u2124\nhm2 : m - 1 = m0 * 2\n\u22a2 2 \u2223 m ^ 2 + n ^ 2 \u2227\n    2 \u2223 m ^ 2 - n ^ 2 \u2227 (m ^ 2 - n ^ 2) / 2 % 2 = 0 \u2227 Int.gcd ((m ^ 2 - n ^ 2) / 2) ((m ^ 2 + n ^ 2) / 2) = 1", "state_after": "case intro.intro\nm n : \u2124\nh : Int.gcd m n = 1\nhm : m % 2 = 1\nhn : n % 2 = 1\nm0 : \u2124\nhm2 : m - 1 = m0 * 2\nn0 : \u2124\nhn2 : n - 1 = n0 * 2\n\u22a2 2 \u2223 m ^ 2 + n ^ 2 \u2227\n    2 \u2223 m ^ 2 - n ^ 2 \u2227 (m ^ 2 - n ^ 2) / 2 % 2 = 0 \u2227 Int.gcd ((m ^ 2 - n ^ 2) / 2) ((m ^ 2 + n ^ 2) / 2) = 1"}, {"tactic": "rw [sub_eq_iff_eq_add] at hm2 hn2", "annotated_tactic": ["rw [<a>sub_eq_iff_eq_add</a>] at hm2 hn2", [{"full_name": "sub_eq_iff_eq_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [829, 3], "def_end_pos": [829, 14]}]], "state_before": "case intro.intro\nm n : \u2124\nh : Int.gcd m n = 1\nhm : m % 2 = 1\nhn : n % 2 = 1\nm0 : \u2124\nhm2 : m - 1 = m0 * 2\nn0 : \u2124\nhn2 : n - 1 = n0 * 2\n\u22a2 2 \u2223 m ^ 2 + n ^ 2 \u2227\n    2 \u2223 m ^ 2 - n ^ 2 \u2227 (m ^ 2 - n ^ 2) / 2 % 2 = 0 \u2227 Int.gcd ((m ^ 2 - n ^ 2) / 2) ((m ^ 2 + n ^ 2) / 2) = 1", "state_after": "case intro.intro\nm n : \u2124\nh : Int.gcd m n = 1\nhm : m % 2 = 1\nhn : n % 2 = 1\nm0 : \u2124\nhm2 : m = m0 * 2 + 1\nn0 : \u2124\nhn2 : n = n0 * 2 + 1\n\u22a2 2 \u2223 m ^ 2 + n ^ 2 \u2227\n    2 \u2223 m ^ 2 - n ^ 2 \u2227 (m ^ 2 - n ^ 2) / 2 % 2 = 0 \u2227 Int.gcd ((m ^ 2 - n ^ 2) / 2) ((m ^ 2 + n ^ 2) / 2) = 1"}, {"tactic": "subst m", "annotated_tactic": ["subst m", []], "state_before": "case intro.intro\nm n : \u2124\nh : Int.gcd m n = 1\nhm : m % 2 = 1\nhn : n % 2 = 1\nm0 : \u2124\nhm2 : m = m0 * 2 + 1\nn0 : \u2124\nhn2 : n = n0 * 2 + 1\n\u22a2 2 \u2223 m ^ 2 + n ^ 2 \u2227\n    2 \u2223 m ^ 2 - n ^ 2 \u2227 (m ^ 2 - n ^ 2) / 2 % 2 = 0 \u2227 Int.gcd ((m ^ 2 - n ^ 2) / 2) ((m ^ 2 + n ^ 2) / 2) = 1", "state_after": "case intro.intro\nn : \u2124\nhn : n % 2 = 1\nm0 n0 : \u2124\nhn2 : n = n0 * 2 + 1\nh : Int.gcd (m0 * 2 + 1) n = 1\nhm : (m0 * 2 + 1) % 2 = 1\n\u22a2 2 \u2223 (m0 * 2 + 1) ^ 2 + n ^ 2 \u2227\n    2 \u2223 (m0 * 2 + 1) ^ 2 - n ^ 2 \u2227\n      ((m0 * 2 + 1) ^ 2 - n ^ 2) / 2 % 2 = 0 \u2227\n        Int.gcd (((m0 * 2 + 1) ^ 2 - n ^ 2) / 2) (((m0 * 2 + 1) ^ 2 + n ^ 2) / 2) = 1"}, {"tactic": "subst n", "annotated_tactic": ["subst n", []], "state_before": "case intro.intro\nn : \u2124\nhn : n % 2 = 1\nm0 n0 : \u2124\nhn2 : n = n0 * 2 + 1\nh : Int.gcd (m0 * 2 + 1) n = 1\nhm : (m0 * 2 + 1) % 2 = 1\n\u22a2 2 \u2223 (m0 * 2 + 1) ^ 2 + n ^ 2 \u2227\n    2 \u2223 (m0 * 2 + 1) ^ 2 - n ^ 2 \u2227\n      ((m0 * 2 + 1) ^ 2 - n ^ 2) / 2 % 2 = 0 \u2227\n        Int.gcd (((m0 * 2 + 1) ^ 2 - n ^ 2) / 2) (((m0 * 2 + 1) ^ 2 + n ^ 2) / 2) = 1", "state_after": "case intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\n\u22a2 2 \u2223 (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 \u2227\n    2 \u2223 (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 \u2227\n      ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 \u2227\n        Int.gcd (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2) (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1"}, {"tactic": "have h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) := by\n  ring", "annotated_tactic": ["have h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) := by\n    ring", []], "state_before": "case intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\n\u22a2 2 \u2223 (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 \u2227\n    2 \u2223 (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 \u2227\n      ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 \u2227\n        Int.gcd (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2) (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1", "state_after": "case intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\n\u22a2 2 \u2223 (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 \u2227\n    2 \u2223 (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 \u2227\n      ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 \u2227\n        Int.gcd (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2) (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1"}, {"tactic": "have h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) := by ring", "annotated_tactic": ["have h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) := by ring", []], "state_before": "case intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\n\u22a2 2 \u2223 (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 \u2227\n    2 \u2223 (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 \u2227\n      ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 \u2227\n        Int.gcd (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2) (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1", "state_after": "case intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\n\u22a2 2 \u2223 (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 \u2227\n    2 \u2223 (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 \u2227\n      ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 \u2227\n        Int.gcd (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2) (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1"}, {"tactic": "have h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 := by\n  rw [h2, Int.mul_ediv_cancel_left, Int.mul_emod_right]\n  exact by decide", "annotated_tactic": ["have h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 := by\n    rw [h2, <a>Int.mul_ediv_cancel_left</a>, <a>Int.mul_emod_right</a>]\n    exact by decide", [{"full_name": "Int.mul_ediv_cancel_left", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [219, 17], "def_end_pos": [219, 37]}, {"full_name": "Int.mul_emod_right", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [471, 17], "def_end_pos": [471, 31]}]], "state_before": "case intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\n\u22a2 2 \u2223 (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 \u2227\n    2 \u2223 (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 \u2227\n      ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 \u2227\n        Int.gcd (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2) (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1", "state_after": "case intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\n\u22a2 2 \u2223 (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 \u2227\n    2 \u2223 (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 \u2227\n      ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 \u2227\n        Int.gcd (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2) (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1"}, {"tactic": "refine' \u27e8\u27e8_, h1\u27e9, \u27e8_, h2\u27e9, h3, _\u27e9", "annotated_tactic": ["refine' \u27e8\u27e8_, h1\u27e9, \u27e8_, h2\u27e9, h3, _\u27e9", []], "state_before": "case intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\n\u22a2 2 \u2223 (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 \u2227\n    2 \u2223 (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 \u2227\n      ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0 \u2227\n        Int.gcd (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2) (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1", "state_after": "case intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\n\u22a2 Int.gcd (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2) (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1"}, {"tactic": "have h20 : (2 : \u2124) \u2260 0 := by decide", "annotated_tactic": ["have h20 : (2 : \u2124) \u2260 0 := by decide", []], "state_before": "case intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\n\u22a2 Int.gcd (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2) (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1", "state_after": "case intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\n\u22a2 Int.gcd (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2) (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1"}, {"tactic": "rw [h1, h2, Int.mul_ediv_cancel_left _ h20, Int.mul_ediv_cancel_left _ h20]", "annotated_tactic": ["rw [h1, h2, <a>Int.mul_ediv_cancel_left</a> _ h20, <a>Int.mul_ediv_cancel_left</a> _ h20]", [{"full_name": "Int.mul_ediv_cancel_left", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [219, 17], "def_end_pos": [219, 37]}, {"full_name": "Int.mul_ediv_cancel_left", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [219, 17], "def_end_pos": [219, 37]}]], "state_before": "case intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\n\u22a2 Int.gcd (((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2) (((m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2) / 2) = 1", "state_after": "case intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\n\u22a2 Int.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1"}, {"tactic": "by_contra h4", "annotated_tactic": ["by_contra h4", []], "state_before": "case intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\n\u22a2 Int.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1", "state_after": "case intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\nh4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1\n\u22a2 False"}, {"tactic": "obtain \u27e8p, hp, hp1, hp2\u27e9 := Nat.Prime.not_coprime_iff_dvd.mp h4", "annotated_tactic": ["obtain \u27e8p, hp, hp1, hp2\u27e9 := Nat.Prime.not_coprime_iff_dvd.mp h4", []], "state_before": "case intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\nh4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\nh4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1\np : \u2115\nhp : Nat.Prime p\nhp1 : p \u2223 Int.natAbs (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nhp2 : p \u2223 Int.natAbs (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\n\u22a2 False"}, {"tactic": "apply hp.not_dvd_one", "annotated_tactic": ["apply hp.not_dvd_one", []], "state_before": "case intro.intro.intro.intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\nh4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1\np : \u2115\nhp : Nat.Prime p\nhp1 : p \u2223 Int.natAbs (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nhp2 : p \u2223 Int.natAbs (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\nh4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1\np : \u2115\nhp : Nat.Prime p\nhp1 : p \u2223 Int.natAbs (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nhp2 : p \u2223 Int.natAbs (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\n\u22a2 p \u2223 1"}, {"tactic": "rw [\u2190 h]", "annotated_tactic": ["rw [\u2190 h]", []], "state_before": "case intro.intro.intro.intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\nh4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1\np : \u2115\nhp : Nat.Prime p\nhp1 : p \u2223 Int.natAbs (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nhp2 : p \u2223 Int.natAbs (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\n\u22a2 p \u2223 1", "state_after": "case intro.intro.intro.intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\nh4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1\np : \u2115\nhp : Nat.Prime p\nhp1 : p \u2223 Int.natAbs (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nhp2 : p \u2223 Int.natAbs (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\n\u22a2 p \u2223 Int.gcd (m0 * 2 + 1) (n0 * 2 + 1)"}, {"tactic": "rw [\u2190 Int.coe_nat_dvd_left] at hp1 hp2", "annotated_tactic": ["rw [\u2190 <a>Int.coe_nat_dvd_left</a>] at hp1 hp2", [{"full_name": "Int.coe_nat_dvd_left", "def_path": "Mathlib/Data/Int/Dvd/Basic.lean", "def_pos": [33, 9], "def_end_pos": [33, 25]}]], "state_before": "case intro.intro.intro.intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\nh4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1\np : \u2115\nhp : Nat.Prime p\nhp1 : p \u2223 Int.natAbs (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nhp2 : p \u2223 Int.natAbs (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\n\u22a2 p \u2223 Int.gcd (m0 * 2 + 1) (n0 * 2 + 1)", "state_after": "case intro.intro.intro.intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\nh4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1\np : \u2115\nhp : Nat.Prime p\nhp1 : \u2191p \u2223 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)\nhp2 : \u2191p \u2223 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1\n\u22a2 p \u2223 Int.gcd (m0 * 2 + 1) (n0 * 2 + 1)"}, {"tactic": "apply Nat.dvd_gcd", "annotated_tactic": ["apply <a>Nat.dvd_gcd</a>", [{"full_name": "Nat.dvd_gcd", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}]], "state_before": "case intro.intro.intro.intro.intro\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\nh4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1\np : \u2115\nhp : Nat.Prime p\nhp1 : \u2191p \u2223 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)\nhp2 : \u2191p \u2223 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1\n\u22a2 p \u2223 Int.gcd (m0 * 2 + 1) (n0 * 2 + 1)", "state_after": "case intro.intro.intro.intro.intro.a\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\nh4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1\np : \u2115\nhp : Nat.Prime p\nhp1 : \u2191p \u2223 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)\nhp2 : \u2191p \u2223 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1\n\u22a2 p \u2223 Int.natAbs (m0 * 2 + 1)\n\ncase intro.intro.intro.intro.intro.a\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\nh4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1\np : \u2115\nhp : Nat.Prime p\nhp1 : \u2191p \u2223 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)\nhp2 : \u2191p \u2223 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1\n\u22a2 p \u2223 Int.natAbs (n0 * 2 + 1)"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "m0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\n\u22a2 (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)", "state_after": "no goals"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "m0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\n\u22a2 (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))", "state_after": "no goals"}, {"tactic": "rw [h2, Int.mul_ediv_cancel_left, Int.mul_emod_right]", "annotated_tactic": ["rw [h2, <a>Int.mul_ediv_cancel_left</a>, <a>Int.mul_emod_right</a>]", [{"full_name": "Int.mul_ediv_cancel_left", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [219, 17], "def_end_pos": [219, 37]}, {"full_name": "Int.mul_emod_right", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [471, 17], "def_end_pos": [471, 31]}]], "state_before": "m0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\n\u22a2 ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0", "state_after": "case H\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\n\u22a2 2 \u2260 0"}, {"tactic": "exact by decide", "annotated_tactic": ["exact by decide", []], "state_before": "case H\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\n\u22a2 2 \u2260 0", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "m0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\n\u22a2 2 \u2260 0", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "m0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\n\u22a2 2 \u2260 0", "state_after": "no goals"}, {"tactic": "apply Int.Prime.dvd_natAbs_of_coe_dvd_sq hp", "annotated_tactic": ["apply <a>Int.Prime.dvd_natAbs_of_coe_dvd_sq</a> hp", [{"full_name": "Int.Prime.dvd_natAbs_of_coe_dvd_sq", "def_path": "Mathlib/Data/Int/NatPrime.lean", "def_pos": [36, 9], "def_end_pos": [36, 39]}]], "state_before": "case intro.intro.intro.intro.intro.a\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\nh4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1\np : \u2115\nhp : Nat.Prime p\nhp1 : \u2191p \u2223 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)\nhp2 : \u2191p \u2223 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1\n\u22a2 p \u2223 Int.natAbs (m0 * 2 + 1)", "state_after": "case intro.intro.intro.intro.intro.a.h\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\nh4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1\np : \u2115\nhp : Nat.Prime p\nhp1 : \u2191p \u2223 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)\nhp2 : \u2191p \u2223 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1\n\u22a2 \u2191p \u2223 (m0 * 2 + 1) ^ 2"}, {"tactic": "convert dvd_add hp1 hp2", "annotated_tactic": ["convert <a>dvd_add</a> hp1 hp2", [{"full_name": "dvd_add", "def_path": "Mathlib/Algebra/Ring/Divisibility/Basic.lean", "def_pos": [27, 9], "def_end_pos": [27, 16]}]], "state_before": "case intro.intro.intro.intro.intro.a.h\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\nh4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1\np : \u2115\nhp : Nat.Prime p\nhp1 : \u2191p \u2223 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)\nhp2 : \u2191p \u2223 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1\n\u22a2 \u2191p \u2223 (m0 * 2 + 1) ^ 2", "state_after": "case h.e'_4\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\nh4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1\np : \u2115\nhp : Nat.Prime p\nhp1 : \u2191p \u2223 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)\nhp2 : \u2191p \u2223 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1\n\u22a2 (m0 * 2 + 1) ^ 2 = 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0) + (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case h.e'_4\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\nh4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1\np : \u2115\nhp : Nat.Prime p\nhp1 : \u2191p \u2223 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)\nhp2 : \u2191p \u2223 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1\n\u22a2 (m0 * 2 + 1) ^ 2 = 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0) + (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)", "state_after": "no goals"}, {"tactic": "apply Int.Prime.dvd_natAbs_of_coe_dvd_sq hp", "annotated_tactic": ["apply <a>Int.Prime.dvd_natAbs_of_coe_dvd_sq</a> hp", [{"full_name": "Int.Prime.dvd_natAbs_of_coe_dvd_sq", "def_path": "Mathlib/Data/Int/NatPrime.lean", "def_pos": [36, 9], "def_end_pos": [36, 39]}]], "state_before": "case intro.intro.intro.intro.intro.a\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\nh4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1\np : \u2115\nhp : Nat.Prime p\nhp1 : \u2191p \u2223 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)\nhp2 : \u2191p \u2223 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1\n\u22a2 p \u2223 Int.natAbs (n0 * 2 + 1)", "state_after": "case intro.intro.intro.intro.intro.a.h\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\nh4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1\np : \u2115\nhp : Nat.Prime p\nhp1 : \u2191p \u2223 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)\nhp2 : \u2191p \u2223 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1\n\u22a2 \u2191p \u2223 (n0 * 2 + 1) ^ 2"}, {"tactic": "convert dvd_sub hp2 hp1", "annotated_tactic": ["convert <a>dvd_sub</a> hp2 hp1", [{"full_name": "dvd_sub", "def_path": "Mathlib/Algebra/Ring/Divisibility/Basic.lean", "def_pos": [86, 9], "def_end_pos": [86, 16]}]], "state_before": "case intro.intro.intro.intro.intro.a.h\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\nh4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1\np : \u2115\nhp : Nat.Prime p\nhp1 : \u2191p \u2223 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)\nhp2 : \u2191p \u2223 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1\n\u22a2 \u2191p \u2223 (n0 * 2 + 1) ^ 2", "state_after": "case h.e'_4\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\nh4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1\np : \u2115\nhp : Nat.Prime p\nhp1 : \u2191p \u2223 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)\nhp2 : \u2191p \u2223 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1\n\u22a2 (n0 * 2 + 1) ^ 2 = 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1 - 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case h.e'_4\nm0 n0 : \u2124\nhm : (m0 * 2 + 1) % 2 = 1\nhn : (n0 * 2 + 1) % 2 = 1\nh : Int.gcd (m0 * 2 + 1) (n0 * 2 + 1) = 1\nh1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)\nh2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))\nh3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0\nh20 : 2 \u2260 0\nh4 : \u00acInt.gcd (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)) (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1\np : \u2115\nhp : Nat.Prime p\nhp1 : \u2191p \u2223 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)\nhp2 : \u2191p \u2223 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1\n\u22a2 (n0 * 2 + 1) ^ 2 = 2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1 - 2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Module/Submodule/Lattice.lean", "full_name": "Submodule.toAddSubmonoid_toNatSubmodule", "start": [431, 1], "end": [433, 49], "traced_tactics": 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with n e", "annotated_tactic": ["cases' <a>Cardinal.lt_aleph0</a>.1 h with n e", [{"full_name": "Cardinal.lt_aleph0", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1429, 9], "def_end_pos": [1429, 18]}]], "state_before": "case inr.inr\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\no : Ordinal.{u_2}\nl : IsLimit o\nh : cof o < \u2135\u2080\n\u22a2 False", "state_after": "case inr.inr.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\no : Ordinal.{u_2}\nl : IsLimit o\nh : cof o < \u2135\u2080\nn : \u2115\ne : cof o = \u2191n\n\u22a2 False"}, {"tactic": "have := cof_cof o", "annotated_tactic": ["have := <a>cof_cof</a> o", [{"full_name": "Ordinal.cof_cof", "def_path": "Mathlib/SetTheory/Cardinal/Cofinality.lean", "def_pos": [649, 9], "def_end_pos": [649, 16]}]], "state_before": "case inr.inr.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\no : Ordinal.{u_2}\nl : IsLimit o\nh : cof o < \u2135\u2080\nn : \u2115\ne : cof o = \u2191n\n\u22a2 False", "state_after": "case inr.inr.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\no : Ordinal.{u_2}\nl : IsLimit o\nh : cof o < \u2135\u2080\nn : \u2115\ne : cof o = \u2191n\nthis : cof (ord (cof o)) = cof o\n\u22a2 False"}, {"tactic": "rw [e, ord_nat] at this", "annotated_tactic": ["rw [e, <a>ord_nat</a>] at this", [{"full_name": "Cardinal.ord_nat", "def_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "def_pos": [1420, 9], "def_end_pos": [1420, 16]}]], "state_before": "case inr.inr.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\no : Ordinal.{u_2}\nl : IsLimit o\nh : cof o < \u2135\u2080\nn : \u2115\ne : cof o = \u2191n\nthis : cof (ord (cof o)) = cof o\n\u22a2 False", "state_after": "case inr.inr.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\no : Ordinal.{u_2}\nl : IsLimit o\nh : cof o < \u2135\u2080\nn : \u2115\ne : cof o = \u2191n\nthis : cof \u2191n = \u2191n\n\u22a2 False"}, {"tactic": "cases n", "annotated_tactic": ["cases n", []], "state_before": "case inr.inr.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\no : Ordinal.{u_2}\nl : IsLimit o\nh : cof o < \u2135\u2080\nn : \u2115\ne : cof o = \u2191n\nthis : cof \u2191n = \u2191n\n\u22a2 False", "state_after": "case inr.inr.intro.zero\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\no : Ordinal.{u_2}\nl : IsLimit o\nh : cof o < \u2135\u2080\ne : cof o = \u2191Nat.zero\nthis : cof \u2191Nat.zero = \u2191Nat.zero\n\u22a2 False\n\ncase inr.inr.intro.succ\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\no : Ordinal.{u_2}\nl : IsLimit o\nh : cof o < \u2135\u2080\nn\u271d : \u2115\ne : cof o = \u2191(Nat.succ n\u271d)\nthis : cof \u2191(Nat.succ n\u271d) = \u2191(Nat.succ n\u271d)\n\u22a2 False"}, {"tactic": "simp at e", "annotated_tactic": ["simp at e", []], "state_before": "case inr.inr.intro.zero\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\no : Ordinal.{u_2}\nl : IsLimit o\nh : cof o < \u2135\u2080\ne : cof o = \u2191Nat.zero\nthis : cof \u2191Nat.zero = \u2191Nat.zero\n\u22a2 False", "state_after": "case inr.inr.intro.zero\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\no : Ordinal.{u_2}\nl : IsLimit o\nh : cof o < \u2135\u2080\nthis : cof \u2191Nat.zero = \u2191Nat.zero\ne : o = 0\n\u22a2 False"}, {"tactic": "simp [e, not_zero_isLimit] at l", "annotated_tactic": ["simp [e, <a>not_zero_isLimit</a>] at l", [{"full_name": "Ordinal.not_zero_isLimit", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [245, 9], "def_end_pos": [245, 25]}]], "state_before": "case inr.inr.intro.zero\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\no : Ordinal.{u_2}\nl : IsLimit o\nh : cof o < \u2135\u2080\nthis : cof \u2191Nat.zero = \u2191Nat.zero\ne : o = 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rw [nat_cast_succ, cof_succ] at this", "annotated_tactic": ["rw [<a>nat_cast_succ</a>, <a>cof_succ</a>] at this", [{"full_name": "Ordinal.nat_cast_succ", "def_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 22]}, {"full_name": "Ordinal.cof_succ", "def_path": "Mathlib/SetTheory/Cardinal/Cofinality.lean", "def_pos": [500, 9], "def_end_pos": [500, 17]}]], "state_before": "case inr.inr.intro.succ\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\no : Ordinal.{u_2}\nl : IsLimit o\nh : cof o < \u2135\u2080\nn\u271d : \u2115\ne : cof o = \u2191(Nat.succ n\u271d)\nthis : cof \u2191(Nat.succ n\u271d) = \u2191(Nat.succ n\u271d)\n\u22a2 False", "state_after": "case inr.inr.intro.succ\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\no : Ordinal.{u_2}\nl : IsLimit o\nh : cof o < \u2135\u2080\nn\u271d : \u2115\ne : cof o = \u2191(Nat.succ n\u271d)\nthis : 1 = \u2191(Nat.succ n\u271d)\n\u22a2 False"}, {"tactic": "rw [\u2190 this, cof_eq_one_iff_is_succ] at e", "annotated_tactic": ["rw [\u2190 this, <a>cof_eq_one_iff_is_succ</a>] at e", [{"full_name": 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\u2191(Nat.succ n\u271d)\n\u22a2 False", "state_after": "case inr.inr.intro.succ.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nn\u271d : \u2115\nthis : 1 = \u2191(Nat.succ n\u271d)\na : Ordinal.{u_2}\nl : IsLimit (succ a)\nh : cof (succ a) < \u2135\u2080\n\u22a2 False"}, {"tactic": "exact not_succ_isLimit _ l", "annotated_tactic": ["exact <a>not_succ_isLimit</a> _ l", [{"full_name": "Ordinal.not_succ_isLimit", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [249, 9], "def_end_pos": [249, 25]}]], "state_before": "case inr.inr.intro.succ.intro\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nn\u271d : \u2115\nthis : 1 = \u2191(Nat.succ n\u271d)\na : Ordinal.{u_2}\nl : IsLimit (succ a)\nh : cof (succ a) < \u2135\u2080\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/ModelTheory/Semantics.lean", "full_name": "FirstOrder.Language.Formula.realize_iExs", "start": [940, 1], "end": [957, 20], "traced_tactics": [{"tactic": "let e := Classical.choice (Classical.choose_spec (Finite.exists_equiv_fin \u03b3))", "annotated_tactic": ["let e := <a>Classical.choice</a> (<a>Classical.choose_spec</a> (<a>Finite.exists_equiv_fin</a> \u03b3))", [{"full_name": "Classical.choice", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [712, 7], "def_end_pos": [712, 23]}, {"full_name": "Classical.choose_spec", "def_path": "lake-packages/lean4/src/lean/Init/Classical.lean", "def_pos": [22, 9], "def_end_pos": [22, 20]}, {"full_name": "Finite.exists_equiv_fin", "def_path": "Mathlib/Data/Finite/Defs.lean", "def_pos": [66, 9], "def_end_pos": [66, 32]}]], "state_before": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\n\u22a2 Formula.Realize (Formula.iExs f \u03c6) v \u2194 \u2203 i, Formula.Realize \u03c6 fun a => Sum.elim v i (f a)", "state_after": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\ne : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) :=\n  Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))\n\u22a2 Formula.Realize (Formula.iExs f \u03c6) v \u2194 \u2203 i, Formula.Realize \u03c6 fun a => Sum.elim v i (f a)"}, {"tactic": "rw [Formula.iExs]", "annotated_tactic": ["rw [<a>Formula.iExs</a>]", [{"full_name": "FirstOrder.Language.Formula.iExs", "def_path": "Mathlib/ModelTheory/Syntax.lean", "def_pos": [1051, 19], "def_end_pos": [1051, 23]}]], "state_before": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\ne : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) :=\n  Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))\n\u22a2 Formula.Realize (Formula.iExs f \u03c6) v \u2194 \u2203 i, Formula.Realize \u03c6 fun a => Sum.elim v i (f a)", "state_after": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\ne : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) :=\n  Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))\n\u22a2 Formula.Realize\n      (let e := Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))));\n      exs (relabel (fun a => Sum.map id (\u2191e) (f a)) \u03c6))\n      v \u2194\n    \u2203 i, Formula.Realize \u03c6 fun a => Sum.elim v i (f a)"}, {"tactic": "simp only [Nat.add_zero, realize_exs, realize_relabel, Function.comp,\n  castAdd_zero, castIso_refl, OrderIso.refl_apply, Sum.elim_map, id_eq]", "annotated_tactic": ["simp only [<a>Nat.add_zero</a>, <a>realize_exs</a>, <a>realize_relabel</a>, <a>Function.comp</a>,\n    <a>castAdd_zero</a>, <a>castIso_refl</a>, <a>OrderIso.refl_apply</a>, <a>Sum.elim_map</a>, <a>id_eq</a>]", [{"full_name": "Nat.add_zero", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [457, 27], "def_end_pos": [457, 39]}, {"full_name": "FirstOrder.Language.BoundedFormula.realize_exs", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [900, 9], "def_end_pos": [900, 20]}, {"full_name": "FirstOrder.Language.BoundedFormula.realize_relabel", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [398, 9], "def_end_pos": [398, 24]}, {"full_name": "Function.comp", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}, {"full_name": "Fin.castAdd_zero", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [310, 17], "def_end_pos": [310, 29]}, {"full_name": "Fin.castIso_refl", "def_path": "Mathlib/Data/Fin/Basic.lean", "def_pos": [832, 9], "def_end_pos": [832, 21]}, {"full_name": "OrderIso.refl_apply", "def_path": "Mathlib/Order/Hom/Basic.lean", "def_pos": [839, 9], "def_end_pos": [839, 19]}, {"full_name": "Sum.elim_map", "def_path": "lake-packages/std/Std/Data/Sum/Lemmas.lean", "def_pos": [130, 9], "def_end_pos": [130, 17]}, {"full_name": "id_eq", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [284, 17], "def_end_pos": [284, 22]}]], "state_before": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\ne : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) :=\n  Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))\n\u22a2 Formula.Realize\n      (let e := Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))));\n      exs (relabel (fun a => Sum.map id (\u2191e) (f a)) \u03c6))\n      v \u2194\n    \u2203 i, Formula.Realize \u03c6 fun a => Sum.elim v i (f a)", "state_after": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\ne : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) :=\n  Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))\n\u22a2 (\u2203 xs,\n      Realize \u03c6\n        (fun x =>\n          Sum.elim (fun x => v x)\n            (fun x =>\n              xs\n                (Fin.cast\n                  (_ :\n                    Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)) = Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))\n                  (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))) x)))\n            (f x))\n        fun x => xs (natAdd (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) x)) \u2194\n    \u2203 i, Formula.Realize \u03c6 fun a => Sum.elim v i (f a)"}, {"tactic": "rw [\u2190 not_iff_not, not_exists, not_exists]", "annotated_tactic": ["rw [\u2190 <a>not_iff_not</a>, <a>not_exists</a>, <a>not_exists</a>]", [{"full_name": "not_iff_not", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [439, 9], "def_end_pos": [439, 20]}, {"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}]], "state_before": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\ne : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) :=\n  Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))\n\u22a2 (\u2203 xs,\n      Realize \u03c6\n        (fun x =>\n          Sum.elim (fun x => v x)\n            (fun x =>\n              xs\n                (Fin.cast\n                  (_ :\n                    Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)) = Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))\n                  (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))) x)))\n            (f x))\n        fun x => xs (natAdd (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) x)) \u2194\n    \u2203 i, Formula.Realize \u03c6 fun a => Sum.elim v i (f a)", "state_after": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\ne : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) :=\n  Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))\n\u22a2 (\u2200 (x : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M),\n      \u00acRealize \u03c6\n          (fun x_1 =>\n            Sum.elim (fun x => v x)\n              (fun x_2 =>\n                x\n                  (Fin.cast\n                    (_ :\n                      Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)) =\n                        Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))\n                    (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))))\n                      x_2)))\n              (f x_1))\n          fun x_1 => x (natAdd (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) x_1)) \u2194\n    \u2200 (x : \u03b3 \u2192 M), \u00acFormula.Realize \u03c6 fun a => Sum.elim v x (f a)"}, {"tactic": "refine Equiv.forall_congr ?_ ?_", "annotated_tactic": ["refine <a>Equiv.forall_congr</a> ?_ ?_", [{"full_name": "Equiv.forall_congr", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [867, 19], "def_end_pos": [867, 31]}]], "state_before": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\ne : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) :=\n  Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))\n\u22a2 (\u2200 (x : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M),\n      \u00acRealize \u03c6\n          (fun x_1 =>\n            Sum.elim (fun x => v x)\n              (fun x_2 =>\n                x\n                  (Fin.cast\n                    (_ :\n                      Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)) =\n                        Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))\n                    (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))))\n                      x_2)))\n              (f x_1))\n          fun x_1 => x (natAdd (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) x_1)) \u2194\n    \u2200 (x : \u03b3 \u2192 M), \u00acFormula.Realize \u03c6 fun a => Sum.elim v x (f a)", "state_after": "case refine_1\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\ne : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) :=\n  Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))\n\u22a2 (Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M) \u2243 (\u03b3 \u2192 M)\n\ncase refine_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\ne : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) :=\n  Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))\n\u22a2 \u2200 {x : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M},\n    (\u00acRealize \u03c6\n          (fun x_1 =>\n            Sum.elim (fun x => v x)\n              (fun x_2 =>\n                x\n                  (Fin.cast\n                    (_ :\n                      Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)) =\n                        Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))\n                    (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))))\n                      x_2)))\n              (f x_1))\n          fun x_1 => x (natAdd (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) x_1)) \u2194\n      \u00acFormula.Realize \u03c6 fun a => Sum.elim v (\u2191?refine_1 x) (f a)"}, {"tactic": "exact \u27e8fun v => v \u2218 e, fun v => v \u2218 e.symm,\n  fun _ => by simp [Function.comp],\n  fun _ => by simp [Function.comp]\u27e9", "annotated_tactic": ["exact \u27e8fun v => v \u2218 e, fun v => v \u2218 e.symm,\n      fun _ => by simp [<a>Function.comp</a>],\n      fun _ => by simp [<a>Function.comp</a>]\u27e9", [{"full_name": "Function.comp", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}, {"full_name": "Function.comp", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}]], "state_before": "case refine_1\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\ne : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) :=\n  Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))\n\u22a2 (Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M) \u2243 (\u03b3 \u2192 M)", "state_after": "no goals"}, {"tactic": "simp [Function.comp]", "annotated_tactic": ["simp [<a>Function.comp</a>]", [{"full_name": "Function.comp", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}]], "state_before": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\ne : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) :=\n  Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))\nx\u271d : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M\n\u22a2 (fun v => v \u2218 \u2191e.symm) ((fun v => v \u2218 \u2191e) x\u271d) = x\u271d", "state_after": "no goals"}, {"tactic": "simp [Function.comp]", "annotated_tactic": ["simp [<a>Function.comp</a>]", [{"full_name": "Function.comp", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}]], "state_before": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\ne : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) :=\n  Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))\nx\u271d : \u03b3 \u2192 M\n\u22a2 (fun v => v \u2218 \u2191e) ((fun v => v \u2218 \u2191e.symm) x\u271d) = x\u271d", "state_after": "no goals"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case refine_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\ne : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) :=\n  Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))\n\u22a2 \u2200 {x : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M},\n    (\u00acRealize \u03c6\n          (fun x_1 =>\n            Sum.elim (fun x => v x)\n              (fun x_2 =>\n                x\n                  (Fin.cast\n                    (_ :\n                      Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)) =\n                        Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))\n                    (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))))\n                      x_2)))\n              (f x_1))\n          fun x_1 => x (natAdd (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) x_1)) \u2194\n      \u00acFormula.Realize \u03c6 fun a =>\n          Sum.elim v\n            (\u2191{ toFun := fun v => v \u2218 \u2191e, invFun := fun v => v \u2218 \u2191e.symm,\n                  left_inv :=\n                    (_ :\n                      \u2200 (x : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M),\n                        (fun x_1 =>\n                            x\n                              (\u2191(Classical.choice\n                                    (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))))\n                                (\u2191(Classical.choice\n                                        (_ :\n                                          Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))).symm\n                                  x_1))) =\n                          x),\n                  right_inv :=\n                    (_ :\n                      \u2200 (x : \u03b3 \u2192 M),\n                        (fun x_1 =>\n                            x\n                              (\u2191(Classical.choice\n                                      (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))).symm\n                                (\u2191(Classical.choice\n                                      (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))))\n                                  x_1))) =\n                          x) }\n              x)\n            (f a)", "state_after": "case refine_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\ne : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) :=\n  Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))\nx : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M\n\u22a2 (\u00acRealize \u03c6\n        (fun x_1 =>\n          Sum.elim (fun x => v x)\n            (fun x_2 =>\n              x\n                (Fin.cast\n                  (_ :\n                    Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)) = Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))\n                  (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))))\n                    x_2)))\n            (f x_1))\n        fun x_1 => x (natAdd (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) x_1)) \u2194\n    \u00acFormula.Realize \u03c6 fun a =>\n        Sum.elim v\n          (\u2191{ toFun := fun v => v \u2218 \u2191e, invFun := fun v => v \u2218 \u2191e.symm,\n                left_inv :=\n                  (_ :\n                    \u2200 (x : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M),\n                      (fun x_1 =>\n                          x\n                            (\u2191(Classical.choice\n                                  (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))))\n                              (\u2191(Classical.choice\n                                      (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))).symm\n                                x_1))) =\n                        x),\n                right_inv :=\n                  (_ :\n                    \u2200 (x : \u03b3 \u2192 M),\n                      (fun x_1 =>\n                          x\n                            (\u2191(Classical.choice\n                                    (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))).symm\n                              (\u2191(Classical.choice\n                                    (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))))\n                                x_1))) =\n                        x) }\n            x)\n          (f a)"}, {"tactic": "rw [Formula.Realize, iff_iff_eq]", "annotated_tactic": ["rw [<a>Formula.Realize</a>, <a>iff_iff_eq</a>]", [{"full_name": "FirstOrder.Language.Formula.Realize", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [609, 12], "def_end_pos": [609, 19]}, {"full_name": "iff_iff_eq", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [51, 9], "def_end_pos": [51, 19]}]], "state_before": "case refine_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\ne : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) :=\n  Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))\nx : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M\n\u22a2 (\u00acRealize \u03c6\n        (fun x_1 =>\n          Sum.elim (fun x => v x)\n            (fun x_2 =>\n              x\n                (Fin.cast\n                  (_ :\n                    Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)) = Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))\n                  (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))))\n                    x_2)))\n            (f x_1))\n        fun x_1 => x (natAdd (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) x_1)) \u2194\n    \u00acFormula.Realize \u03c6 fun a =>\n        Sum.elim v\n          (\u2191{ toFun := fun v => v \u2218 \u2191e, invFun := fun v => v \u2218 \u2191e.symm,\n                left_inv :=\n                  (_ :\n                    \u2200 (x : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M),\n                      (fun x_1 =>\n                          x\n                            (\u2191(Classical.choice\n                                  (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))))\n                              (\u2191(Classical.choice\n                                      (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))).symm\n                                x_1))) =\n                        x),\n                right_inv :=\n                  (_ :\n                    \u2200 (x : \u03b3 \u2192 M),\n                      (fun x_1 =>\n                          x\n                            (\u2191(Classical.choice\n                                    (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))).symm\n                              (\u2191(Classical.choice\n                                    (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))))\n                                x_1))) =\n                        x) }\n            x)\n          (f a)", "state_after": "case refine_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\ne : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) :=\n  Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))\nx : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M\n\u22a2 (\u00acRealize \u03c6\n        (fun x_1 =>\n          Sum.elim (fun x => v x)\n            (fun x_2 =>\n              x\n                (Fin.cast\n                  (_ :\n                    Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)) = Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))\n                  (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))))\n                    x_2)))\n            (f x_1))\n        fun x_1 => x (natAdd (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) x_1)) =\n    \u00acRealize \u03c6\n        (fun a =>\n          Sum.elim v\n            (\u2191{ toFun := fun v => v \u2218 \u2191e, invFun := fun v => v \u2218 \u2191e.symm,\n                  left_inv :=\n                    (_ :\n                      \u2200 (x : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M),\n                        (fun x_1 =>\n                            x\n                              (\u2191(Classical.choice\n                                    (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))))\n                                (\u2191(Classical.choice\n                                        (_ :\n                                          Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))).symm\n                                  x_1))) =\n                          x),\n                  right_inv :=\n                    (_ :\n                      \u2200 (x : \u03b3 \u2192 M),\n                        (fun x_1 =>\n                            x\n                              (\u2191(Classical.choice\n                                      (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))).symm\n                                (\u2191(Classical.choice\n                                      (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))))\n                                  x_1))) =\n                          x) }\n              x)\n            (f a))\n        default"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case refine_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\ne : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) :=\n  Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))\nx : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M\n\u22a2 (\u00acRealize \u03c6\n        (fun x_1 =>\n          Sum.elim (fun x => v x)\n            (fun x_2 =>\n              x\n                (Fin.cast\n                  (_ :\n                    Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)) = Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))\n                  (\u2191(Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))))\n                    x_2)))\n            (f x_1))\n        fun x_1 => x (natAdd (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) x_1)) =\n    \u00acRealize \u03c6\n        (fun a =>\n          Sum.elim v\n            (\u2191{ toFun := fun v => v \u2218 \u2191e, invFun := fun v => v \u2218 \u2191e.symm,\n                  left_inv :=\n                    (_ :\n                      \u2200 (x : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M),\n                        (fun x_1 =>\n                            x\n                              (\u2191(Classical.choice\n                                    (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))))\n                                (\u2191(Classical.choice\n                                        (_ :\n                                          Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))).symm\n                                  x_1))) =\n                          x),\n                  right_inv :=\n                    (_ :\n                      \u2200 (x : \u03b3 \u2192 M),\n                        (fun x_1 =>\n                            x\n                              (\u2191(Classical.choice\n                                      (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))).symm\n                                (\u2191(Classical.choice\n                                      (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))))))\n                                  x_1))) =\n                          x) }\n              x)\n            (f a))\n        default", "state_after": "case refine_2.e_a.e__xs\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\ne : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) :=\n  Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))\nx : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M\n\u22a2 (fun x_1 => x (natAdd (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) x_1)) = default"}, {"tactic": "funext i", "annotated_tactic": ["funext i", []], "state_before": "case refine_2.e_a.e__xs\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\ne : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) :=\n  Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))\nx : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M\n\u22a2 (fun x_1 => x (natAdd (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) x_1)) = default", "state_after": "case refine_2.e_a.e__xs.h\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\ne : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) :=\n  Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))\nx : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M\ni : Fin 0\n\u22a2 x (natAdd (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) i) = default i"}, {"tactic": "exact i.elim0", "annotated_tactic": ["exact i.elim0", []], "state_before": "case refine_2.e_a.e__xs.h\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nT : Theory L\ninst\u271d : Finite \u03b3\nf : \u03b1 \u2192 \u03b2 \u2295 \u03b3\n\u03c6 : Formula L \u03b1\nv : \u03b2 \u2192 M\ne : \u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) :=\n  Classical.choice (_ : Nonempty (\u03b3 \u2243 Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n)))))\nx : Fin (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) \u2192 M\ni : Fin 0\n\u22a2 x (natAdd (Classical.choose (_ : \u2203 n, Nonempty (\u03b3 \u2243 Fin n))) i) = default i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/IntegralClosure.lean", "full_name": "IsIntegral.sum", "start": [678, 1], "end": [680, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/CoprodI.lean", "full_name": "Monoid.CoprodI.lift_word_prod_nontrivial_of_head_card", "start": [912, 1], "end": [925, 14], "traced_tactics": [{"tactic": "obtain \u27e8h, hn1, hnh\u27e9 := Cardinal.three_le hcard 1 w.head\u207b\u00b9", "annotated_tactic": ["obtain \u27e8h, hn1, hnh\u27e9 := <a>Cardinal.three_le</a> hcard 1 w.head\u207b\u00b9", [{"full_name": "Cardinal.three_le", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [2401, 9], "def_end_pos": [2401, 17]}]], "state_before": "\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type u_3\ninst\u271d\u00b3 : Monoid N\nhnontriv : Nontrivial \u03b9\nG : Type u_4\ninst\u271d\u00b2 : Group G\nH : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i)\nf : (i : \u03b9) \u2192 H i \u2192* G\nhcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i)\n\u03b1 : Type u_6\ninst\u271d : MulAction G \u03b1\nX : \u03b9 \u2192 Set \u03b1\nhXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i\ni j : \u03b9\nw : NeWord H i j\nhcard : 3 \u2264 #(H i)\nhheadtail : i \u2260 j\n\u22a2 \u2191(\u2191lift f) (NeWord.prod w) \u2260 1", "state_after": "case intro.intro\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type u_3\ninst\u271d\u00b3 : Monoid N\nhnontriv : Nontrivial \u03b9\nG : Type u_4\ninst\u271d\u00b2 : Group G\nH : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i)\nf : (i : \u03b9) \u2192 H i \u2192* G\nhcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i)\n\u03b1 : Type u_6\ninst\u271d : MulAction G \u03b1\nX : \u03b9 \u2192 Set \u03b1\nhXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i\ni j : \u03b9\nw : NeWord H i j\nhcard : 3 \u2264 #(H i)\nhheadtail : i \u2260 j\nh : H i\nhn1 : h \u2260 1\nhnh : h \u2260 (NeWord.head w)\u207b\u00b9\n\u22a2 \u2191(\u2191lift f) (NeWord.prod w) \u2260 1"}, {"tactic": "have hnot1 : h * w.head \u2260 1 := by\n  rw [\u2190 div_inv_eq_mul]\n  exact div_ne_one_of_ne hnh", "annotated_tactic": ["have hnot1 : h * w.head \u2260 1 := by\n    rw [\u2190 <a>div_inv_eq_mul</a>]\n    exact <a>div_ne_one_of_ne</a> hnh", [{"full_name": "div_inv_eq_mul", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [454, 9], "def_end_pos": [454, 23]}, {"full_name": "div_ne_one_of_ne", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [383, 9], "def_end_pos": [383, 25]}]], "state_before": "case intro.intro\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type u_3\ninst\u271d\u00b3 : Monoid N\nhnontriv : Nontrivial \u03b9\nG : Type u_4\ninst\u271d\u00b2 : Group G\nH : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i)\nf : (i : \u03b9) \u2192 H i \u2192* G\nhcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i)\n\u03b1 : Type u_6\ninst\u271d : MulAction G \u03b1\nX : \u03b9 \u2192 Set \u03b1\nhXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i\ni j : \u03b9\nw : NeWord H i j\nhcard : 3 \u2264 #(H i)\nhheadtail : i \u2260 j\nh : H i\nhn1 : h \u2260 1\nhnh : h \u2260 (NeWord.head w)\u207b\u00b9\n\u22a2 \u2191(\u2191lift f) (NeWord.prod w) \u2260 1", "state_after": "case intro.intro\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type u_3\ninst\u271d\u00b3 : Monoid N\nhnontriv : Nontrivial \u03b9\nG : Type u_4\ninst\u271d\u00b2 : Group G\nH : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i)\nf : (i : \u03b9) \u2192 H i \u2192* G\nhcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i)\n\u03b1 : Type u_6\ninst\u271d : MulAction G \u03b1\nX : \u03b9 \u2192 Set \u03b1\nhXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i\ni j : \u03b9\nw : NeWord H i j\nhcard : 3 \u2264 #(H i)\nhheadtail : i \u2260 j\nh : H i\nhn1 : h \u2260 1\nhnh : h \u2260 (NeWord.head w)\u207b\u00b9\nhnot1 : h * NeWord.head w \u2260 1\n\u22a2 \u2191(\u2191lift f) (NeWord.prod w) \u2260 1"}, {"tactic": "let w' : NeWord H i i :=\n  NeWord.append (NeWord.mulHead w h hnot1) hheadtail.symm\n    (NeWord.singleton h\u207b\u00b9 (inv_ne_one.mpr hn1))", "annotated_tactic": ["let w' : <a>NeWord</a> H i i :=\n    <a>NeWord.append</a> (<a>NeWord.mulHead</a> w h hnot1) hheadtail.symm\n      (<a>NeWord.singleton</a> h\u207b\u00b9 (inv_ne_one.mpr hn1))", [{"full_name": "Monoid.CoprodI.NeWord", "def_path": "Mathlib/GroupTheory/CoprodI.lean", "def_pos": [646, 11], "def_end_pos": [646, 17]}, {"full_name": "Monoid.CoprodI.NeWord.append", "def_path": "Mathlib/GroupTheory/CoprodI.lean", "def_pos": [648, 5], "def_end_pos": [648, 11]}, {"full_name": "Monoid.CoprodI.NeWord.mulHead", "def_path": "Mathlib/GroupTheory/CoprodI.lean", "def_pos": [801, 5], "def_end_pos": [801, 12]}, {"full_name": "Monoid.CoprodI.NeWord.singleton", "def_path": "Mathlib/GroupTheory/CoprodI.lean", "def_pos": [647, 5], "def_end_pos": [647, 14]}]], "state_before": "case intro.intro\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type u_3\ninst\u271d\u00b3 : Monoid N\nhnontriv : Nontrivial \u03b9\nG : Type u_4\ninst\u271d\u00b2 : Group G\nH : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i)\nf : (i : \u03b9) \u2192 H i \u2192* G\nhcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i)\n\u03b1 : Type u_6\ninst\u271d : MulAction G \u03b1\nX : \u03b9 \u2192 Set \u03b1\nhXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i\ni j : \u03b9\nw : NeWord H i j\nhcard : 3 \u2264 #(H i)\nhheadtail : i \u2260 j\nh : H i\nhn1 : h \u2260 1\nhnh : h \u2260 (NeWord.head w)\u207b\u00b9\nhnot1 : h * NeWord.head w \u2260 1\n\u22a2 \u2191(\u2191lift f) (NeWord.prod w) \u2260 1", "state_after": "case intro.intro\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type u_3\ninst\u271d\u00b3 : Monoid N\nhnontriv : Nontrivial \u03b9\nG : Type u_4\ninst\u271d\u00b2 : Group G\nH : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i)\nf : (i : \u03b9) \u2192 H i \u2192* G\nhcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i)\n\u03b1 : Type u_6\ninst\u271d : MulAction G \u03b1\nX : \u03b9 \u2192 Set \u03b1\nhXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i\ni j : \u03b9\nw : NeWord H i j\nhcard : 3 \u2264 #(H i)\nhheadtail : i \u2260 j\nh : H i\nhn1 : h \u2260 1\nhnh : h \u2260 (NeWord.head w)\u207b\u00b9\nhnot1 : h * NeWord.head w \u2260 1\nw' : NeWord H i i := NeWord.append (NeWord.mulHead w h hnot1) (_ : j \u2260 i) (NeWord.singleton h\u207b\u00b9 (_ : h\u207b\u00b9 \u2260 1))\n\u22a2 \u2191(\u2191lift f) (NeWord.prod w) \u2260 1"}, {"tactic": "have hw' : lift f w'.prod \u2260 1 :=\n  lift_word_prod_nontrivial_of_head_eq_last f X hXnonempty hXdisj hpp w'", "annotated_tactic": ["have hw' : <a>lift</a> f w'.prod \u2260 1 :=\n    <a>lift_word_prod_nontrivial_of_head_eq_last</a> f X hXnonempty hXdisj hpp w'", [{"full_name": "Monoid.CoprodI.lift", "def_path": "Mathlib/GroupTheory/CoprodI.lean", "def_pos": [151, 5], "def_end_pos": [151, 9]}, {"full_name": "Monoid.CoprodI.lift_word_prod_nontrivial_of_head_eq_last", "def_path": "Mathlib/GroupTheory/CoprodI.lean", "def_pos": [907, 9], "def_end_pos": [907, 50]}]], "state_before": "case intro.intro\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type u_3\ninst\u271d\u00b3 : Monoid N\nhnontriv : Nontrivial \u03b9\nG : Type u_4\ninst\u271d\u00b2 : Group G\nH : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i)\nf : (i : \u03b9) \u2192 H i \u2192* G\nhcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i)\n\u03b1 : Type u_6\ninst\u271d : MulAction G \u03b1\nX : \u03b9 \u2192 Set \u03b1\nhXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i\ni j : \u03b9\nw : NeWord H i j\nhcard : 3 \u2264 #(H i)\nhheadtail : i \u2260 j\nh : H i\nhn1 : h \u2260 1\nhnh : h \u2260 (NeWord.head w)\u207b\u00b9\nhnot1 : h * NeWord.head w \u2260 1\nw' : NeWord H i i := NeWord.append (NeWord.mulHead w h hnot1) (_ : j \u2260 i) (NeWord.singleton h\u207b\u00b9 (_ : h\u207b\u00b9 \u2260 1))\n\u22a2 \u2191(\u2191lift f) (NeWord.prod w) \u2260 1", "state_after": "case intro.intro\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type u_3\ninst\u271d\u00b3 : Monoid N\nhnontriv : Nontrivial \u03b9\nG : Type u_4\ninst\u271d\u00b2 : Group G\nH : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i)\nf : (i : \u03b9) \u2192 H i \u2192* G\nhcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i)\n\u03b1 : Type u_6\ninst\u271d : MulAction G \u03b1\nX : \u03b9 \u2192 Set \u03b1\nhXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i\ni j : \u03b9\nw : NeWord H i j\nhcard : 3 \u2264 #(H i)\nhheadtail : i \u2260 j\nh : H i\nhn1 : h \u2260 1\nhnh : h \u2260 (NeWord.head w)\u207b\u00b9\nhnot1 : h * NeWord.head w \u2260 1\nw' : NeWord H i i := NeWord.append (NeWord.mulHead w h hnot1) (_ : j \u2260 i) (NeWord.singleton h\u207b\u00b9 (_ : h\u207b\u00b9 \u2260 1))\nhw' : \u2191(\u2191lift f) (NeWord.prod w') \u2260 1\n\u22a2 \u2191(\u2191lift f) (NeWord.prod w) \u2260 1"}, {"tactic": "intro heq1", "annotated_tactic": ["intro heq1", []], "state_before": "case intro.intro\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type u_3\ninst\u271d\u00b3 : Monoid N\nhnontriv : Nontrivial \u03b9\nG : Type u_4\ninst\u271d\u00b2 : Group G\nH : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i)\nf : (i : \u03b9) \u2192 H i \u2192* G\nhcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i)\n\u03b1 : Type u_6\ninst\u271d : MulAction G \u03b1\nX : \u03b9 \u2192 Set \u03b1\nhXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i\ni j : \u03b9\nw : NeWord H i j\nhcard : 3 \u2264 #(H i)\nhheadtail : i \u2260 j\nh : H i\nhn1 : h \u2260 1\nhnh : h \u2260 (NeWord.head w)\u207b\u00b9\nhnot1 : h * NeWord.head w \u2260 1\nw' : NeWord H i i := NeWord.append (NeWord.mulHead w h hnot1) (_ : j \u2260 i) (NeWord.singleton h\u207b\u00b9 (_ : h\u207b\u00b9 \u2260 1))\nhw' : \u2191(\u2191lift f) (NeWord.prod w') \u2260 1\n\u22a2 \u2191(\u2191lift f) (NeWord.prod w) \u2260 1", "state_after": "case intro.intro\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type u_3\ninst\u271d\u00b3 : Monoid N\nhnontriv : Nontrivial \u03b9\nG : Type u_4\ninst\u271d\u00b2 : Group G\nH : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i)\nf : (i : \u03b9) \u2192 H i \u2192* G\nhcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i)\n\u03b1 : Type u_6\ninst\u271d : MulAction G \u03b1\nX : \u03b9 \u2192 Set \u03b1\nhXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i\ni j : \u03b9\nw : NeWord H i j\nhcard : 3 \u2264 #(H i)\nhheadtail : i \u2260 j\nh : H i\nhn1 : h \u2260 1\nhnh : h \u2260 (NeWord.head w)\u207b\u00b9\nhnot1 : h * NeWord.head w \u2260 1\nw' : NeWord H i i := NeWord.append (NeWord.mulHead w h hnot1) (_ : j \u2260 i) (NeWord.singleton h\u207b\u00b9 (_ : h\u207b\u00b9 \u2260 1))\nhw' : \u2191(\u2191lift f) (NeWord.prod w') \u2260 1\nheq1 : \u2191(\u2191lift f) (NeWord.prod w) = 1\n\u22a2 False"}, {"tactic": "apply hw'", "annotated_tactic": ["apply hw'", []], "state_before": "case intro.intro\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type u_3\ninst\u271d\u00b3 : Monoid N\nhnontriv : Nontrivial \u03b9\nG : Type u_4\ninst\u271d\u00b2 : Group G\nH : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i)\nf : (i : \u03b9) \u2192 H i \u2192* G\nhcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i)\n\u03b1 : Type u_6\ninst\u271d : MulAction G \u03b1\nX : \u03b9 \u2192 Set \u03b1\nhXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i\ni j : \u03b9\nw : NeWord H i j\nhcard : 3 \u2264 #(H i)\nhheadtail : i \u2260 j\nh : H i\nhn1 : h \u2260 1\nhnh : h \u2260 (NeWord.head w)\u207b\u00b9\nhnot1 : h * NeWord.head w \u2260 1\nw' : NeWord H i i := NeWord.append (NeWord.mulHead w h hnot1) (_ : j \u2260 i) (NeWord.singleton h\u207b\u00b9 (_ : h\u207b\u00b9 \u2260 1))\nhw' : \u2191(\u2191lift f) (NeWord.prod w') \u2260 1\nheq1 : \u2191(\u2191lift f) (NeWord.prod w) = 1\n\u22a2 False", "state_after": "case intro.intro\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type u_3\ninst\u271d\u00b3 : Monoid N\nhnontriv : Nontrivial \u03b9\nG : Type u_4\ninst\u271d\u00b2 : Group G\nH : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i)\nf : (i : \u03b9) \u2192 H i \u2192* G\nhcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i)\n\u03b1 : Type u_6\ninst\u271d : MulAction G \u03b1\nX : \u03b9 \u2192 Set \u03b1\nhXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i\ni j : \u03b9\nw : NeWord H i j\nhcard : 3 \u2264 #(H i)\nhheadtail : i \u2260 j\nh : H i\nhn1 : h \u2260 1\nhnh : h \u2260 (NeWord.head w)\u207b\u00b9\nhnot1 : h * NeWord.head w \u2260 1\nw' : NeWord H i i := NeWord.append (NeWord.mulHead w h hnot1) (_ : j \u2260 i) (NeWord.singleton h\u207b\u00b9 (_ : h\u207b\u00b9 \u2260 1))\nhw' : \u2191(\u2191lift f) (NeWord.prod w') \u2260 1\nheq1 : \u2191(\u2191lift f) (NeWord.prod w) = 1\n\u22a2 \u2191(\u2191lift f) (NeWord.prod w') = 1"}, {"tactic": "simp [heq1]", "annotated_tactic": ["simp [heq1]", []], "state_before": "case intro.intro\n\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type u_3\ninst\u271d\u00b3 : Monoid N\nhnontriv : Nontrivial \u03b9\nG : Type u_4\ninst\u271d\u00b2 : Group G\nH : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i)\nf : (i : \u03b9) \u2192 H i \u2192* G\nhcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i)\n\u03b1 : Type u_6\ninst\u271d : MulAction G \u03b1\nX : \u03b9 \u2192 Set \u03b1\nhXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i\ni j : \u03b9\nw : NeWord H i j\nhcard : 3 \u2264 #(H i)\nhheadtail : i \u2260 j\nh : H i\nhn1 : h \u2260 1\nhnh : h \u2260 (NeWord.head w)\u207b\u00b9\nhnot1 : h * NeWord.head w \u2260 1\nw' : NeWord H i i := NeWord.append (NeWord.mulHead w h hnot1) (_ : j \u2260 i) (NeWord.singleton h\u207b\u00b9 (_ : h\u207b\u00b9 \u2260 1))\nhw' : \u2191(\u2191lift f) (NeWord.prod w') \u2260 1\nheq1 : \u2191(\u2191lift f) (NeWord.prod w) = 1\n\u22a2 \u2191(\u2191lift f) (NeWord.prod w') = 1", "state_after": "no goals"}, {"tactic": "rw [\u2190 div_inv_eq_mul]", "annotated_tactic": ["rw [\u2190 <a>div_inv_eq_mul</a>]", [{"full_name": "div_inv_eq_mul", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [454, 9], "def_end_pos": [454, 23]}]], "state_before": "\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type u_3\ninst\u271d\u00b3 : Monoid N\nhnontriv : Nontrivial \u03b9\nG : Type u_4\ninst\u271d\u00b2 : Group G\nH : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i)\nf : (i : \u03b9) \u2192 H i \u2192* G\nhcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i)\n\u03b1 : Type u_6\ninst\u271d : MulAction G \u03b1\nX : \u03b9 \u2192 Set \u03b1\nhXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i\ni j : \u03b9\nw : NeWord H i j\nhcard : 3 \u2264 #(H i)\nhheadtail : i \u2260 j\nh : H i\nhn1 : h \u2260 1\nhnh : h \u2260 (NeWord.head w)\u207b\u00b9\n\u22a2 h * NeWord.head w \u2260 1", "state_after": "\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type u_3\ninst\u271d\u00b3 : Monoid N\nhnontriv : Nontrivial \u03b9\nG : Type u_4\ninst\u271d\u00b2 : Group G\nH : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i)\nf : (i : \u03b9) \u2192 H i \u2192* G\nhcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i)\n\u03b1 : Type u_6\ninst\u271d : MulAction G \u03b1\nX : \u03b9 \u2192 Set \u03b1\nhXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i\ni j : \u03b9\nw : NeWord H i j\nhcard : 3 \u2264 #(H i)\nhheadtail : i \u2260 j\nh : H i\nhn1 : h \u2260 1\nhnh : h \u2260 (NeWord.head w)\u207b\u00b9\n\u22a2 h / (NeWord.head w)\u207b\u00b9 \u2260 1"}, {"tactic": "exact div_ne_one_of_ne hnh", "annotated_tactic": ["exact <a>div_ne_one_of_ne</a> hnh", [{"full_name": "div_ne_one_of_ne", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [383, 9], "def_end_pos": [383, 25]}]], "state_before": "\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type u_3\ninst\u271d\u00b3 : Monoid N\nhnontriv : Nontrivial \u03b9\nG : Type u_4\ninst\u271d\u00b2 : Group G\nH : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Group (H i)\nf : (i : \u03b9) \u2192 H i \u2192* G\nhcard\u271d : 3 \u2264 #\u03b9 \u2228 \u2203 i, 3 \u2264 #(H i)\n\u03b1 : Type u_6\ninst\u271d : MulAction G \u03b1\nX : \u03b9 \u2192 Set \u03b1\nhXnonempty : \u2200 (i : \u03b9), Set.Nonempty (X i)\nhXdisj : Pairwise fun i j => Disjoint (X i) (X j)\nhpp : Pairwise fun i j => \u2200 (h : H i), h \u2260 1 \u2192 \u2191(f i) h \u2022 X j \u2286 X i\ni j : \u03b9\nw : NeWord H i j\nhcard : 3 \u2264 #(H i)\nhheadtail : i \u2260 j\nh : H i\nhn1 : h \u2260 1\nhnh : h \u2260 (NeWord.head w)\u207b\u00b9\n\u22a2 h / (NeWord.head w)\u207b\u00b9 \u2260 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Sites/Sheaf.lean", "full_name": "CategoryTheory.Presheaf.isSheaf_of_isTerminal", "start": [266, 1], "end": [268, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.sup'_lt_iff", "start": [1187, 1], "end": [1189, 51], "traced_tactics": [{"tactic": "rw [\u2190 WithBot.coe_lt_coe, coe_sup', Finset.sup_lt_iff (WithBot.bot_lt_coe a)]", "annotated_tactic": ["rw [\u2190 <a>WithBot.coe_lt_coe</a>, <a>coe_sup'</a>, <a>Finset.sup_lt_iff</a> (<a>WithBot.bot_lt_coe</a> a)]", [{"full_name": "WithBot.coe_lt_coe", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [280, 9], "def_end_pos": [280, 19]}, {"full_name": "Finset.coe_sup'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [772, 9], "def_end_pos": [772, 17]}, {"full_name": "Finset.sup_lt_iff", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [695, 19], "def_end_pos": [695, 29]}, {"full_name": "WithBot.bot_lt_coe", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [289, 9], "def_end_pos": [289, 19]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns : Finset \u03b9\nH : Finset.Nonempty s\nf : \u03b9 \u2192 \u03b1\na : \u03b1\n\u22a2 sup' s H f < a \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 f i < a", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns : Finset \u03b9\nH : Finset.Nonempty s\nf : \u03b9 \u2192 \u03b1\na : \u03b1\n\u22a2 (\u2200 (b : \u03b9), b \u2208 s \u2192 (WithBot.some \u2218 f) b < \u2191a) \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 f i < a"}, {"tactic": "exact ball_congr (fun _ _ => WithBot.coe_lt_coe)", "annotated_tactic": ["exact <a>ball_congr</a> (fun _ _ => <a>WithBot.coe_lt_coe</a>)", [{"full_name": "ball_congr", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1039, 9], "def_end_pos": [1039, 19]}, {"full_name": "WithBot.coe_lt_coe", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [280, 9], "def_end_pos": [280, 19]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : LinearOrder \u03b1\ns : Finset \u03b9\nH : Finset.Nonempty s\nf : \u03b9 \u2192 \u03b1\na : \u03b1\n\u22a2 (\u2200 (b : \u03b9), b \u2208 s \u2192 (WithBot.some \u2218 f) b < \u2191a) \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 f i < a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Interval.lean", "full_name": "NonemptyInterval.ext_iff", "start": [60, 1], "end": [61, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean", "full_name": "TopCat.snd_openEmbedding_of_left_openEmbedding", "start": [317, 1], "end": [323, 7], "traced_tactics": [{"tactic": "convert (homeoOfIso (asIso (pullback.snd : pullback (\ud835\udfd9 S) g \u27f6 _))).openEmbedding.comp\n    (pullback_map_openEmbedding_of_open_embeddings (i\u2082 := \ud835\udfd9 Y) f g (\ud835\udfd9 _) g H\n      (homeoOfIso (Iso.refl _)).openEmbedding (\ud835\udfd9 _) rfl (by simp))", "annotated_tactic": ["convert (<a>homeoOfIso</a> (<a>asIso</a> (<a>pullback.snd</a> : <a>pullback</a> (\ud835\udfd9 S) g \u27f6 _))).openEmbedding.comp\n      (<a>pullback_map_openEmbedding_of_open_embeddings</a> (i\u2082 := \ud835\udfd9 Y) f g (\ud835\udfd9 _) g H\n        (<a>homeoOfIso</a> (<a>Iso.refl</a> _)).<a>openEmbedding</a> (\ud835\udfd9 _) <a>rfl</a> (by simp))", [{"full_name": "TopCat.homeoOfIso", "def_path": "Mathlib/Topology/Category/TopCat/Basic.lean", "def_pos": [133, 5], "def_end_pos": [133, 15]}, {"full_name": "CategoryTheory.asIso", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [307, 19], "def_end_pos": [307, 24]}, {"full_name": "CategoryTheory.Limits.pullback.snd", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [1118, 8], "def_end_pos": [1118, 20]}, {"full_name": "CategoryTheory.Limits.pullback", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [1103, 8], "def_end_pos": [1103, 16]}, {"full_name": "TopCat.pullback_map_openEmbedding_of_open_embeddings", "def_path": "Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean", "def_pos": [275, 9], "def_end_pos": [275, 54]}, {"full_name": "TopCat.homeoOfIso", "def_path": "Mathlib/Topology/Category/TopCat/Basic.lean", "def_pos": [133, 5], "def_end_pos": [133, 15]}, {"full_name": "CategoryTheory.Iso.refl", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [134, 5], "def_end_pos": [134, 9]}, {"full_name": "Homeomorph.openEmbedding", "def_path": "Mathlib/Topology/Homeomorph.lean", "def_pos": [370, 19], "def_end_pos": [370, 32]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "J : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\nH : OpenEmbedding \u2191f\ng : Y \u27f6 S\n\u22a2 OpenEmbedding \u2191pullback.snd", "state_after": "case h.e'_5.h\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\nH : OpenEmbedding \u2191f\ng : Y \u27f6 S\ne_2\u271d : (forget TopCat).obj Y = \u2191Y\n\u22a2 \u2191pullback.snd =\n    \u2191(homeoOfIso (asIso pullback.snd)) \u2218\n      \u2191(pullback.map f g (\ud835\udfd9 S) g f (\ud835\udfd9 Y) (\ud835\udfd9 S) (_ : f \u226b \ud835\udfd9 S = f \u226b \ud835\udfd9 S) (_ : g \u226b \ud835\udfd9 S = \ud835\udfd9 Y \u226b g))"}, {"tactic": "erw [\u2190 coe_comp]", "annotated_tactic": ["erw [\u2190 <a>coe_comp</a>]", [{"full_name": "CategoryTheory.coe_comp", "def_path": "Mathlib/CategoryTheory/ConcreteCategory/Basic.lean", "def_pos": [129, 9], "def_end_pos": [129, 17]}]], "state_before": "case h.e'_5.h\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\nH : OpenEmbedding \u2191f\ng : Y \u27f6 S\ne_2\u271d : (forget TopCat).obj Y = \u2191Y\n\u22a2 \u2191pullback.snd =\n    \u2191(homeoOfIso (asIso pullback.snd)) \u2218\n      \u2191(pullback.map f g (\ud835\udfd9 S) g f (\ud835\udfd9 Y) (\ud835\udfd9 S) (_ : f \u226b \ud835\udfd9 S = f \u226b \ud835\udfd9 S) (_ : g \u226b \ud835\udfd9 S = \ud835\udfd9 Y \u226b g))", "state_after": "case h.e'_5.h\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\nH : OpenEmbedding \u2191f\ng : Y \u27f6 S\ne_2\u271d : (forget TopCat).obj Y = \u2191Y\n\u22a2 \u2191pullback.snd =\n    \u2191(pullback.map f g (\ud835\udfd9 S) g f (\ud835\udfd9 Y) (\ud835\udfd9 S) (_ : f \u226b \ud835\udfd9 S = f \u226b \ud835\udfd9 S) (_ : g \u226b \ud835\udfd9 S = \ud835\udfd9 Y \u226b g) \u226b (asIso pullback.snd).hom)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_5.h\nJ : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\nH : OpenEmbedding \u2191f\ng : Y \u27f6 S\ne_2\u271d : (forget TopCat).obj Y = \u2191Y\n\u22a2 \u2191pullback.snd =\n    \u2191(pullback.map f g (\ud835\udfd9 S) g f (\ud835\udfd9 Y) (\ud835\udfd9 S) (_ : f \u226b \ud835\udfd9 S = f \u226b \ud835\udfd9 S) (_ : g \u226b \ud835\udfd9 S = \ud835\udfd9 Y \u226b g) \u226b (asIso pullback.snd).hom)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "J : Type v\ninst\u271d : SmallCategory J\nX\u271d Y\u271d Z : TopCat\nX Y S : TopCat\nf : X \u27f6 S\nH : OpenEmbedding \u2191f\ng : Y \u27f6 S\n\u22a2 g \u226b \ud835\udfd9 S = \ud835\udfd9 Y \u226b g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.coe_valMinAbs", "start": [971, 1], "end": [977, 100], "traced_tactics": [{"tactic": "rw [valMinAbs_def_pos]", "annotated_tactic": ["rw [<a>valMinAbs_def_pos</a>]", [{"full_name": "ZMod.valMinAbs_def_pos", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [963, 9], "def_end_pos": [963, 26]}]], "state_before": "k n : \u2115\nh\u271d : k = n + 1\nx : ZMod (namedPattern k (n + 1) h\u271d)\n\u22a2 \u2191(valMinAbs x) = x", "state_after": "k n : \u2115\nh\u271d : k = n + 1\nx : ZMod (namedPattern k (n + 1) h\u271d)\n\u22a2 \u2191(if val x \u2264 namedPattern k (n + 1) h\u271d / 2 then \u2191(val x) else \u2191(val x) - \u2191(namedPattern k (n + 1) h\u271d)) = x"}, {"tactic": "split_ifs", "annotated_tactic": ["split_ifs", []], "state_before": "k n : \u2115\nh\u271d : k = n + 1\nx : ZMod (namedPattern k (n + 1) h\u271d)\n\u22a2 \u2191(if val x \u2264 namedPattern k (n + 1) h\u271d / 2 then \u2191(val x) else \u2191(val x) - \u2191(namedPattern k (n + 1) h\u271d)) = x", "state_after": "case pos\nk n : \u2115\nh\u271d\u00b9 : k = n + 1\nx : ZMod (namedPattern k (n + 1) h\u271d\u00b9)\nh\u271d : val x \u2264 namedPattern k (n + 1) h\u271d\u00b9 / 2\n\u22a2 \u2191\u2191(val x) = x\n\ncase neg\nk n : \u2115\nh\u271d\u00b9 : k = n + 1\nx : ZMod (namedPattern k (n + 1) h\u271d\u00b9)\nh\u271d : \u00acval x \u2264 namedPattern k (n + 1) h\u271d\u00b9 / 2\n\u22a2 \u2191(\u2191(val x) - \u2191(namedPattern k (n + 1) h\u271d\u00b9)) = x"}, {"tactic": "rw [Int.cast_ofNat, nat_cast_zmod_val]", "annotated_tactic": ["rw [<a>Int.cast_ofNat</a>, <a>nat_cast_zmod_val</a>]", [{"full_name": "Int.cast_ofNat", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [65, 9], "def_end_pos": [65, 19]}, {"full_name": "ZMod.nat_cast_zmod_val", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [195, 9], "def_end_pos": [195, 26]}]], "state_before": "case pos\nk n : \u2115\nh\u271d\u00b9 : k = n + 1\nx : ZMod (namedPattern k (n + 1) h\u271d\u00b9)\nh\u271d : val x \u2264 namedPattern k (n + 1) h\u271d\u00b9 / 2\n\u22a2 \u2191\u2191(val x) = x", "state_after": "no goals"}, {"tactic": "rw [Int.cast_sub, Int.cast_ofNat, nat_cast_zmod_val, Int.cast_ofNat, nat_cast_self, sub_zero]", "annotated_tactic": ["rw [<a>Int.cast_sub</a>, <a>Int.cast_ofNat</a>, <a>nat_cast_zmod_val</a>, <a>Int.cast_ofNat</a>, <a>nat_cast_self</a>, <a>sub_zero</a>]", [{"full_name": "Int.cast_sub", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 17]}, {"full_name": "Int.cast_ofNat", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [65, 9], "def_end_pos": [65, 19]}, {"full_name": "ZMod.nat_cast_zmod_val", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [195, 9], "def_end_pos": [195, 26]}, {"full_name": "Int.cast_ofNat", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [65, 9], "def_end_pos": [65, 19]}, {"full_name": "ZMod.nat_cast_self", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [132, 9], "def_end_pos": [132, 22]}, {"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [339, 3], "def_end_pos": [339, 14]}]], "state_before": "case neg\nk n : \u2115\nh\u271d\u00b9 : k = n + 1\nx : ZMod (namedPattern k (n + 1) h\u271d\u00b9)\nh\u271d : \u00acval x \u2264 namedPattern k (n + 1) h\u271d\u00b9 / 2\n\u22a2 \u2191(\u2191(val x) - \u2191(namedPattern k (n + 1) h\u271d\u00b9)) = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Hom/NonUnitalAlg.lean", "full_name": "NonUnitalAlgHom.coe_injective", "start": [148, 1], "end": [149, 52], "traced_tactics": [{"tactic": "rintro \u27e8\u27e8\u27e8f, _\u27e9, _\u27e9, _\u27e9 \u27e8\u27e8\u27e8g, _\u27e9, _\u27e9, _\u27e9 h", "annotated_tactic": ["rintro \u27e8\u27e8\u27e8f, _\u27e9, _\u27e9, _\u27e9 \u27e8\u27e8\u27e8g, _\u27e9, _\u27e9, _\u27e9 h", []], "state_before": "R : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\n\u22a2 Function.Injective FunLike.coe", "state_after": "case mk.mk.mk.mk.mk.mk\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nf : A \u2192 B\nmap_smul'\u271d\u00b9 : \u2200 (m : R) (x : A), f (m \u2022 x) = m \u2022 f x\nmap_zero'\u271d\u00b9 : MulActionHom.toFun { toFun := f, map_smul' := map_smul'\u271d\u00b9 } 0 = 0\nmap_add'\u271d\u00b9 :\n  \u2200 (x y : A),\n    MulActionHom.toFun { toFun := f, map_smul' := map_smul'\u271d\u00b9 } (x + y) =\n      MulActionHom.toFun { toFun := f, map_smul' := map_smul'\u271d\u00b9 } x +\n        MulActionHom.toFun { toFun := f, map_smul' := map_smul'\u271d\u00b9 } y\nmap_mul'\u271d\u00b9 :\n  \u2200 (x y : A),\n    MulActionHom.toFun\n        { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9,\n            map_add' := map_add'\u271d\u00b9 }.toMulActionHom\n        (x * y) =\n      MulActionHom.toFun\n          { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9,\n              map_add' := map_add'\u271d\u00b9 }.toMulActionHom\n          x *\n        MulActionHom.toFun\n          { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9,\n              map_add' := map_add'\u271d\u00b9 }.toMulActionHom\n          y\ng : A \u2192 B\nmap_smul'\u271d : \u2200 (m : R) (x : A), g (m \u2022 x) = m \u2022 g x\nmap_zero'\u271d : MulActionHom.toFun { toFun := g, map_smul' := map_smul'\u271d } 0 = 0\nmap_add'\u271d :\n  \u2200 (x y : A),\n    MulActionHom.toFun { toFun := g, map_smul' := map_smul'\u271d } (x + y) =\n      MulActionHom.toFun { toFun := g, map_smul' := map_smul'\u271d } x +\n        MulActionHom.toFun { toFun := g, map_smul' := map_smul'\u271d } y\nmap_mul'\u271d :\n  \u2200 (x y : A),\n    MulActionHom.toFun\n        { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d,\n            map_add' := map_add'\u271d }.toMulActionHom\n        (x * y) =\n      MulActionHom.toFun\n          { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d,\n              map_add' := map_add'\u271d }.toMulActionHom\n          x *\n        MulActionHom.toFun\n          { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d,\n              map_add' := map_add'\u271d }.toMulActionHom\n          y\nh :\n  \u2191{\n        toDistribMulActionHom :=\n          { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9,\n            map_add' := map_add'\u271d\u00b9 },\n        map_mul' := map_mul'\u271d\u00b9 } =\n    \u2191{\n        toDistribMulActionHom :=\n          { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d },\n        map_mul' := map_mul'\u271d }\n\u22a2 {\n      toDistribMulActionHom :=\n        { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9,\n          map_add' := map_add'\u271d\u00b9 },\n      map_mul' := map_mul'\u271d\u00b9 } =\n    {\n      toDistribMulActionHom :=\n        { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d },\n      map_mul' := map_mul'\u271d }"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case mk.mk.mk.mk.mk.mk\nR : Type u\nA : Type v\nB : Type w\nC : Type w\u2081\ninst\u271d\u2076 : Monoid R\ninst\u271d\u2075 : NonUnitalNonAssocSemiring A\ninst\u271d\u2074 : DistribMulAction R A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : DistribMulAction R B\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring C\ninst\u271d : DistribMulAction R C\nf : A \u2192 B\nmap_smul'\u271d\u00b9 : \u2200 (m : R) (x : A), f (m \u2022 x) = m \u2022 f x\nmap_zero'\u271d\u00b9 : MulActionHom.toFun { toFun := f, map_smul' := map_smul'\u271d\u00b9 } 0 = 0\nmap_add'\u271d\u00b9 :\n  \u2200 (x y : A),\n    MulActionHom.toFun { toFun := f, map_smul' := map_smul'\u271d\u00b9 } (x + y) =\n      MulActionHom.toFun { toFun := f, map_smul' := map_smul'\u271d\u00b9 } x +\n        MulActionHom.toFun { toFun := f, map_smul' := map_smul'\u271d\u00b9 } y\nmap_mul'\u271d\u00b9 :\n  \u2200 (x y : A),\n    MulActionHom.toFun\n        { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9,\n            map_add' := map_add'\u271d\u00b9 }.toMulActionHom\n        (x * y) =\n      MulActionHom.toFun\n          { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9,\n              map_add' := map_add'\u271d\u00b9 }.toMulActionHom\n          x *\n        MulActionHom.toFun\n          { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9,\n              map_add' := map_add'\u271d\u00b9 }.toMulActionHom\n          y\ng : A \u2192 B\nmap_smul'\u271d : \u2200 (m : R) (x : A), g (m \u2022 x) = m \u2022 g x\nmap_zero'\u271d : MulActionHom.toFun { toFun := g, map_smul' := map_smul'\u271d } 0 = 0\nmap_add'\u271d :\n  \u2200 (x y : A),\n    MulActionHom.toFun { toFun := g, map_smul' := map_smul'\u271d } (x + y) =\n      MulActionHom.toFun { toFun := g, map_smul' := map_smul'\u271d } x +\n        MulActionHom.toFun { toFun := g, map_smul' := map_smul'\u271d } y\nmap_mul'\u271d :\n  \u2200 (x y : A),\n    MulActionHom.toFun\n        { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d,\n            map_add' := map_add'\u271d }.toMulActionHom\n        (x * y) =\n      MulActionHom.toFun\n          { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d,\n              map_add' := map_add'\u271d }.toMulActionHom\n          x *\n        MulActionHom.toFun\n          { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d,\n              map_add' := map_add'\u271d }.toMulActionHom\n          y\nh :\n  \u2191{\n        toDistribMulActionHom :=\n          { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9,\n            map_add' := map_add'\u271d\u00b9 },\n        map_mul' := map_mul'\u271d\u00b9 } =\n    \u2191{\n        toDistribMulActionHom :=\n          { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d },\n        map_mul' := map_mul'\u271d }\n\u22a2 {\n      toDistribMulActionHom :=\n        { toMulActionHom := { toFun := f, map_smul' := map_smul'\u271d\u00b9 }, map_zero' := map_zero'\u271d\u00b9,\n          map_add' := map_add'\u271d\u00b9 },\n      map_mul' := map_mul'\u271d\u00b9 } =\n    {\n      toDistribMulActionHom :=\n        { toMulActionHom := { toFun := g, map_smul' := map_smul'\u271d }, map_zero' := map_zero'\u271d, map_add' := map_add'\u271d },\n      map_mul' := map_mul'\u271d }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Limits/Constructions/EpiMono.lean", "full_name": "CategoryTheory.preserves_mono_of_preservesLimit", "start": [33, 1], "end": [37, 50], "traced_tactics": [{"tactic": "have := isLimitPullbackConeMapOfIsLimit F _ (PullbackCone.isLimitMkIdId f)", "annotated_tactic": ["have := <a>isLimitPullbackConeMapOfIsLimit</a> F _ (<a>PullbackCone.isLimitMkIdId</a> f)", [{"full_name": "CategoryTheory.Limits.isLimitPullbackConeMapOfIsLimit", "def_path": "Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean", "def_pos": [62, 5], "def_end_pos": [62, 36]}, {"full_name": "CategoryTheory.Limits.PullbackCone.isLimitMkIdId", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [703, 5], "def_end_pos": [703, 18]}]], "state_before": "C : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : PreservesLimit (cospan f f) F\ninst\u271d : Mono f\n\u22a2 Mono (F.map f)", "state_after": "C : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : PreservesLimit (cospan f f) F\ninst\u271d : Mono f\nthis :\n  let_fun this := (_ : F.map (\ud835\udfd9 X) \u226b F.map f = F.map (\ud835\udfd9 X) \u226b F.map f);\n  IsLimit (PullbackCone.mk (F.map (\ud835\udfd9 X)) (F.map (\ud835\udfd9 X)) this)\n\u22a2 Mono (F.map f)"}, {"tactic": "simp_rw [F.map_id] at this", "annotated_tactic": ["simp_rw [F.map_id] at this", []], "state_before": "C : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : PreservesLimit (cospan f f) F\ninst\u271d : Mono f\nthis :\n  let_fun this := (_ : F.map (\ud835\udfd9 X) \u226b F.map f = F.map (\ud835\udfd9 X) \u226b F.map f);\n  IsLimit (PullbackCone.mk (F.map (\ud835\udfd9 X)) (F.map (\ud835\udfd9 X)) this)\n\u22a2 Mono (F.map f)", "state_after": "C : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : PreservesLimit (cospan f f) F\ninst\u271d : Mono f\nthis : IsLimit (PullbackCone.mk (\ud835\udfd9 (F.obj X)) (\ud835\udfd9 (F.obj X)) (_ : \ud835\udfd9 (F.obj X) \u226b F.map f = \ud835\udfd9 (F.obj X) \u226b F.map f))\n\u22a2 Mono (F.map f)"}, {"tactic": "apply PullbackCone.mono_of_isLimitMkIdId _ this", "annotated_tactic": ["apply <a>PullbackCone.mono_of_isLimitMkIdId</a> _ this", [{"full_name": "CategoryTheory.Limits.PullbackCone.mono_of_isLimitMkIdId", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [713, 9], "def_end_pos": [713, 30]}]], "state_before": "C : Type u\u2081\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : PreservesLimit (cospan f f) F\ninst\u271d : Mono f\nthis : IsLimit (PullbackCone.mk (\ud835\udfd9 (F.obj X)) (\ud835\udfd9 (F.obj X)) (_ : \ud835\udfd9 (F.obj X) \u226b F.map f = \ud835\udfd9 (F.obj X) \u226b F.map f))\n\u22a2 Mono (F.map f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Basic.lean", "full_name": "MeasureTheory.martingale_of_condexp_sub_eq_zero_nat", "start": [464, 1], "end": [472, 56], "traced_tactics": [{"tactic": "refine' martingale_iff.2 \u27e8supermartingale_of_condexp_sub_nonneg_nat hadp hint fun i => _,\n  submartingale_of_condexp_sub_nonneg_nat hadp hint fun i => (hf i).symm.le\u27e9", "annotated_tactic": ["refine' <a>martingale_iff</a>.2 \u27e8<a>supermartingale_of_condexp_sub_nonneg_nat</a> hadp hint fun i => _,\n    <a>submartingale_of_condexp_sub_nonneg_nat</a> hadp hint fun i => (hf i).symm.le\u27e9", [{"full_name": "MeasureTheory.martingale_iff", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [145, 9], "def_end_pos": [145, 23]}, {"full_name": "MeasureTheory.supermartingale_of_condexp_sub_nonneg_nat", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [456, 9], "def_end_pos": [456, 50]}, {"full_name": "MeasureTheory.submartingale_of_condexp_sub_nonneg_nat", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [448, 9], "def_end_pos": [448, 48]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\n\u22a2 Martingale f \ud835\udca2 \u03bc", "state_after": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\ni : \u2115\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[f i - f (i + 1)|\u2191\ud835\udca2 i]"}, {"tactic": "rw [\u2190 neg_sub]", "annotated_tactic": ["rw [\u2190 <a>neg_sub</a>]", [{"full_name": "neg_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [400, 3], "def_end_pos": [400, 14]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\ni : \u2115\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[f i - f (i + 1)|\u2191\ud835\udca2 i]", "state_after": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\ni : \u2115\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[-(f (i + 1) - f i)|\u2191\ud835\udca2 i]"}, {"tactic": "refine' (EventuallyEq.trans _ (condexp_neg _).symm).le", "annotated_tactic": ["refine' (<a>EventuallyEq.trans</a> _ (<a>condexp_neg</a> _).<a>symm</a>).<a>le</a>", [{"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1503, 9], "def_end_pos": [1503, 27]}, {"full_name": "MeasureTheory.condexp_neg", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [316, 9], "def_end_pos": [316, 20]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}, {"full_name": "Filter.EventuallyEq.le", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1661, 9], "def_end_pos": [1661, 24]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\ni : \u2115\n\u22a2 0 \u2264\u1d50[\u03bc] \u03bc[-(f (i + 1) - f i)|\u2191\ud835\udca2 i]", "state_after": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\ni : \u2115\n\u22a2 0 =\u1d50[\u03bc] -\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]"}, {"tactic": "filter_upwards [hf i] with x hx", "annotated_tactic": ["filter_upwards [hf i] with x hx", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\ni : \u2115\n\u22a2 0 =\u1d50[\u03bc] -\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]", "state_after": "case h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\ni : \u2115\nx : \u03a9\nhx : (\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]) x = OfNat.ofNat 0 x\n\u22a2 OfNat.ofNat 0 x = (-\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]) x"}, {"tactic": "simpa only [Pi.zero_apply, Pi.neg_apply, zero_eq_neg]", "annotated_tactic": ["simpa only [<a>Pi.zero_apply</a>, <a>Pi.neg_apply</a>, <a>zero_eq_neg</a>]", [{"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Pi.neg_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [170, 3], "def_end_pos": [170, 14]}, {"full_name": "zero_eq_neg", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [428, 3], "def_end_pos": [428, 14]}]], "state_before": "case h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i)\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1d50[\u03bc] 0\ni : \u2115\nx : \u03a9\nhx : (\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]) x = OfNat.ofNat 0 x\n\u22a2 OfNat.ofNat 0 x = (-\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean", "full_name": "ProjectiveSpectrum.mem_coe_basicOpen", "start": [393, 1], "end": [395, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/LocallyConvex/WithSeminorms.lean", "full_name": "WithSeminorms.continuous_seminorm", "start": [454, 1], "end": [458, 76], "traced_tactics": [{"tactic": "have := hp.topologicalAddGroup", "annotated_tactic": ["have := hp.topologicalAddGroup", []], "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5d : Type u_3\n\ud835\udd5d\u2082 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\n\u03b9 : Type u_8\n\u03b9' : Type u_9\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Nonempty \u03b9\nt : TopologicalSpace E\np : SeminormFamily \ud835\udd5c E \u03b9\nhp : WithSeminorms p\ni : \u03b9\n\u22a2 Continuous \u2191(p i)", "state_after": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5d : Type u_3\n\ud835\udd5d\u2082 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\n\u03b9 : Type u_8\n\u03b9' : Type u_9\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Nonempty \u03b9\nt : TopologicalSpace E\np : SeminormFamily \ud835\udd5c E \u03b9\nhp : WithSeminorms p\ni : \u03b9\nthis : TopologicalAddGroup E\n\u22a2 Continuous \u2191(p i)"}, {"tactic": "exact continuous_iInf_dom (@continuous_norm _ (p i).toSeminormedAddGroup)", "annotated_tactic": ["exact <a>continuous_iInf_dom</a> (@<a>continuous_norm</a> _ (p i).<a>toSeminormedAddGroup</a>)", [{"full_name": "continuous_iInf_dom", "def_path": "Mathlib/Topology/Order.lean", "def_pos": [823, 9], "def_end_pos": [823, 28]}, {"full_name": "continuous_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1145, 36], "def_end_pos": [1145, 51]}, {"full_name": "AddGroupSeminorm.toSeminormedAddGroup", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [299, 3], "def_end_pos": [299, 14]}]], "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5d : Type u_3\n\ud835\udd5d\u2082 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\n\u03b9 : Type u_8\n\u03b9' : Type u_9\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Nonempty \u03b9\nt : TopologicalSpace E\np : SeminormFamily \ud835\udd5c E \u03b9\nhp : WithSeminorms p\ni : \u03b9\nthis : TopologicalAddGroup E\n\u22a2 Continuous \u2191(p i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Category/ModuleCat/Products.lean", "full_name": "ModuleCat.piIsoPi_inv_kernel_\u03b9", "start": [60, 1], "end": [62, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Module/Zlattice.lean", "full_name": "Zspan.fract_fract", "start": [153, 1], "end": [154, 87], "traced_tactics": [{"tactic": "classical simp only [repr_fract_apply, Int.fract_fract]", "annotated_tactic": ["classical simp only [<a>repr_fract_apply</a>, <a>Int.fract_fract</a>]", [{"full_name": "Zspan.repr_fract_apply", "def_path": "Mathlib/Algebra/Module/Zlattice.lean", "def_pos": [148, 9], "def_end_pos": [148, 25]}, {"full_name": "Int.fract_fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [1018, 9], "def_end_pos": [1018, 20]}]], "state_before": "E : Type u_1\n\u03b9 : Type u_2\nK : Type u_3\ninst\u271d\u2074 : NormedLinearOrderedField K\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace K E\nb : Basis \u03b9 K E\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Fintype \u03b9\nm : E\nx\u271d : \u03b9\n\u22a2 \u2191(\u2191b.repr (fract b (fract b m))) x\u271d = \u2191(\u2191b.repr (fract b m)) x\u271d", "state_after": "no goals"}, {"tactic": "simp only [repr_fract_apply, Int.fract_fract]", "annotated_tactic": ["simp only [<a>repr_fract_apply</a>, <a>Int.fract_fract</a>]", [{"full_name": "Zspan.repr_fract_apply", "def_path": "Mathlib/Algebra/Module/Zlattice.lean", "def_pos": [148, 9], "def_end_pos": [148, 25]}, {"full_name": "Int.fract_fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [1018, 9], "def_end_pos": [1018, 20]}]], "state_before": "E : Type u_1\n\u03b9 : Type u_2\nK : Type u_3\ninst\u271d\u2074 : NormedLinearOrderedField K\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace K E\nb : Basis \u03b9 K E\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Fintype \u03b9\nm : E\nx\u271d : \u03b9\n\u22a2 \u2191(\u2191b.repr (fract b (fract b m))) x\u271d = \u2191(\u2191b.repr (fract b m)) x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/RingDivision.lean", "full_name": "Polynomial.mem_roots_sub_C'", "start": [657, 1], "end": [658, 74], "traced_tactics": [{"tactic": "rw [mem_roots', IsRoot.def, sub_ne_zero, eval_sub, sub_eq_zero, eval_C]", "annotated_tactic": ["rw [<a>mem_roots'</a>, <a>IsRoot.def</a>, <a>sub_ne_zero</a>, <a>eval_sub</a>, <a>sub_eq_zero</a>, <a>eval_C</a>]", [{"full_name": "Polynomial.mem_roots'", "def_path": "Mathlib/Data/Polynomial/RingDivision.lean", "def_pos": [594, 9], "def_end_pos": [594, 19]}, {"full_name": "Polynomial.IsRoot.def", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [501, 9], "def_end_pos": [501, 19]}, {"full_name": "sub_ne_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [813, 3], "def_end_pos": [813, 14]}, {"full_name": "Polynomial.eval_sub", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [1286, 9], "def_end_pos": [1286, 17]}, {"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [801, 3], "def_end_pos": [801, 14]}, {"full_name": "Polynomial.eval_C", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [359, 9], "def_end_pos": [359, 15]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q p : R[X]\na x : R\n\u22a2 x \u2208 roots (p - \u2191C a) \u2194 p \u2260 \u2191C a \u2227 eval x p = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Logic/Function/Basic.lean", "full_name": "Function.Injective.hasLeftInverse", "start": [475, 1], "end": [476, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Module/LocalizedModule.lean", "full_name": "LocalizedModule.r.isEquiv", "start": [59, 1], "end": [68, 60], "traced_tactics": [{"tactic": "rw [one_smul]", "annotated_tactic": ["rw [<a>one_smul</a>]", [{"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}]], "state_before": "R : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Submonoid R\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nx\u271d : M \u00d7 { x // x \u2208 S }\nm : M\ns : { x // x \u2208 S }\n\u22a2 1 \u2022 (m, s).2 \u2022 (m, s).1 = 1 \u2022 (m, s).2 \u2022 (m, s).1", "state_after": "no goals"}, {"tactic": "use u1 * u2 * s2", "annotated_tactic": ["use u1 * u2 * s2", []], "state_before": "R : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Submonoid R\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : M \u00d7 { x // x \u2208 S }\nm1 : M\ns1 : { x // x \u2208 S }\nm2 : M\ns2 : { x // x \u2208 S }\nx\u271d\u00b9 : r S M (m1, s1) (m2, s2)\nm3 : M\ns3 : { x // x \u2208 S }\nx\u271d : r S M (m2, s2) (m3, s3)\nu1 : { x // x \u2208 S }\nhu1 : u1 \u2022 (m2, s2).2 \u2022 (m1, s1).1 = u1 \u2022 (m1, s1).2 \u2022 (m2, s2).1\nu2 : { x // x \u2208 S }\nhu2 : u2 \u2022 (m3, s3).2 \u2022 (m2, s2).1 = u2 \u2022 (m2, s2).2 \u2022 (m3, s3).1\n\u22a2 r S M (m1, s1) (m3, s3)", "state_after": "case h\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Submonoid R\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : M \u00d7 { x // x \u2208 S }\nm1 : M\ns1 : { x // x \u2208 S }\nm2 : M\ns2 : { x // x \u2208 S }\nx\u271d\u00b9 : r S M (m1, s1) (m2, s2)\nm3 : M\ns3 : { x // x \u2208 S }\nx\u271d : r S M (m2, s2) (m3, s3)\nu1 : { x // x \u2208 S }\nhu1 : u1 \u2022 (m2, s2).2 \u2022 (m1, s1).1 = u1 \u2022 (m1, s1).2 \u2022 (m2, s2).1\nu2 : { x // x \u2208 S }\nhu2 : u2 \u2022 (m3, s3).2 \u2022 (m2, s2).1 = u2 \u2022 (m2, s2).2 \u2022 (m3, s3).1\n\u22a2 (u1 * u2 * s2) \u2022 (m3, s3).2 \u2022 (m1, s1).1 = (u1 * u2 * s2) \u2022 (m1, s1).2 \u2022 (m3, s3).1"}, {"tactic": "simp only [\u2190 mul_smul, smul_assoc, mul_assoc, mul_comm, mul_left_comm] at hu1' hu2' \u22a2", "annotated_tactic": ["simp only [\u2190 <a>mul_smul</a>, <a>smul_assoc</a>, <a>mul_assoc</a>, <a>mul_comm</a>, <a>mul_left_comm</a>] at hu1' hu2' \u22a2", [{"full_name": "MulAction.mul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [112, 3], "def_end_pos": [112, 11]}, {"full_name": "smul_assoc", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 19]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "mul_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [96, 9], "def_end_pos": [96, 22]}]], "state_before": "case h\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Submonoid R\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : M \u00d7 { x // x \u2208 S }\nm1 : M\ns1 : { x // x \u2208 S }\nm2 : M\ns2 : { x // x \u2208 S }\nx\u271d\u00b9 : r S M (m1, s1) (m2, s2)\nm3 : M\ns3 : { x // x \u2208 S }\nx\u271d : r S M (m2, s2) (m3, s3)\nu1 : { x // x \u2208 S }\nhu1 : u1 \u2022 (m2, s2).2 \u2022 (m1, s1).1 = u1 \u2022 (m1, s1).2 \u2022 (m2, s2).1\nu2 : { x // x \u2208 S }\nhu2 : u2 \u2022 (m3, s3).2 \u2022 (m2, s2).1 = u2 \u2022 (m2, s2).2 \u2022 (m3, s3).1\nhu1' :\n  (fun x x_1 => x \u2022 x_1) (u2 * s3) (u1 \u2022 (m1, s1).2 \u2022 (m2, s2).1) =\n    (fun x x_1 => x \u2022 x_1) (u2 * s3) (u1 \u2022 (m2, s2).2 \u2022 (m1, s1).1)\nhu2' :\n  (fun x x_1 => x \u2022 x_1) (u1 * s1) (u2 \u2022 (m2, s2).2 \u2022 (m3, s3).1) =\n    (fun x x_1 => x \u2022 x_1) (u1 * s1) (u2 \u2022 (m3, s3).2 \u2022 (m2, s2).1)\n\u22a2 (u1 * u2 * s2) \u2022 (m3, s3).2 \u2022 (m1, s1).1 = (u1 * u2 * s2) \u2022 (m1, s1).2 \u2022 (m3, s3).1", "state_after": "case h\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Submonoid R\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : M \u00d7 { x // x \u2208 S }\nm1 : M\ns1 : { x // x \u2208 S }\nm2 : M\ns2 : { x // x \u2208 S }\nx\u271d\u00b9 : r S M (m1, s1) (m2, s2)\nm3 : M\ns3 : { x // x \u2208 S }\nx\u271d : r S M (m2, s2) (m3, s3)\nu1 : { x // x \u2208 S }\nhu1 : u1 \u2022 (m2, s2).2 \u2022 (m1, s1).1 = u1 \u2022 (m1, s1).2 \u2022 (m2, s2).1\nu2 : { x // x \u2208 S }\nhu2 : u2 \u2022 (m3, s3).2 \u2022 (m2, s2).1 = u2 \u2022 (m2, s2).2 \u2022 (m3, s3).1\nhu1' : (s1 * (s3 * (u1 * u2))) \u2022 m2 = (s2 * (s3 * (u1 * u2))) \u2022 m1\nhu2' : (s1 * (s2 * (u1 * u2))) \u2022 m3 = (s1 * (s3 * (u1 * u2))) \u2022 m2\n\u22a2 (s2 * (s3 * (u1 * u2))) \u2022 m1 = (s1 * (s2 * (u1 * u2))) \u2022 m3"}, {"tactic": "rw [hu2', hu1']", "annotated_tactic": ["rw [hu2', hu1']", []], "state_before": "case h\nR : Type u\ninst\u271d\u00b2 : CommSemiring R\nS : Submonoid R\nM : Type v\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : M \u00d7 { x // x \u2208 S }\nm1 : M\ns1 : { x // x \u2208 S }\nm2 : M\ns2 : { x // x \u2208 S }\nx\u271d\u00b9 : r S M (m1, s1) (m2, s2)\nm3 : M\ns3 : { x // x \u2208 S }\nx\u271d : r S M (m2, s2) (m3, s3)\nu1 : { x // x \u2208 S }\nhu1 : u1 \u2022 (m2, s2).2 \u2022 (m1, s1).1 = u1 \u2022 (m1, s1).2 \u2022 (m2, s2).1\nu2 : { x // x \u2208 S }\nhu2 : u2 \u2022 (m3, s3).2 \u2022 (m2, s2).1 = u2 \u2022 (m2, s2).2 \u2022 (m3, s3).1\nhu1' : (s1 * (s3 * (u1 * u2))) \u2022 m2 = (s2 * (s3 * (u1 * u2))) \u2022 m1\nhu2' : (s1 * (s2 * (u1 * u2))) \u2022 m3 = (s1 * (s3 * (u1 * u2))) \u2022 m2\n\u22a2 (s2 * (s3 * (u1 * u2))) \u2022 m1 = (s1 * (s2 * (u1 * u2))) \u2022 m3", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "full_name": "Associates.mem_factors_of_dvd", "start": [1607, 1], "end": [1610, 63], "traced_tactics": [{"tactic": "rw [factors_mk _ ha0]", "annotated_tactic": ["rw [<a>factors_mk</a> _ ha0]", [{"full_name": "Associates.factors_mk", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [1427, 9], "def_end_pos": [1427, 19]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\nhd : p \u2223 a\n\u22a2 Associates.mk p \u2208 factors (Associates.mk a)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\nhd : p \u2223 a\n\u22a2 Associates.mk p \u2208 \u2191(factors' a)"}, {"tactic": "exact mem_factorSet_some.mpr (mem_factors'_of_dvd ha0 hp hd)", "annotated_tactic": ["exact mem_factorSet_some.mpr (<a>mem_factors'_of_dvd</a> ha0 hp hd)", [{"full_name": "Associates.mem_factors'_of_dvd", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [1591, 9], "def_end_pos": [1591, 28]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ndec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p)\ninst\u271d : UniqueFactorizationMonoid \u03b1\ndec : DecidableEq \u03b1\ndec' : DecidableEq (Associates \u03b1)\na p : \u03b1\nha0 : a \u2260 0\nhp : Irreducible p\nhd : p \u2223 a\n\u22a2 Associates.mk p \u2208 \u2191(factors' a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean", "full_name": "Real.exists_extension_norm_eq", "start": [42, 1], "end": [57, 26], "traced_tactics": [{"tactic": "rcases exists_extension_of_le_sublinear \u27e8p, f\u27e9 (fun x => \u2016f\u2016 * \u2016x\u2016)\n    (fun c hc x => by simp only [norm_smul c x, Real.norm_eq_abs, abs_of_pos hc, mul_left_comm])\n    (fun x y => by rw [\u2190 left_distrib]\n      exact mul_le_mul_of_nonneg_left (norm_add_le x y) (@norm_nonneg _ _ f))\n    fun x => le_trans (le_abs_self _) (f.le_op_norm _) with \u27e8g, g_eq, g_le\u27e9", "annotated_tactic": ["rcases <a>exists_extension_of_le_sublinear</a> \u27e8p, f\u27e9 (fun x => \u2016f\u2016 * \u2016x\u2016)\n      (fun c hc x => by simp only [<a>norm_smul</a> c x, <a>Real.norm_eq_abs</a>, <a>abs_of_pos</a> hc, <a>mul_left_comm</a>])\n      (fun x y => by -- Porting note: placeholder filled here\n        rw [\u2190 <a>left_distrib</a>]\n        exact <a>mul_le_mul_of_nonneg_left</a> (<a>norm_add_le</a> x y) (@<a>norm_nonneg</a> _ _ f))\n      fun x => <a>le_trans</a> (<a>le_abs_self</a> _) (f.le_op_norm _) with \u27e8g, g_eq, g_le\u27e9", [{"full_name": "exists_extension_of_le_sublinear", "def_path": "Mathlib/Analysis/Convex/Cone/Basic.lean", "def_pos": [833, 9], "def_end_pos": [833, 41]}, {"full_name": "norm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [89, 9], "def_end_pos": [89, 18]}, {"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 20]}, {"full_name": "abs_of_pos", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [111, 9], "def_end_pos": [111, 19]}, {"full_name": "mul_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [96, 9], "def_end_pos": [96, 22]}, {"full_name": "left_distrib", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [78, 9], "def_end_pos": [78, 21]}, {"full_name": "mul_le_mul_of_nonneg_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [152, 9], "def_end_pos": [152, 34]}, {"full_name": "norm_add_le", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [482, 15], "def_end_pos": [482, 26]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_abs_self", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [57, 9], "def_end_pos": [57, 20]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\n\u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\n\u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016"}, {"tactic": "set g' :=\n  g.mkContinuous \u2016f\u2016 fun x => abs_le.2 \u27e8neg_le.1 <| g.map_neg x \u25b8 norm_neg x \u25b8 g_le (-x), g_le x\u27e9", "annotated_tactic": ["set g' :=\n    g.mkContinuous \u2016f\u2016 fun x => <a>abs_le</a>.2 \u27e8<a>neg_le</a>.1 <| g.map_neg x \u25b8 <a>norm_neg</a> x \u25b8 g_le (-x), g_le x\u27e9", [{"full_name": "abs_le", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [234, 9], "def_end_pos": [234, 15]}, {"full_name": "neg_le", "def_path": "Mathlib/Algebra/Order/Group/OrderIso.lean", "def_pos": [47, 15], "def_end_pos": [47, 21]}, {"full_name": "norm_neg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [437, 30], "def_end_pos": [437, 38]}]], "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\n\u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\n\u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016"}, {"tactic": "simp only [norm_smul c x, Real.norm_eq_abs, abs_of_pos hc, mul_left_comm]", "annotated_tactic": ["simp only [<a>norm_smul</a> c x, <a>Real.norm_eq_abs</a>, <a>abs_of_pos</a> hc, <a>mul_left_comm</a>]", [{"full_name": "norm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [89, 9], "def_end_pos": [89, 18]}, {"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 20]}, {"full_name": "abs_of_pos", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [111, 9], "def_end_pos": [111, 19]}, {"full_name": "mul_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [96, 9], "def_end_pos": [96, 22]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\nc : \u211d\nhc : 0 < c\nx : E\n\u22a2 (fun x => \u2016f\u2016 * \u2016x\u2016) (c \u2022 x) = c * (fun x => \u2016f\u2016 * \u2016x\u2016) x", "state_after": "no goals"}, {"tactic": "rw [\u2190 left_distrib]", "annotated_tactic": ["rw [\u2190 <a>left_distrib</a>]", [{"full_name": "left_distrib", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [78, 9], "def_end_pos": [78, 21]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\nx y : E\n\u22a2 (fun x => \u2016f\u2016 * \u2016x\u2016) (x + y) \u2264 (fun x => \u2016f\u2016 * \u2016x\u2016) x + (fun x => \u2016f\u2016 * \u2016x\u2016) y", "state_after": "E : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\nx y : E\n\u22a2 (fun x => \u2016f\u2016 * \u2016x\u2016) (x + y) \u2264 \u2016f\u2016 * (\u2016x\u2016 + \u2016y\u2016)"}, {"tactic": "exact mul_le_mul_of_nonneg_left (norm_add_le x y) (@norm_nonneg _ _ f)", "annotated_tactic": ["exact <a>mul_le_mul_of_nonneg_left</a> (<a>norm_add_le</a> x y) (@<a>norm_nonneg</a> _ _ f)", [{"full_name": "mul_le_mul_of_nonneg_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [152, 9], "def_end_pos": [152, 34]}, {"full_name": "norm_add_le", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [482, 15], "def_end_pos": [482, 26]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\nx y : E\n\u22a2 (fun x => \u2016f\u2016 * \u2016x\u2016) (x + y) \u2264 \u2016f\u2016 * (\u2016x\u2016 + \u2016y\u2016)", "state_after": "no goals"}, {"tactic": "refine' \u27e8g', g_eq, _\u27e9", "annotated_tactic": ["refine' \u27e8g', g_eq, _\u27e9", []], "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\n\u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\n\u22a2 \u2016g'\u2016 = \u2016f\u2016"}, {"tactic": "apply le_antisymm (g.mkContinuous_norm_le (norm_nonneg f) _)", "annotated_tactic": ["apply <a>le_antisymm</a> (g.mkContinuous_norm_le (<a>norm_nonneg</a> f) _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\n\u22a2 \u2016g'\u2016 = \u2016f\u2016", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\n\u22a2 \u2016f\u2016 \u2264 \u2016LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\u2016"}, {"tactic": "refine' f.op_norm_le_bound (norm_nonneg _) fun x => _", "annotated_tactic": ["refine' f.op_norm_le_bound (<a>norm_nonneg</a> _) fun x => _", [{"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\n\u22a2 \u2016f\u2016 \u2264 \u2016LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\u2016", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\nx : { x // x \u2208 p }\n\u22a2 \u2016\u2191f x\u2016 \u2264 \u2016LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\u2016 * \u2016x\u2016"}, {"tactic": "dsimp at g_eq", "annotated_tactic": ["dsimp at g_eq", []], "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 { domain := p, toFun := \u2191f }.domain }), \u2191g \u2191x = \u2191{ domain := p, toFun := \u2191f } x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\nx : { x // x \u2208 p }\n\u22a2 \u2016\u2191f x\u2016 \u2264 \u2016LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\u2016 * \u2016x\u2016", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\nx : { x // x \u2208 p }\n\u22a2 \u2016\u2191f x\u2016 \u2264 \u2016LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\u2016 * \u2016x\u2016"}, {"tactic": "rw [\u2190 g_eq]", "annotated_tactic": ["rw [\u2190 g_eq]", []], "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\nx : { x // x \u2208 p }\n\u22a2 \u2016\u2191f x\u2016 \u2264 \u2016LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\u2016 * \u2016x\u2016", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\nx : { x // x \u2208 p }\n\u22a2 \u2016\u2191g \u2191x\u2016 \u2264 \u2016LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\u2016 * \u2016x\u2016"}, {"tactic": "apply g'.le_op_norm", "annotated_tactic": ["apply g'.le_op_norm", []], "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\np : Subspace \u211d E\nf : { x // x \u2208 p } \u2192L[\u211d] \u211d\ng : E \u2192\u2097[\u211d] \u211d\ng_eq : \u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x\ng_le : \u2200 (x : E), \u2191g x \u2264 \u2016f\u2016 * \u2016x\u2016\ng' : E \u2192L[\u211d] \u211d := LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\nx : { x // x \u2208 p }\n\u22a2 \u2016\u2191g \u2191x\u2016 \u2264 \u2016LinearMap.mkContinuous g \u2016f\u2016 (_ : \u2200 (x : E), |\u2191g x| \u2264 \u2016f\u2016 * \u2016x\u2016)\u2016 * \u2016x\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Subsemigroup/Basic.lean", "full_name": "Subsemigroup.mem_closure", "start": [305, 1], "end": [306, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/LocallyFinite.lean", "full_name": "WithBot.Icc_bot_coe", "start": [1178, 1], "end": [1179, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/DirectSum/Internal.lean", "full_name": "DirectSum.coe_of_mul_apply_of_not_le", "start": [227, 1], "end": [236, 64], "traced_tactics": [{"tactic": "rw [coe_mul_apply_eq_dfinsupp_sum]", "annotated_tactic": ["rw [<a>coe_mul_apply_eq_dfinsupp_sum</a>]", [{"full_name": "DirectSum.coe_mul_apply_eq_dfinsupp_sum", "def_path": "Mathlib/Algebra/DirectSum/Internal.lean", "def_pos": [164, 9], "def_end_pos": [164, 38]}]], "state_before": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 \u2191(\u2191(\u2191(of (fun i => { x // x \u2208 A i }) i) r * r') n) = 0", "state_after": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum (\u2191(of (fun i => { x // x \u2208 A i }) i) r) fun i ri =>\n      DFinsupp.sum r' fun j rj => if i + j = n then \u2191ri * \u2191rj else 0) =\n    0"}, {"tactic": "apply (DFinsupp.sum_single_index _).trans", "annotated_tactic": ["apply (<a>DFinsupp.sum_single_index</a> _).<a>trans</a>", [{"full_name": "DFinsupp.sum_single_index", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [1758, 3], "def_end_pos": [1758, 14]}, {"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}]], "state_before": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum (\u2191(of (fun i => { x // x \u2208 A i }) i) r) fun i ri =>\n      DFinsupp.sum r' fun j rj => if i + j = n then \u2191ri * \u2191rj else 0) =\n    0", "state_after": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum r' fun j rj => if i + j = n then \u2191r * \u2191rj else 0) = 0\n\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum r' fun j rj => if i + j = n then \u21910 * \u2191rj else 0) = 0"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum r' fun j rj => if i + j = n then \u2191r * \u2191rj else 0) = 0\n\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum r' fun j rj => if i + j = n then \u21910 * \u2191rj else 0) = 0", "state_after": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum r' fun j rj => if i + j = n then \u21910 * \u2191rj else 0) = 0\n\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum r' fun j rj => if i + j = n then \u2191r * \u2191rj else 0) = 0"}, {"tactic": "simp_rw [ZeroMemClass.coe_zero, zero_mul, ite_self]", "annotated_tactic": ["simp_rw [<a>ZeroMemClass.coe_zero</a>, <a>zero_mul</a>, <a>ite_self</a>]", [{"full_name": "ZeroMemClass.coe_zero", "def_path": "Mathlib/GroupTheory/Submonoid/Operations.lean", "def_pos": [520, 3], "def_end_pos": [520, 14]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "ite_self", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [81, 17], "def_end_pos": [81, 25]}]], "state_before": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum r' fun j rj => if i + j = n then \u21910 * \u2191rj else 0) = 0", "state_after": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum r' fun j rj => 0) = 0"}, {"tactic": "exact DFinsupp.sum_zero", "annotated_tactic": ["exact <a>DFinsupp.sum_zero</a>", [{"full_name": "DFinsupp.sum_zero", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [1808, 3], "def_end_pos": [1808, 14]}]], "state_before": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum r' fun j rj => 0) = 0", "state_after": "no goals"}, {"tactic": "rw [DFinsupp.sum, Finset.sum_ite_of_false _ _ fun x _ H => _, Finset.sum_const_zero]", "annotated_tactic": ["rw [<a>DFinsupp.sum</a>, <a>Finset.sum_ite_of_false</a> _ _ fun x _ H => _, <a>Finset.sum_const_zero</a>]", [{"full_name": "DFinsupp.sum", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [1718, 3], "def_end_pos": [1718, 14]}, {"full_name": "Finset.sum_ite_of_false", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1002, 3], "def_end_pos": [1002, 14]}, {"full_name": "Finset.sum_const_zero", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [356, 3], "def_end_pos": [356, 14]}]], "state_before": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 (DFinsupp.sum r' fun j rj => if i + j = n then \u2191r * \u2191rj else 0) = 0", "state_after": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 \u2200 (x : \u03b9), x \u2208 DFinsupp.support r' \u2192 i + x = n \u2192 False"}, {"tactic": "exact fun x _ H => h ((self_le_add_right i x).trans_eq H)", "annotated_tactic": ["exact fun x _ H => h ((<a>self_le_add_right</a> i x).<a>trans_eq</a> H)", [{"full_name": "self_le_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [167, 3], "def_end_pos": [167, 14]}, {"full_name": "LE.le.trans_eq", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [211, 7], "def_end_pos": [211, 21]}]], "state_before": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nS : Type u_3\nR : Type u_4\ninst\u271d\u2075 : DecidableEq \u03b9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : SetLike \u03c3 R\ninst\u271d\u00b2 : AddSubmonoidClass \u03c3 R\nA : \u03b9 \u2192 \u03c3\ninst\u271d\u00b9 : CanonicallyOrderedAddCommMonoid \u03b9\ninst\u271d : SetLike.GradedMonoid A\ni : \u03b9\nr : { x // x \u2208 A i }\nr' : \u2a01 (i : \u03b9), { x // x \u2208 A i }\nn : \u03b9\nh : \u00aci \u2264 n\n\u22a2 \u2200 (x : \u03b9), x \u2208 DFinsupp.support r' \u2192 i + x = n \u2192 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Logic/Function/Basic.lean", "full_name": "eq_mpr_bijective", "start": [1013, 1], "end": [1015, 42], "traced_tactics": [{"tactic": "cases h", "annotated_tactic": ["cases h", []], "state_before": "\u03b1 \u03b2 : Sort u_1\nh : \u03b1 = \u03b2\n\u22a2 Bijective (Eq.mpr h)", "state_after": "case refl\n\u03b1 : Sort u_1\n\u22a2 Bijective (Eq.mpr (_ : \u03b1 = \u03b1))"}, {"tactic": "refine \u27e8fun _ _ \u21a6 id, fun x \u21a6 \u27e8x, rfl\u27e9\u27e9", "annotated_tactic": ["refine \u27e8fun _ _ \u21a6 <a>id</a>, fun x \u21a6 \u27e8x, <a>rfl</a>\u27e9\u27e9", [{"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case refl\n\u03b1 : Sort u_1\n\u22a2 Bijective (Eq.mpr (_ : \u03b1 = \u03b1))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "full_name": "Std.RBNode.Path.Balanced.insert", "start": [163, 11], "end": [167, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Preadditive/Basic.lean", "full_name": "CategoryTheory.Preadditive.hasCoequalizers_of_hasCokernels", "start": [457, 1], "end": [459, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Sites/InducedTopology.lean", "full_name": "CategoryTheory.LocallyCoverDense.pushforward_cover_iff_cover_pullback", "start": [61, 1], "end": [67, 16], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "C : Type u_1\ninst\u271d\u2074 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\nS : Sieve X\n\u22a2 GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S) \u2194 \u2203 T, Sieve.functorPullback G \u2191T = S", "state_after": "case mp\nC : Type u_1\ninst\u271d\u2074 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\nS : Sieve X\n\u22a2 GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S) \u2192 \u2203 T, Sieve.functorPullback G \u2191T = S\n\ncase mpr\nC : Type u_1\ninst\u271d\u2074 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\nS : Sieve X\n\u22a2 (\u2203 T, Sieve.functorPullback G \u2191T = S) \u2192 GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)"}, {"tactic": "intro hS", "annotated_tactic": ["intro hS", []], "state_before": "case mp\nC : Type u_1\ninst\u271d\u2074 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\nS : Sieve X\n\u22a2 GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S) \u2192 \u2203 T, Sieve.functorPullback G \u2191T = S", "state_after": "case mp\nC : Type u_1\ninst\u271d\u2074 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\nS : Sieve X\nhS : GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)\n\u22a2 \u2203 T, Sieve.functorPullback G \u2191T = S"}, {"tactic": "exact \u27e8\u27e8_, hS\u27e9, (Sieve.fullyFaithfulFunctorGaloisCoinsertion G X).u_l_eq S\u27e9", "annotated_tactic": ["exact \u27e8\u27e8_, hS\u27e9, (<a>Sieve.fullyFaithfulFunctorGaloisCoinsertion</a> G X).<a>u_l_eq</a> S\u27e9", [{"full_name": "CategoryTheory.Sieve.fullyFaithfulFunctorGaloisCoinsertion", "def_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "def_pos": [745, 5], "def_end_pos": [745, 42]}, {"full_name": "GaloisCoinsertion.u_l_eq", "def_path": "Mathlib/Order/GaloisConnection.lean", "def_pos": [789, 9], "def_end_pos": [789, 15]}]], "state_before": "case mp\nC : Type u_1\ninst\u271d\u2074 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\nS : Sieve X\nhS : GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)\n\u22a2 \u2203 T, Sieve.functorPullback G \u2191T = S", "state_after": "no goals"}, {"tactic": "rintro \u27e8T, rfl\u27e9", "annotated_tactic": ["rintro \u27e8T, rfl\u27e9", []], "state_before": "case mpr\nC : Type u_1\ninst\u271d\u2074 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\nS : Sieve X\n\u22a2 (\u2203 T, Sieve.functorPullback G \u2191T = S) \u2192 GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G S)", "state_after": "case mpr.intro\nC : Type u_1\ninst\u271d\u2074 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\nT : \u2191(GrothendieckTopology.sieves K (G.obj X))\n\u22a2 GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G (Sieve.functorPullback G \u2191T))"}, {"tactic": "exact Hld T", "annotated_tactic": ["exact Hld T", []], "state_before": "case mpr.intro\nC : Type u_1\ninst\u271d\u2074 : Category.{u_3, u_1} C\nD : Type u_2\ninst\u271d\u00b3 : Category.{u_4, u_2} D\nG : C \u2964 D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst\u271d\u00b2 : Category.{u, v} A\ninst\u271d\u00b9 : Full G\ninst\u271d : Faithful G\nHld : LocallyCoverDense K G\nX : C\nT : \u2191(GrothendieckTopology.sieves K (G.obj X))\n\u22a2 GrothendieckTopology.sieves K (G.obj X) (Sieve.functorPushforward G (Sieve.functorPullback G \u2191T))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.Tendsto.mono_right", "start": [3041, 1], "end": [3043, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/Eval.lean", "full_name": "Polynomial.C_mul_comp", "start": [626, 1], "end": [632, 21], "traced_tactics": [{"tactic": "induction p using Polynomial.induction_on' with\n| h_add p q hp hq =>\n  simp [hp, hq, mul_add]\n| h_monomial n b =>\n  simp [mul_assoc]", "annotated_tactic": ["induction p using <a>Polynomial.induction_on'</a> with\n  | h_add p q hp hq =>\n    simp [hp, hq, <a>mul_add</a>]\n  | h_monomial n b =>\n    simp [<a>mul_assoc</a>]", [{"full_name": "Polynomial.induction_on'", "def_path": "Mathlib/Data/Polynomial/Induction.lean", "def_pos": [63, 19], "def_end_pos": [63, 32]}, {"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 comp (\u2191C a * p) r = \u2191C a * comp p r", "state_after": "no goals"}, {"tactic": "simp [hp, hq, mul_add]", "annotated_tactic": ["simp [hp, hq, <a>mul_add</a>]", [{"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}]], "state_before": "case h_add\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q\u271d r p q : R[X]\nhp : comp (\u2191C a * p) r = \u2191C a * comp p r\nhq : comp (\u2191C a * q) r = \u2191C a * comp q r\n\u22a2 comp (\u2191C a * (p + q)) r = \u2191C a * comp (p + q) r", "state_after": "no goals"}, {"tactic": "simp [mul_assoc]", "annotated_tactic": ["simp [<a>mul_assoc</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "case h_monomial\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b\u271d : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nn : \u2115\nb : R\n\u22a2 comp (\u2191C a * \u2191(monomial n) b) r = \u2191C a * comp (\u2191(monomial n) b) r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "full_name": "EMetric.subset_countable_closure_of_compact", "start": [838, 1], "end": [842, 88], "traced_tactics": [{"tactic": "refine' subset_countable_closure_of_almost_dense_set s fun \u03b5 h\u03b5 => _", "annotated_tactic": ["refine' <a>subset_countable_closure_of_almost_dense_set</a> s fun \u03b5 h\u03b5 => _", [{"full_name": "EMetric.subset_countable_closure_of_almost_dense_set", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [781, 9], "def_end_pos": [781, 53]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\ninst\u271d : PseudoEMetricSpace \u03b1\nx y z : \u03b1\n\u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns\u271d t s : Set \u03b1\nhs : IsCompact s\n\u22a2 \u2203 t, t \u2286 s \u2227 Set.Countable t \u2227 s \u2286 closure t", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\ninst\u271d : PseudoEMetricSpace \u03b1\nx y z : \u03b1\n\u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns\u271d t s : Set \u03b1\nhs : IsCompact s\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 t, Set.Countable t \u2227 s \u2286 \u22c3 x \u2208 t, closedBall x \u03b5"}, {"tactic": "rcases totallyBounded_iff'.1 hs.totallyBounded \u03b5 h\u03b5 with \u27e8t, -, htf, hst\u27e9", "annotated_tactic": ["rcases <a>totallyBounded_iff'</a>.1 hs.totallyBounded \u03b5 h\u03b5 with \u27e8t, -, htf, hst\u27e9", [{"full_name": "EMetric.totallyBounded_iff'", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [768, 9], "def_end_pos": [768, 28]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\ninst\u271d : PseudoEMetricSpace \u03b1\nx y z : \u03b1\n\u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns\u271d t s : Set \u03b1\nhs : IsCompact s\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 t, Set.Countable t \u2227 s \u2286 \u22c3 x \u2208 t, closedBall x \u03b5", "state_after": "case intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\ninst\u271d : PseudoEMetricSpace \u03b1\nx y z : \u03b1\n\u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns\u271d t\u271d s : Set \u03b1\nhs : IsCompact s\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nt : Set \u03b1\nhtf : Set.Finite t\nhst : s \u2286 \u22c3 y \u2208 t, ball y \u03b5\n\u22a2 \u2203 t, Set.Countable t \u2227 s \u2286 \u22c3 x \u2208 t, closedBall x \u03b5"}, {"tactic": "exact \u27e8t, htf.countable, hst.trans <| iUnion\u2082_mono fun _ _ => ball_subset_closedBall\u27e9", "annotated_tactic": ["exact \u27e8t, htf.countable, hst.trans <| <a>iUnion\u2082_mono</a> fun _ _ => <a>ball_subset_closedBall</a>\u27e9", [{"full_name": "Set.iUnion\u2082_mono", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [488, 9], "def_end_pos": [488, 21]}, {"full_name": "EMetric.ball_subset_closedBall", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [554, 9], "def_end_pos": [554, 31]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\ninst\u271d : PseudoEMetricSpace \u03b1\nx y z : \u03b1\n\u03b5\u271d \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns\u271d t\u271d s : Set \u03b1\nhs : IsCompact s\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 > 0\nt : Set \u03b1\nhtf : Set.Finite t\nhst : s \u2286 \u22c3 y \u2208 t, ball y \u03b5\n\u22a2 \u2203 t, Set.Countable t \u2227 s \u2286 \u22c3 x \u2208 t, closedBall x \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Variables.lean", "full_name": "MvPolynomial.exists_degree_lt", "start": [740, 1], "end": [751, 43], "traced_tactics": [{"tactic": "contrapose! h", "annotated_tactic": ["contrapose! h", []], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn\u271d m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d : Fintype \u03c3\nf : MvPolynomial \u03c3 R\nn : \u2115\nh : totalDegree f < n * Fintype.card \u03c3\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support f\n\u22a2 \u2203 i, \u2191d i < n", "state_after": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn\u271d m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d : Fintype \u03c3\nf : MvPolynomial \u03c3 R\nn : \u2115\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support f\nh : \u2200 (i : \u03c3), n \u2264 \u2191d i\n\u22a2 n * Fintype.card \u03c3 \u2264 totalDegree f"}, {"tactic": "calc\n  n * Fintype.card \u03c3 = \u2211 _s : \u03c3, n := by\n    rw [Finset.sum_const, Nat.nsmul_eq_mul, mul_comm, Finset.card_univ]\n  _ \u2264 \u2211 s, d s := (Finset.sum_le_sum fun s _ => h s)\n  _ \u2264 d.sum fun _ e => e := by\n    rw [Finsupp.sum_fintype]\n    intros\n    rfl\n  _ \u2264 f.totalDegree := le_totalDegree hd", "annotated_tactic": ["calc\n    n * <a>Fintype.card</a> \u03c3 = \u2211 _s : \u03c3, n := by\n      rw [<a>Finset.sum_const</a>, <a>Nat.nsmul_eq_mul</a>, <a>mul_comm</a>, <a>Finset.card_univ</a>]\n    _ \u2264 \u2211 s, d s := (<a>Finset.sum_le_sum</a> fun s _ => h s)\n    _ \u2264 d.sum fun _ e => e := by\n      rw [<a>Finsupp.sum_fintype</a>]\n      intros\n      rfl\n    _ \u2264 f.totalDegree := <a>le_totalDegree</a> hd", [{"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "Finset.sum_const", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1440, 3], "def_end_pos": [1440, 14]}, {"full_name": "Nat.nsmul_eq_mul", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [106, 19], "def_end_pos": [106, 31]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "Finset.card_univ", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [247, 9], "def_end_pos": [247, 25]}, {"full_name": "Finset.sum_le_sum", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [111, 15], "def_end_pos": [111, 25]}, {"full_name": "Finsupp.sum_fintype", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}, {"full_name": "MvPolynomial.le_totalDegree", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [602, 9], "def_end_pos": [602, 23]}]], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn\u271d m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d : Fintype \u03c3\nf : MvPolynomial \u03c3 R\nn : \u2115\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support f\nh : \u2200 (i : \u03c3), n \u2264 \u2191d i\n\u22a2 n * Fintype.card \u03c3 \u2264 totalDegree f", "state_after": "no goals"}, {"tactic": "rw [Finset.sum_const, Nat.nsmul_eq_mul, mul_comm, Finset.card_univ]", "annotated_tactic": ["rw [<a>Finset.sum_const</a>, <a>Nat.nsmul_eq_mul</a>, <a>mul_comm</a>, <a>Finset.card_univ</a>]", [{"full_name": "Finset.sum_const", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1440, 3], "def_end_pos": [1440, 14]}, {"full_name": "Nat.nsmul_eq_mul", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [106, 19], "def_end_pos": [106, 31]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "Finset.card_univ", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [247, 9], "def_end_pos": [247, 25]}]], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn\u271d m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d : Fintype \u03c3\nf : MvPolynomial \u03c3 R\nn : \u2115\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support f\nh : \u2200 (i : \u03c3), n \u2264 \u2191d i\n\u22a2 n * Fintype.card \u03c3 = \u2211 _s : \u03c3, n", "state_after": "no goals"}, {"tactic": "rw [Finsupp.sum_fintype]", "annotated_tactic": ["rw [<a>Finsupp.sum_fintype</a>]", [{"full_name": "Finsupp.sum_fintype", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}]], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn\u271d m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d : Fintype \u03c3\nf : MvPolynomial \u03c3 R\nn : \u2115\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support f\nh : \u2200 (i : \u03c3), n \u2264 \u2191d i\n\u22a2 \u2211 s : \u03c3, \u2191d s \u2264 sum d fun x e => e", "state_after": "case h\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn\u271d m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d : Fintype \u03c3\nf : MvPolynomial \u03c3 R\nn : \u2115\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support f\nh : \u2200 (i : \u03c3), n \u2264 \u2191d i\n\u22a2 \u03c3 \u2192 0 = 0"}, {"tactic": "intros", "annotated_tactic": ["intros", []], "state_before": "case h\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn\u271d m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d : Fintype \u03c3\nf : MvPolynomial \u03c3 R\nn : \u2115\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support f\nh : \u2200 (i : \u03c3), n \u2264 \u2191d i\n\u22a2 \u03c3 \u2192 0 = 0", "state_after": "case h\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn\u271d m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d : Fintype \u03c3\nf : MvPolynomial \u03c3 R\nn : \u2115\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support f\nh : \u2200 (i : \u03c3), n \u2264 \u2191d i\ni\u271d : \u03c3\n\u22a2 0 = 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h\nR : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr : R\ne : \u2115\nn\u271d m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d : Fintype \u03c3\nf : MvPolynomial \u03c3 R\nn : \u2115\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 support f\nh : \u2200 (i : \u03c3), n \u2264 \u2191d i\ni\u271d : \u03c3\n\u22a2 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Basic.lean", "full_name": "Filter.Tendsto.nndist", "start": [1880, 11], "end": [1883, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/ContinuousFunction/Basic.lean", "full_name": "ContinuousMap.congr_fun", "start": [151, 11], "end": [152, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/NormedSpace/CompactOperator.lean", "full_name": "isCompactOperator_iff_isCompact_closure_image_closedBall", "start": [187, 1], "end": [192, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Logic/Relation.lean", "full_name": "Relation.TransGen.head'", "start": [375, 1], "end": [376, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/GeomSum.lean", "full_name": "Commute.geom_sum\u2082_mul", "start": [168, 11], "end": [172, 30], "traced_tactics": [{"tactic": "have := (h.sub_left (Commute.refl y)).geom_sum\u2082_mul_add n", "annotated_tactic": ["have := (h.sub_left (<a>Commute.refl</a> y)).<a>geom_sum\u2082_mul_add</a> n", [{"full_name": "Commute.refl", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [54, 19], "def_end_pos": [54, 23]}, {"full_name": "Commute.geom_sum\u2082_mul_add", "def_path": "Mathlib/Algebra/GeomSum.lean", "def_pos": [103, 19], "def_end_pos": [103, 44]}]], "state_before": "\u03b1 : Type u\ninst\u271d : Ring \u03b1\nx y : \u03b1\nh : Commute x y\nn : \u2115\n\u22a2 (\u2211 i in range n, x ^ i * y ^ (n - 1 - i)) * (x - y) = x ^ n - y ^ n", "state_after": "\u03b1 : Type u\ninst\u271d : Ring \u03b1\nx y : \u03b1\nh : Commute x y\nn : \u2115\nthis : (\u2211 i in range n, (x - y + y) ^ i * y ^ (n - 1 - i)) * (x - y) + y ^ n = (x - y + y) ^ n\n\u22a2 (\u2211 i in range n, x ^ i * y ^ (n - 1 - i)) * (x - y) = x ^ n - y ^ n"}, {"tactic": "rw [sub_add_cancel] at this", "annotated_tactic": ["rw [<a>sub_add_cancel</a>] at this", [{"full_name": "sub_add_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [728, 30], "def_end_pos": [728, 44]}]], "state_before": "\u03b1 : Type u\ninst\u271d : Ring \u03b1\nx y : \u03b1\nh : Commute x y\nn : \u2115\nthis : (\u2211 i in range n, (x - y + y) ^ i * y ^ (n - 1 - i)) * (x - y) + y ^ n = (x - y + y) ^ n\n\u22a2 (\u2211 i in range n, x ^ i * y ^ (n - 1 - i)) * (x - y) = x ^ n - y ^ n", "state_after": "\u03b1 : Type u\ninst\u271d : Ring \u03b1\nx y : \u03b1\nh : Commute x y\nn : \u2115\nthis : (\u2211 i in range n, x ^ i * y ^ (n - 1 - i)) * (x - y) + y ^ n = x ^ n\n\u22a2 (\u2211 i in range n, x ^ i * y ^ (n - 1 - i)) * (x - y) = x ^ n - y ^ n"}, {"tactic": "rw [\u2190 this, add_sub_cancel]", "annotated_tactic": ["rw [\u2190 this, <a>add_sub_cancel</a>]", [{"full_name": "add_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [739, 30], "def_end_pos": [739, 44]}]], "state_before": "\u03b1 : Type u\ninst\u271d : Ring \u03b1\nx y : \u03b1\nh : Commute x y\nn : \u2115\nthis : (\u2211 i in range n, x ^ i * y ^ (n - 1 - i)) * (x - y) + y ^ n = x ^ n\n\u22a2 (\u2211 i in range n, x ^ i * y ^ (n - 1 - i)) * (x - y) = x ^ n - y ^ n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/SimpleGraph/Clique.lean", "full_name": "SimpleGraph.cliqueFree_completeMultipartiteGraph", "start": [235, 1], "end": [241, 21], "traced_tactics": [{"tactic": "rw [cliqueFree_iff, isEmpty_iff]", "annotated_tactic": ["rw [<a>cliqueFree_iff</a>, <a>isEmpty_iff</a>]", [{"full_name": "SimpleGraph.cliqueFree_iff", "def_path": "Mathlib/Combinatorics/SimpleGraph/Clique.lean", "def_pos": [200, 9], "def_end_pos": [200, 23]}, {"full_name": "isEmpty_iff", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [94, 9], "def_end_pos": [94, 20]}]], "state_before": "\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\nV : \u03b9 \u2192 Type u_3\nhc : Fintype.card \u03b9 < n\n\u22a2 CliqueFree (completeMultipartiteGraph V) n", "state_after": "\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\nV : \u03b9 \u2192 Type u_3\nhc : Fintype.card \u03b9 < n\n\u22a2 \u22a4 \u21aag completeMultipartiteGraph V \u2192 False"}, {"tactic": "intro f", "annotated_tactic": ["intro f", []], "state_before": "\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\nV : \u03b9 \u2192 Type u_3\nhc : Fintype.card \u03b9 < n\n\u22a2 \u22a4 \u21aag completeMultipartiteGraph V \u2192 False", "state_after": "\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\nV : \u03b9 \u2192 Type u_3\nhc : Fintype.card \u03b9 < n\nf : \u22a4 \u21aag completeMultipartiteGraph V\n\u22a2 False"}, {"tactic": "obtain \u27e8v, w, hn, he\u27e9 := exists_ne_map_eq_of_card_lt (Sigma.fst \u2218 f) (by simp [hc])", "annotated_tactic": ["obtain \u27e8v, w, hn, he\u27e9 := <a>exists_ne_map_eq_of_card_lt</a> (<a>Sigma.fst</a> \u2218 f) (by simp [hc])", [{"full_name": "Fintype.exists_ne_map_eq_of_card_lt", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [518, 9], "def_end_pos": [518, 36]}, {"full_name": "Sigma.fst", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [142, 3], "def_end_pos": [142, 6]}]], "state_before": "\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\nV : \u03b9 \u2192 Type u_3\nhc : Fintype.card \u03b9 < n\nf : \u22a4 \u21aag completeMultipartiteGraph V\n\u22a2 False", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\nV : \u03b9 \u2192 Type u_3\nhc : Fintype.card \u03b9 < n\nf : \u22a4 \u21aag completeMultipartiteGraph V\nv w : Fin n\nhn : v \u2260 w\nhe : (Sigma.fst \u2218 \u2191f) v = (Sigma.fst \u2218 \u2191f) w\n\u22a2 False"}, {"tactic": "rw [\u2190 top_adj, \u2190 f.map_adj_iff, comap_Adj, top_adj] at hn", "annotated_tactic": ["rw [\u2190 <a>top_adj</a>, \u2190 f.map_adj_iff, <a>comap_Adj</a>, <a>top_adj</a>] at hn", [{"full_name": "SimpleGraph.top_adj", "def_path": "Mathlib/Combinatorics/SimpleGraph/Basic.lean", "def_pos": [398, 9], "def_end_pos": [398, 16]}, {"full_name": "SimpleGraph.comap_Adj", "def_path": "Mathlib/Combinatorics/SimpleGraph/Basic.lean", "def_pos": [1312, 3], "def_end_pos": [1312, 8]}, {"full_name": "SimpleGraph.top_adj", "def_path": "Mathlib/Combinatorics/SimpleGraph/Basic.lean", "def_pos": [398, 9], "def_end_pos": [398, 16]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\nV : \u03b9 \u2192 Type u_3\nhc : Fintype.card \u03b9 < n\nf : \u22a4 \u21aag completeMultipartiteGraph V\nv w : Fin n\nhn : v \u2260 w\nhe : (Sigma.fst \u2218 \u2191f) v = (Sigma.fst \u2218 \u2191f) w\n\u22a2 False", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\nV : \u03b9 \u2192 Type u_3\nhc : Fintype.card \u03b9 < n\nf : \u22a4 \u21aag completeMultipartiteGraph V\nv w : Fin n\nhn : (\u2191f v).fst \u2260 (\u2191f w).fst\nhe : (Sigma.fst \u2218 \u2191f) v = (Sigma.fst \u2218 \u2191f) w\n\u22a2 False"}, {"tactic": "exact absurd he hn", "annotated_tactic": ["exact <a>absurd</a> he hn", [{"full_name": "absurd", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [233, 21], "def_end_pos": [233, 27]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\nV : \u03b9 \u2192 Type u_3\nhc : Fintype.card \u03b9 < n\nf : \u22a4 \u21aag completeMultipartiteGraph V\nv w : Fin n\nhn : (\u2191f v).fst \u2260 (\u2191f w).fst\nhe : (Sigma.fst \u2218 \u2191f) v = (Sigma.fst \u2218 \u2191f) w\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simp [hc]", "annotated_tactic": ["simp [hc]", []], "state_before": "\u03b1 : Type u_1\nG H : SimpleGraph \u03b1\nm n : \u2115\ns : Finset \u03b1\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\nV : \u03b9 \u2192 Type u_3\nhc : Fintype.card \u03b9 < n\nf : \u22a4 \u21aag completeMultipartiteGraph V\n\u22a2 Fintype.card \u03b9 < Fintype.card (Fin n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "full_name": "CategoryTheory.Limits.coequalizer.condition", "start": [958, 1], "end": [959, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Basic.lean", "full_name": "Set.Ioc_subset_Ioo_union_Icc", "start": [1565, 1], "end": [1566, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/BigOperators/Basic.lean", "full_name": "List.one_le_prod_of_one_le", "start": [361, 1], "end": [368, 95], "traced_tactics": [{"tactic": "induction' l with hd tl ih", "annotated_tactic": ["induction' l with hd tl ih", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl : List M\nhl\u2081 : \u2200 (x : M), x \u2208 l \u2192 1 \u2264 x\n\u22a2 1 \u2264 prod l", "state_after": "case nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl : List M\nhl\u2081\u271d : \u2200 (x : M), x \u2208 l \u2192 1 \u2264 x\nhl\u2081 : \u2200 (x : M), x \u2208 [] \u2192 1 \u2264 x\n\u22a2 1 \u2264 prod []\n\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl : List M\nhl\u2081\u271d : \u2200 (x : M), x \u2208 l \u2192 1 \u2264 x\nhd : M\ntl : List M\nih : (\u2200 (x : M), x \u2208 tl \u2192 1 \u2264 x) \u2192 1 \u2264 prod tl\nhl\u2081 : \u2200 (x : M), x \u2208 hd :: tl \u2192 1 \u2264 x\n\u22a2 1 \u2264 prod (hd :: tl)"}, {"tactic": "rw [prod_cons]", "annotated_tactic": ["rw [<a>prod_cons</a>]", [{"full_name": "List.prod_cons", "def_path": "Mathlib/Data/List/BigOperators/Basic.lean", "def_pos": [41, 9], "def_end_pos": [41, 18]}]], "state_before": "case cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl : List M\nhl\u2081\u271d : \u2200 (x : M), x \u2208 l \u2192 1 \u2264 x\nhd : M\ntl : List M\nih : (\u2200 (x : M), x \u2208 tl \u2192 1 \u2264 x) \u2192 1 \u2264 prod tl\nhl\u2081 : \u2200 (x : M), x \u2208 hd :: tl \u2192 1 \u2264 x\n\u22a2 1 \u2264 prod (hd :: tl)", "state_after": "case cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl : List M\nhl\u2081\u271d : \u2200 (x : M), x \u2208 l \u2192 1 \u2264 x\nhd : M\ntl : List M\nih : (\u2200 (x : M), x \u2208 tl \u2192 1 \u2264 x) \u2192 1 \u2264 prod tl\nhl\u2081 : \u2200 (x : M), x \u2208 hd :: tl \u2192 1 \u2264 x\n\u22a2 1 \u2264 hd * prod tl"}, {"tactic": "exact one_le_mul (hl\u2081 hd (mem_cons_self hd tl)) (ih fun x h => hl\u2081 x (mem_cons_of_mem hd h))", "annotated_tactic": ["exact <a>one_le_mul</a> (hl\u2081 hd (<a>mem_cons_self</a> hd tl)) (ih fun x h => hl\u2081 x (<a>mem_cons_of_mem</a> hd h))", [{"full_name": "one_le_mul", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [1077, 7], "def_end_pos": [1077, 17]}, {"full_name": "List.mem_cons_self", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [66, 9], "def_end_pos": [66, 22]}, {"full_name": "List.mem_cons_of_mem", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [68, 9], "def_end_pos": [68, 24]}]], "state_before": "case cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl : List M\nhl\u2081\u271d : \u2200 (x : M), x \u2208 l \u2192 1 \u2264 x\nhd : M\ntl : List M\nih : (\u2200 (x : M), x \u2208 tl \u2192 1 \u2264 x) \u2192 1 \u2264 prod tl\nhl\u2081 : \u2200 (x : M), x \u2208 hd :: tl \u2192 1 \u2264 x\n\u22a2 1 \u2264 hd * prod tl", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl : List M\nhl\u2081\u271d : \u2200 (x : M), x \u2208 l \u2192 1 \u2264 x\nhl\u2081 : \u2200 (x : M), x \u2208 [] \u2192 1 \u2264 x\n\u22a2 1 \u2264 prod []", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "full_name": "Subgroup.mem_closure", "start": [1108, 1], "end": [1109, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "full_name": "isOpen_extChartAt_source", "start": [1258, 1], "end": [1259, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Associated.lean", "full_name": "associated_zero_iff_eq_zero", "start": [440, 1], "end": [445, 35], "traced_tactics": [{"tactic": "let \u27e8u, h\u27e9 := h.symm", "annotated_tactic": ["let \u27e8u, h\u27e9 := h.symm", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MonoidWithZero \u03b1\na : \u03b1\nh : a ~\u1d64 0\n\u22a2 a = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MonoidWithZero \u03b1\na : \u03b1\nh\u271d : a ~\u1d64 0\nu : \u03b1\u02e3\nh : 0 * \u2191u = a\n\u22a2 a = 0"}, {"tactic": "simpa using h.symm", "annotated_tactic": ["simpa using h.symm", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MonoidWithZero \u03b1\na : \u03b1\nh\u271d : a ~\u1d64 0\nu : \u03b1\u02e3\nh : 0 * \u2191u = a\n\u22a2 a = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Ring/Semiconj.lean", "full_name": "SemiconjBy.neg_left", "start": [57, 1], "end": [58, 49], "traced_tactics": [{"tactic": "simp only [SemiconjBy, h.eq, neg_mul, mul_neg]", "annotated_tactic": ["simp only [<a>SemiconjBy</a>, h.eq, <a>neg_mul</a>, <a>mul_neg</a>]", [{"full_name": "SemiconjBy", "def_path": "Mathlib/Algebra/Group/Semiconj/Defs.lean", "def_pos": [36, 5], "def_end_pos": [36, 15]}, {"full_name": "neg_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 16]}, {"full_name": "mul_neg", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [294, 9], "def_end_pos": [294, 16]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b9 : Mul R\ninst\u271d : HasDistribNeg R\na x y : R\nh : SemiconjBy a x y\n\u22a2 SemiconjBy (-a) x y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/MonoidAlgebra/Degree.lean", "full_name": "AddMonoidAlgebra.le_infDegree_add", "start": [266, 1], "end": [268, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/DFinsupp/Basic.lean", "full_name": "DFinsupp.sumAddHom_zero", "start": [2078, 1], "end": [2080, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.tendsto_add_atTop_nat", "start": [1707, 1], "end": [1708, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/InnerProductSpace/Dual.lean", "full_name": "InnerProductSpace.continuousLinearMapOfBilin_apply", "start": [176, 1], "end": [178, 93], "traced_tactics": [{"tactic": "rw [continuousLinearMapOfBilin, coe_comp', ContinuousLinearEquiv.coe_coe,\n  LinearIsometryEquiv.coe_toContinuousLinearEquiv, Function.comp_apply, toDual_symm_apply]", "annotated_tactic": ["rw [<a>continuousLinearMapOfBilin</a>, <a>coe_comp'</a>, <a>ContinuousLinearEquiv.coe_coe</a>,\n    <a>LinearIsometryEquiv.coe_toContinuousLinearEquiv</a>, <a>Function.comp_apply</a>, <a>toDual_symm_apply</a>]", [{"full_name": "InnerProductSpace.continuousLinearMapOfBilin", "def_path": "Mathlib/Analysis/InnerProductSpace/Dual.lean", "def_pos": [167, 5], "def_end_pos": [167, 31]}, {"full_name": "ContinuousLinearMap.coe_comp'", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [804, 9], "def_end_pos": [804, 18]}, {"full_name": "ContinuousLinearEquiv.coe_coe", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1873, 9], "def_end_pos": [1873, 16]}, {"full_name": "LinearIsometryEquiv.coe_toContinuousLinearEquiv", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [709, 9], "def_end_pos": [709, 36]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "InnerProductSpace.toDual_symm_apply", "def_path": "Mathlib/Analysis/InnerProductSpace/Dual.lean", "def_pos": [158, 9], "def_end_pos": [158, 26]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : CompleteSpace E\nB : E \u2192L\u22c6[\ud835\udd5c] E \u2192L[\ud835\udd5c] \ud835\udd5c\nv w : E\n\u22a2 inner (\u2191B\u266f v) w = \u2191(\u2191B v) w", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/ODE/PicardLindelof.lean", "full_name": "PicardLindelof.FunSpace.uniformInducing_toContinuousMap", "start": [193, 1], "end": [194, 8], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/SimpleGraph/Density.lean", "full_name": "SimpleGraph.edgeDensity_add_edgeDensity_compl", "start": [386, 1], "end": [391, 87], "traced_tactics": [{"tactic": "rw [edgeDensity_def, edgeDensity_def, div_add_div_same, div_eq_one_iff_eq]", "annotated_tactic": ["rw [<a>edgeDensity_def</a>, <a>edgeDensity_def</a>, <a>div_add_div_same</a>, <a>div_eq_one_iff_eq</a>]", [{"full_name": "SimpleGraph.edgeDensity_def", "def_path": "Mathlib/Combinatorics/SimpleGraph/Density.lean", "def_pos": [317, 9], "def_end_pos": [317, 24]}, {"full_name": "SimpleGraph.edgeDensity_def", "def_path": "Mathlib/Combinatorics/SimpleGraph/Density.lean", "def_pos": [317, 9], "def_end_pos": [317, 24]}, {"full_name": "div_add_div_same", "def_path": "Mathlib/Algebra/Field/Basic.lean", "def_pos": [33, 9], "def_end_pos": [33, 25]}, {"full_name": "div_eq_one_iff_eq", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [113, 9], "def_end_pos": [113, 26]}]], "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : DecidableRel G.Adj\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nhs : Finset.Nonempty s\nht : Finset.Nonempty t\nh : Disjoint s t\n\u22a2 edgeDensity G s t + edgeDensity G\u1d9c s t = 1", "state_after": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : DecidableRel G.Adj\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nhs : Finset.Nonempty s\nht : Finset.Nonempty t\nh : Disjoint s t\n\u22a2 \u2191(card (interedges G s t)) + \u2191(card (interedges G\u1d9c s t)) = \u2191(card s) * \u2191(card t)\n\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : DecidableRel G.Adj\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nhs : Finset.Nonempty s\nht : Finset.Nonempty t\nh : Disjoint s t\n\u22a2 \u2191(card s) * \u2191(card t) \u2260 0"}, {"tactic": "exact_mod_cast card_interedges_add_card_interedges_compl _ h", "annotated_tactic": ["exact_mod_cast <a>card_interedges_add_card_interedges_compl</a> _ h", [{"full_name": "SimpleGraph.card_interedges_add_card_interedges_compl", "def_path": "Mathlib/Combinatorics/SimpleGraph/Density.lean", "def_pos": [375, 9], "def_end_pos": [375, 50]}]], "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : DecidableRel G.Adj\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nhs : Finset.Nonempty s\nht : Finset.Nonempty t\nh : Disjoint s t\n\u22a2 \u2191(card (interedges G s t)) + \u2191(card (interedges G\u1d9c s t)) = \u2191(card s) * \u2191(card t)", "state_after": "no goals"}, {"tactic": "apply mul_ne_zero <;> exact_mod_cast Nat.pos_iff_ne_zero.1 (Nonempty.card_pos \u2039_\u203a)", "annotated_tactic": ["apply <a>mul_ne_zero</a> <;> exact_mod_cast <a>Nat.pos_iff_ne_zero</a>.1 (<a>Nonempty.card_pos</a> \u2039_\u203a)", [{"full_name": "mul_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [88, 9], "def_end_pos": [88, 20]}, {"full_name": "Nat.pos_iff_ne_zero", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [204, 19], "def_end_pos": [204, 34]}, {"full_name": "Finset.Nonempty.card_pos", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [78, 11], "def_end_pos": [78, 28]}]], "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : SimpleGraph \u03b1\ninst\u271d\u00b9 : DecidableRel G.Adj\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nhs : Finset.Nonempty s\nht : Finset.Nonempty t\nh : Disjoint s t\n\u22a2 \u2191(card s) * \u2191(card t) \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Prod.lean", "full_name": "Submodule.fst_map_fst", "start": [625, 1], "end": [629, 44], "traced_tactics": [{"tactic": "rw [eq_top_iff]", "annotated_tactic": ["rw [<a>eq_top_iff</a>]", [{"full_name": "eq_top_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [165, 9], "def_end_pos": [165, 19]}]], "state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 map (LinearMap.fst R M M\u2082) (fst R M M\u2082) = \u22a4", "state_after": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 \u22a4 \u2264 map (LinearMap.fst R M M\u2082) (fst R M M\u2082)"}, {"tactic": "rintro x -", "annotated_tactic": ["rintro x -", []], "state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 \u22a4 \u2264 map (LinearMap.fst R M M\u2082) (fst R M M\u2082)", "state_after": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx : M\n\u22a2 x \u2208 map (LinearMap.fst R M M\u2082) (fst R M M\u2082)"}, {"tactic": "simp only [fst, comap_bot, mem_map, mem_ker, snd_apply, fst_apply,\n  Prod.exists, exists_eq_left, exists_eq]", "annotated_tactic": ["simp only [<a>fst</a>, <a>comap_bot</a>, <a>mem_map</a>, <a>mem_ker</a>, <a>snd_apply</a>, <a>fst_apply</a>,\n    <a>Prod.exists</a>, <a>exists_eq_left</a>, <a>exists_eq</a>]", [{"full_name": "Submodule.fst", "def_path": "Mathlib/LinearAlgebra/Prod.lean", "def_pos": [604, 5], "def_end_pos": [604, 8]}, {"full_name": "Submodule.comap_bot", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [1535, 9], "def_end_pos": [1535, 18]}, {"full_name": "Submodule.mem_map", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [598, 9], "def_end_pos": [598, 16]}, {"full_name": "LinearMap.mem_ker", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 16]}, {"full_name": "LinearMap.snd_apply", "def_path": "Mathlib/LinearAlgebra/Prod.lean", "def_pos": [80, 9], "def_end_pos": [80, 18]}, {"full_name": "LinearMap.fst_apply", "def_path": "Mathlib/LinearAlgebra/Prod.lean", "def_pos": [75, 9], "def_end_pos": [75, 18]}, {"full_name": "Prod.exists", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [41, 9], "def_end_pos": [41, 17]}, {"full_name": "exists_eq_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [459, 17], "def_end_pos": [459, 31]}, {"full_name": "exists_eq", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [455, 17], "def_end_pos": [455, 26]}]], "state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx : M\n\u22a2 x \u2208 map (LinearMap.fst R M M\u2082) (fst R M M\u2082)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "full_name": "MulAction.orbit_smul_subset", "start": [88, 1], "end": [89, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "full_name": "max_mul_mul_le_max_mul_max'", "start": [365, 1], "end": [367, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Bicategory/Basic.lean", "full_name": "CategoryTheory.Bicategory.inv_whiskerLeft", "start": [228, 1], "end": [231, 67], "traced_tactics": [{"tactic": "apply IsIso.inv_eq_of_hom_inv_id", "annotated_tactic": ["apply <a>IsIso.inv_eq_of_hom_inv_id</a>", [{"full_name": "CategoryTheory.IsIso.inv_eq_of_hom_inv_id", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [344, 9], "def_end_pos": [344, 29]}]], "state_before": "B : Type u\ninst\u271d\u00b9 : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng h : b \u27f6 c\n\u03b7 : g \u27f6 h\ninst\u271d : IsIso \u03b7\n\u22a2 inv (f \u25c1 \u03b7) = f \u25c1 inv \u03b7", "state_after": "case hom_inv_id\nB : Type u\ninst\u271d\u00b9 : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng h : b \u27f6 c\n\u03b7 : g \u27f6 h\ninst\u271d : IsIso \u03b7\n\u22a2 f \u25c1 \u03b7 \u226b f \u25c1 inv \u03b7 = \ud835\udfd9 (f \u226b g)"}, {"tactic": "simp only [\u2190 whiskerLeft_comp, whiskerLeft_id, IsIso.hom_inv_id]", "annotated_tactic": ["simp only [\u2190 <a>whiskerLeft_comp</a>, <a>whiskerLeft_id</a>, <a>IsIso.hom_inv_id</a>]", [{"full_name": "CategoryTheory.Bicategory.whiskerLeft_comp", "def_path": "Mathlib/CategoryTheory/Bicategory/Basic.lean", "def_pos": [75, 3], "def_end_pos": [75, 19]}, {"full_name": "CategoryTheory.Bicategory.whiskerLeft_id", "def_path": "Mathlib/CategoryTheory/Bicategory/Basic.lean", "def_pos": [73, 3], "def_end_pos": [73, 17]}, {"full_name": "CategoryTheory.IsIso.hom_inv_id", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [273, 9], "def_end_pos": [273, 19]}]], "state_before": "case hom_inv_id\nB : Type u\ninst\u271d\u00b9 : Bicategory B\na b c d e : B\nf : a \u27f6 b\ng h : b \u27f6 c\n\u03b7 : g \u27f6 h\ninst\u271d : IsIso \u03b7\n\u22a2 f \u25c1 \u03b7 \u226b f \u25c1 inv \u03b7 = \ud835\udfd9 (f \u226b g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "full_name": "CategoryTheory.MonoidalCategory.tensor_id_comp_id_tensor", "start": [282, 1], "end": [284, 7], "traced_tactics": [{"tactic": "rw [\u2190 tensor_comp]", "annotated_tactic": ["rw [\u2190 <a>tensor_comp</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.tensor_comp", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [93, 3], "def_end_pos": [93, 14]}]], "state_before": "C\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\nf : W \u27f6 X\ng : Y \u27f6 Z\n\u22a2 (g \u2297 \ud835\udfd9 W) \u226b (\ud835\udfd9 Z \u2297 f) = g \u2297 f", "state_after": "C\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\nf : W \u27f6 X\ng : Y \u27f6 Z\n\u22a2 g \u226b \ud835\udfd9 Z \u2297 \ud835\udfd9 W \u226b f = g \u2297 f"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C\u271d : Type u\n\ud835\udc9e : Category.{v, u} C\u271d\ninst\u271d\u00b2 : MonoidalCategory C\u271d\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X Y Z : C\nf : W \u27f6 X\ng : Y \u27f6 Z\n\u22a2 g \u226b \ud835\udfd9 Z \u2297 \ud835\udfd9 W \u226b f = g \u2297 f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Span.lean", "full_name": "Submodule.span_insert_eq_span", "start": [497, 1], "end": [498, 96], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/SimpleGraph/Clique.lean", "full_name": "SimpleGraph.mem_cliqueFinset_iff", "start": [333, 1], "end": [334, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecialFunctions/Arsinh.lean", "full_name": "HasDerivAt.arsinh", "start": [301, 1], "end": [303, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/UnitaryGroup.lean", "full_name": "Matrix.UnitaryGroup.toLin'_one", "start": [142, 1], "end": [143, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Heyting/Basic.lean", "full_name": "Codisjoint.himp_le_of_right_le", "start": [451, 1], "end": [452, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/CharZero/Lemmas.lean", "full_name": "bit0_eq_zero", "start": [77, 1], "end": [78, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Fin/Basic.lean", "full_name": "Fin.pos", "start": [10, 11], "end": [11, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBNode.insert_toList_zoom_node", "start": [551, 1], "end": [553, 90], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/FieldTheory/IntermediateField.lean", "full_name": "IntermediateField.mem_mk", "start": [107, 1], "end": [109, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "full_name": "MeasureTheory.measurableEmbedding_of_fderivWithin", "start": [786, 1], "end": [790, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/InitialSeg.lean", "full_name": "InitialSeg.trans_apply", "start": [130, 1], "end": [131, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/GradedAlgebra/Radical.lean", "full_name": "Ideal.IsPrime.homogeneousCore", "start": [155, 1], "end": [163, 61], "traced_tactics": [{"tactic": "apply (Ideal.homogeneousCore \ud835\udc9c I).is_homogeneous'.isPrime_of_homogeneous_mem_or_mem", "annotated_tactic": ["apply (<a>Ideal.homogeneousCore</a> \ud835\udc9c I).is_homogeneous'.isPrime_of_homogeneous_mem_or_mem", [{"full_name": "Ideal.homogeneousCore", "def_path": "Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean", "def_pos": [183, 5], "def_end_pos": [183, 26]}]], "state_before": "\u03b9 : Type u_1\n\u03c3 : Type u_2\nA : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nh : IsPrime I\n\u22a2 IsPrime (HomogeneousIdeal.toIdeal (Ideal.homogeneousCore \ud835\udc9c I))", "state_after": "case I_ne_top\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nA : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nh : IsPrime I\n\u22a2 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule \u2260 \u22a4\n\ncase homogeneous_mem_or_mem\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nA : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nh : IsPrime I\n\u22a2 \u2200 {x y : A},\n    Homogeneous \ud835\udc9c x \u2192\n      Homogeneous \ud835\udc9c y \u2192\n        x * y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule \u2192\n          x \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule \u2228 y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule"}, {"tactic": "rintro x y hx hy hxy", "annotated_tactic": ["rintro x y hx hy hxy", []], "state_before": "case homogeneous_mem_or_mem\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nA : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nh : IsPrime I\n\u22a2 \u2200 {x y : A},\n    Homogeneous \ud835\udc9c x \u2192\n      Homogeneous \ud835\udc9c y \u2192\n        x * y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule \u2192\n          x \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule \u2228 y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule", "state_after": "case homogeneous_mem_or_mem\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nA : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nh : IsPrime I\nx y : A\nhx : Homogeneous \ud835\udc9c x\nhy : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule\n\u22a2 x \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule \u2228 y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule"}, {"tactic": "have H := h.mem_or_mem (Ideal.toIdeal_homogeneousCore_le \ud835\udc9c I hxy)", "annotated_tactic": ["have H := h.mem_or_mem (<a>Ideal.toIdeal_homogeneousCore_le</a> \ud835\udc9c I hxy)", [{"full_name": "Ideal.toIdeal_homogeneousCore_le", "def_path": "Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean", "def_pos": [192, 9], "def_end_pos": [192, 41]}]], "state_before": "case homogeneous_mem_or_mem\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nA : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nh : IsPrime I\nx y : A\nhx : Homogeneous \ud835\udc9c x\nhy : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule\n\u22a2 x \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule \u2228 y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule", "state_after": "case homogeneous_mem_or_mem\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nA : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nh : IsPrime I\nx y : A\nhx : Homogeneous \ud835\udc9c x\nhy : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule\nH : x \u2208 I \u2228 y \u2208 I\n\u22a2 x \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule \u2228 y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule"}, {"tactic": "refine' H.imp _ _", "annotated_tactic": ["refine' H.imp _ _", []], "state_before": "case homogeneous_mem_or_mem\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nA : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nh : IsPrime I\nx y : A\nhx : Homogeneous \ud835\udc9c x\nhy : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule\nH : x \u2208 I \u2228 y \u2208 I\n\u22a2 x \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule \u2228 y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule", "state_after": "case homogeneous_mem_or_mem.refine'_1\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nA : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nh : IsPrime I\nx y : A\nhx : Homogeneous \ud835\udc9c x\nhy : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule\nH : x \u2208 I \u2228 y \u2208 I\n\u22a2 x \u2208 I \u2192 x \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule\n\ncase homogeneous_mem_or_mem.refine'_2\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nA : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nh : IsPrime I\nx y : A\nhx : Homogeneous \ud835\udc9c x\nhy : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule\nH : x \u2208 I \u2228 y \u2208 I\n\u22a2 y \u2208 I \u2192 y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule"}, {"tactic": "exact ne_top_of_le_ne_top h.ne_top (Ideal.toIdeal_homogeneousCore_le \ud835\udc9c I)", "annotated_tactic": ["exact <a>ne_top_of_le_ne_top</a> h.ne_top (<a>Ideal.toIdeal_homogeneousCore_le</a> \ud835\udc9c I)", [{"full_name": "ne_top_of_le_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [194, 9], "def_end_pos": [194, 28]}, {"full_name": "Ideal.toIdeal_homogeneousCore_le", "def_path": "Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean", "def_pos": [192, 9], "def_end_pos": [192, 41]}]], "state_before": "case I_ne_top\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nA : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nh : IsPrime I\n\u22a2 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "exact Ideal.mem_homogeneousCore_of_homogeneous_of_mem hx", "annotated_tactic": ["exact <a>Ideal.mem_homogeneousCore_of_homogeneous_of_mem</a> hx", [{"full_name": "Ideal.mem_homogeneousCore_of_homogeneous_of_mem", "def_path": "Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean", "def_pos": [198, 9], "def_end_pos": [198, 56]}]], "state_before": "case homogeneous_mem_or_mem.refine'_1\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nA : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nh : IsPrime I\nx y : A\nhx : Homogeneous \ud835\udc9c x\nhy : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule\nH : x \u2208 I \u2228 y \u2208 I\n\u22a2 x \u2208 I \u2192 x \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule", "state_after": "no goals"}, {"tactic": "exact Ideal.mem_homogeneousCore_of_homogeneous_of_mem hy", "annotated_tactic": ["exact <a>Ideal.mem_homogeneousCore_of_homogeneous_of_mem</a> hy", [{"full_name": "Ideal.mem_homogeneousCore_of_homogeneous_of_mem", "def_path": "Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean", "def_pos": [198, 9], "def_end_pos": [198, 56]}]], "state_before": "case homogeneous_mem_or_mem.refine'_2\n\u03b9 : Type u_1\n\u03c3 : Type u_2\nA : Type u_3\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : LinearOrderedCancelAddCommMonoid \u03b9\ninst\u271d\u00b2 : SetLike \u03c3 A\ninst\u271d\u00b9 : AddSubmonoidClass \u03c3 A\n\ud835\udc9c : \u03b9 \u2192 \u03c3\ninst\u271d : GradedRing \ud835\udc9c\nI : Ideal A\nh : IsPrime I\nx y : A\nhx : Homogeneous \ud835\udc9c x\nhy : Homogeneous \ud835\udc9c y\nhxy : x * y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule\nH : x \u2208 I \u2228 y \u2208 I\n\u22a2 y \u2208 I \u2192 y \u2208 (Ideal.homogeneousCore \ud835\udc9c I).toSubmodule", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PNat/Basic.lean", "full_name": "PNat.div_add_mod'", "start": [371, 1], "end": [373, 24], "traced_tactics": [{"tactic": "rw [mul_comm]", "annotated_tactic": ["rw [<a>mul_comm</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "m k : \u2115+\n\u22a2 div m k * \u2191k + \u2191(mod m k) = \u2191m", "state_after": "m k : \u2115+\n\u22a2 \u2191k * div m k + \u2191(mod m k) = \u2191m"}, {"tactic": "exact div_add_mod _ _", "annotated_tactic": ["exact <a>div_add_mod</a> _ _", [{"full_name": "PNat.div_add_mod", "def_path": "Mathlib/Data/PNat/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 20]}]], "state_before": "m k : \u2115+\n\u22a2 \u2191k * div m k + \u2191(mod m k) = \u2191m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Subgroup/Pointwise.lean", "full_name": "Subgroup.mul_normal", "start": [167, 1], "end": [180, 49], "traced_tactics": [{"tactic": "rw [sup_eq_closure_mul]", "annotated_tactic": ["rw [<a>sup_eq_closure_mul</a>]", [{"full_name": "Subgroup.sup_eq_closure_mul", "def_path": "Mathlib/GroupTheory/Subgroup/Pointwise.lean", "def_pos": [148, 9], "def_end_pos": [148, 27]}]], "state_before": "\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\n\u22a2 \u2191(H \u2294 N) = \u2191H * \u2191N", "state_after": "\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\n\u22a2 \u2191(closure (\u2191H * \u2191N)) = \u2191H * \u2191N"}, {"tactic": "refine Set.Subset.antisymm (fun x hx => ?_) subset_closure", "annotated_tactic": ["refine <a>Set.Subset.antisymm</a> (fun x hx => ?_) <a>subset_closure</a>", [{"full_name": "Set.Subset.antisymm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [370, 9], "def_end_pos": [370, 24]}, {"full_name": "Subgroup.subset_closure", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [1116, 9], "def_end_pos": [1116, 23]}]], "state_before": "\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\n\u22a2 \u2191(closure (\u2191H * \u2191N)) = \u2191H * \u2191N", "state_after": "\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx : G\nhx : x \u2208 \u2191(closure (\u2191H * \u2191N))\n\u22a2 x \u2208 \u2191H * \u2191N"}, {"tactic": "refine closure_induction'' (p := fun x => x \u2208 (H : Set G) * (N : Set G)) hx ?_ ?_ ?_ ?_", "annotated_tactic": ["refine <a>closure_induction''</a> (p := fun x => x \u2208 (H : <a>Set</a> G) * (N : <a>Set</a> G)) hx ?_ ?_ ?_ ?_", [{"full_name": "Subgroup.closure_induction''", "def_path": "Mathlib/GroupTheory/Subgroup/Pointwise.lean", "def_pos": [100, 9], "def_end_pos": [100, 28]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx : G\nhx : x \u2208 \u2191(closure (\u2191H * \u2191N))\n\u22a2 x \u2208 \u2191H * \u2191N", "state_after": "case refine_1\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx : G\nhx : x \u2208 \u2191(closure (\u2191H * \u2191N))\n\u22a2 \u2200 (x : G), x \u2208 \u2191H * \u2191N \u2192 (fun x => x \u2208 \u2191H * \u2191N) x\n\ncase refine_2\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx : G\nhx : x \u2208 \u2191(closure (\u2191H * \u2191N))\n\u22a2 \u2200 (x : G), x \u2208 \u2191H * \u2191N \u2192 (fun x => x \u2208 \u2191H * \u2191N) x\u207b\u00b9\n\ncase refine_3\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx : G\nhx : x \u2208 \u2191(closure (\u2191H * \u2191N))\n\u22a2 (fun x => x \u2208 \u2191H * \u2191N) 1\n\ncase refine_4\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx : G\nhx : x \u2208 \u2191(closure (\u2191H * \u2191N))\n\u22a2 \u2200 (x y : G), (fun x => x \u2208 \u2191H * \u2191N) x \u2192 (fun x => x \u2208 \u2191H * \u2191N) y \u2192 (fun x => x \u2208 \u2191H * \u2191N) (x * y)"}, {"tactic": "rintro _ \u27e8x, y, hx, hy, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8x, y, hx, hy, rfl\u27e9", []], "state_before": "case refine_1\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx : G\nhx : x \u2208 \u2191(closure (\u2191H * \u2191N))\n\u22a2 \u2200 (x : G), x \u2208 \u2191H * \u2191N \u2192 (fun x => x \u2208 \u2191H * \u2191N) x", "state_after": "case refine_1.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx\u271d : G\nhx\u271d : x\u271d \u2208 \u2191(closure (\u2191H * \u2191N))\nx y : G\nhx : x \u2208 \u2191H\nhy : y \u2208 \u2191N\n\u22a2 (fun x x_1 => x * x_1) x y \u2208 \u2191H * \u2191N"}, {"tactic": "exact mul_mem_mul hx hy", "annotated_tactic": ["exact <a>mul_mem_mul</a> hx hy", [{"full_name": "Set.mul_mem_mul", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [342, 9], "def_end_pos": [342, 20]}]], "state_before": "case refine_1.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx\u271d : G\nhx\u271d : x\u271d \u2208 \u2191(closure (\u2191H * \u2191N))\nx y : G\nhx : x \u2208 \u2191H\nhy : y \u2208 \u2191N\n\u22a2 (fun x x_1 => x * x_1) x y \u2208 \u2191H * \u2191N", "state_after": "no goals"}, {"tactic": "rintro _ \u27e8x, y, hx, hy, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8x, y, hx, hy, rfl\u27e9", []], "state_before": "case refine_2\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx : G\nhx : x \u2208 \u2191(closure (\u2191H * \u2191N))\n\u22a2 \u2200 (x : G), x \u2208 \u2191H * \u2191N \u2192 (fun x => x \u2208 \u2191H * \u2191N) x\u207b\u00b9", "state_after": "case refine_2.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx\u271d : G\nhx\u271d : x\u271d \u2208 \u2191(closure (\u2191H * \u2191N))\nx y : G\nhx : x \u2208 \u2191H\nhy : y \u2208 \u2191N\n\u22a2 ((fun x x_1 => x * x_1) x y)\u207b\u00b9 \u2208 \u2191H * \u2191N"}, {"tactic": "simpa only [mul_inv_rev, mul_assoc, inv_inv, inv_mul_cancel_left]\n  using mul_mem_mul (inv_mem hx) (hN.conj_mem _ (inv_mem hy) x)", "annotated_tactic": ["simpa only [<a>mul_inv_rev</a>, <a>mul_assoc</a>, <a>inv_inv</a>, <a>inv_mul_cancel_left</a>]\n      using <a>mul_mem_mul</a> (<a>inv_mem</a> hx) (hN.conj_mem _ (<a>inv_mem</a> hy) x)", [{"full_name": "mul_inv_rev", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1050, 9], "def_end_pos": [1050, 20]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "inv_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [800, 9], "def_end_pos": [800, 16]}, {"full_name": "inv_mul_cancel_left", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1147, 9], "def_end_pos": [1147, 28]}, {"full_name": "Set.mul_mem_mul", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [342, 9], "def_end_pos": [342, 20]}, {"full_name": "InvMemClass.inv_mem", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [101, 3], "def_end_pos": [101, 10]}, {"full_name": "InvMemClass.inv_mem", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [101, 3], "def_end_pos": [101, 10]}]], "state_before": "case refine_2.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx\u271d : G\nhx\u271d : x\u271d \u2208 \u2191(closure (\u2191H * \u2191N))\nx y : G\nhx : x \u2208 \u2191H\nhy : y \u2208 \u2191N\n\u22a2 ((fun x x_1 => x * x_1) x y)\u207b\u00b9 \u2208 \u2191H * \u2191N", "state_after": "no goals"}, {"tactic": "exact \u27e81, 1, one_mem _, one_mem _, mul_one 1\u27e9", "annotated_tactic": ["exact \u27e81, 1, <a>one_mem</a> _, <a>one_mem</a> _, <a>mul_one</a> 1\u27e9", [{"full_name": "OneMemClass.one_mem", "def_path": "Mathlib/GroupTheory/Submonoid/Basic.lean", "def_pos": [73, 3], "def_end_pos": [73, 10]}, {"full_name": "OneMemClass.one_mem", "def_path": "Mathlib/GroupTheory/Submonoid/Basic.lean", "def_pos": [73, 3], "def_end_pos": [73, 10]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "case refine_3\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx : G\nhx : x \u2208 \u2191(closure (\u2191H * \u2191N))\n\u22a2 (fun x => x \u2208 \u2191H * \u2191N) 1", "state_after": "no goals"}, {"tactic": "rintro _ _ \u27e8x, y, hx, hy, rfl\u27e9 \u27e8x', y', hx', hy', rfl\u27e9", "annotated_tactic": ["rintro _ _ \u27e8x, y, hx, hy, rfl\u27e9 \u27e8x', y', hx', hy', rfl\u27e9", []], "state_before": "case refine_4\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx : G\nhx : x \u2208 \u2191(closure (\u2191H * \u2191N))\n\u22a2 \u2200 (x y : G), (fun x => x \u2208 \u2191H * \u2191N) x \u2192 (fun x => x \u2208 \u2191H * \u2191N) y \u2192 (fun x => x \u2208 \u2191H * \u2191N) (x * y)", "state_after": "case refine_4.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx\u271d : G\nhx\u271d : x\u271d \u2208 \u2191(closure (\u2191H * \u2191N))\nx y : G\nhx : x \u2208 \u2191H\nhy : y \u2208 \u2191N\nx' y' : G\nhx' : x' \u2208 \u2191H\nhy' : y' \u2208 \u2191N\n\u22a2 (fun x x_1 => x * x_1) x y * (fun x x_1 => x * x_1) x' y' \u2208 \u2191H * \u2191N"}, {"tactic": "refine \u27e8x * x', x'\u207b\u00b9 * y * x' * y', mul_mem hx hx', mul_mem ?_ hy', ?_\u27e9", "annotated_tactic": ["refine \u27e8x * x', x'\u207b\u00b9 * y * x' * y', <a>mul_mem</a> hx hx', <a>mul_mem</a> ?_ hy', ?_\u27e9", [{"full_name": "MulMemClass.mul_mem", "def_path": "Mathlib/GroupTheory/Subsemigroup/Basic.lean", "def_pos": [64, 3], "def_end_pos": [64, 10]}, {"full_name": "MulMemClass.mul_mem", "def_path": "Mathlib/GroupTheory/Subsemigroup/Basic.lean", "def_pos": [64, 3], "def_end_pos": [64, 10]}]], "state_before": "case refine_4.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx\u271d : G\nhx\u271d : x\u271d \u2208 \u2191(closure (\u2191H * \u2191N))\nx y : G\nhx : x \u2208 \u2191H\nhy : y \u2208 \u2191N\nx' y' : G\nhx' : x' \u2208 \u2191H\nhy' : y' \u2208 \u2191N\n\u22a2 (fun x x_1 => x * x_1) x y * (fun x x_1 => x * x_1) x' y' \u2208 \u2191H * \u2191N", "state_after": "case refine_4.intro.intro.intro.intro.intro.intro.intro.intro.refine_1\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx\u271d : G\nhx\u271d : x\u271d \u2208 \u2191(closure (\u2191H * \u2191N))\nx y : G\nhx : x \u2208 \u2191H\nhy : y \u2208 \u2191N\nx' y' : G\nhx' : x' \u2208 \u2191H\nhy' : y' \u2208 \u2191N\n\u22a2 x'\u207b\u00b9 * y * x' \u2208 N\n\ncase refine_4.intro.intro.intro.intro.intro.intro.intro.intro.refine_2\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx\u271d : G\nhx\u271d : x\u271d \u2208 \u2191(closure (\u2191H * \u2191N))\nx y : G\nhx : x \u2208 \u2191H\nhy : y \u2208 \u2191N\nx' y' : G\nhx' : x' \u2208 \u2191H\nhy' : y' \u2208 \u2191N\n\u22a2 (fun x x_1 => x * x_1) (x * x') (x'\u207b\u00b9 * y * x' * y') = (fun x x_1 => x * x_1) x y * (fun x x_1 => x * x_1) x' y'"}, {"tactic": "simpa using hN.conj_mem _ hy x'\u207b\u00b9", "annotated_tactic": ["simpa using hN.conj_mem _ hy x'\u207b\u00b9", []], "state_before": "case refine_4.intro.intro.intro.intro.intro.intro.intro.intro.refine_1\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx\u271d : G\nhx\u271d : x\u271d \u2208 \u2191(closure (\u2191H * \u2191N))\nx y : G\nhx : x \u2208 \u2191H\nhy : y \u2208 \u2191N\nx' y' : G\nhx' : x' \u2208 \u2191H\nhy' : y' \u2208 \u2191N\n\u22a2 x'\u207b\u00b9 * y * x' \u2208 N", "state_after": "no goals"}, {"tactic": "simp only [mul_assoc, mul_inv_cancel_left]", "annotated_tactic": ["simp only [<a>mul_assoc</a>, <a>mul_inv_cancel_left</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_inv_cancel_left", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1153, 9], "def_end_pos": [1153, 28]}]], "state_before": "case refine_4.intro.intro.intro.intro.intro.intro.intro.intro.refine_2\n\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nH N : Subgroup G\nhN : Normal N\nx\u271d : G\nhx\u271d : x\u271d \u2208 \u2191(closure (\u2191H * \u2191N))\nx y : G\nhx : x \u2208 \u2191H\nhy : y \u2208 \u2191N\nx' y' : G\nhx' : x' \u2208 \u2191H\nhy' : y' \u2208 \u2191N\n\u22a2 (fun x x_1 => x * x_1) (x * x') (x'\u207b\u00b9 * y * x' * y') = (fun x x_1 => x * x_1) x y * (fun x x_1 => x * x_1) x' y'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.sup_neBot", "start": [518, 9], "end": [519, 60], "traced_tactics": [{"tactic": "simp only [neBot_iff, not_and_or, Ne.def, sup_eq_bot_iff]", "annotated_tactic": ["simp only [<a>neBot_iff</a>, <a>not_and_or</a>, <a>Ne.def</a>, <a>sup_eq_bot_iff</a>]", [{"full_name": "Filter.neBot_iff", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [500, 9], "def_end_pos": [500, 18]}, {"full_name": "not_and_or", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [473, 9], "def_end_pos": [473, 19]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "sup_eq_bot_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [464, 9], "def_end_pos": [464, 23]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\n\u03b9 : Sort x\nf\u271d g\u271d : Filter \u03b1\ns t : Set \u03b1\nf g : Filter \u03b1\n\u22a2 NeBot (f \u2294 g) \u2194 NeBot f \u2228 NeBot g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/Sections.lean", "full_name": "List.rel_sections", "start": [41, 1], "end": [45, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/BigOperators/Basic.lean", "full_name": "Finset.prod_disjUnion", "start": [383, 1], "end": [387, 6], "traced_tactics": [{"tactic": "refine' Eq.trans _ (fold_disjUnion h)", "annotated_tactic": ["refine' <a>Eq.trans</a> _ (<a>fold_disjUnion</a> h)", [{"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}, {"full_name": "Finset.fold_disjUnion", "def_path": "Mathlib/Data/Finset/Fold.lean", "def_pos": [102, 9], "def_end_pos": [102, 23]}]], "state_before": "\u03b9 : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nh : Disjoint s\u2081 s\u2082\n\u22a2 \u220f x in disjUnion s\u2081 s\u2082 h, f x = (\u220f x in s\u2081, f x) * \u220f x in s\u2082, f x", "state_after": "\u03b9 : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nh : Disjoint s\u2081 s\u2082\n\u22a2 \u220f x in disjUnion s\u2081 s\u2082 h, f x = fold (fun x x_1 => x * x_1) (1 * 1) (fun x => f x) (disjUnion s\u2081 s\u2082 h)"}, {"tactic": "rw [one_mul]", "annotated_tactic": ["rw [<a>one_mul</a>]", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "\u03b9 : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nh : Disjoint s\u2081 s\u2082\n\u22a2 \u220f x in disjUnion s\u2081 s\u2082 h, f x = fold (fun x x_1 => x * x_1) (1 * 1) (fun x => f x) (disjUnion s\u2081 s\u2082 h)", "state_after": "\u03b9 : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nh : Disjoint s\u2081 s\u2082\n\u22a2 \u220f x in disjUnion s\u2081 s\u2082 h, f x = fold (fun x x_1 => x * x_1) 1 (fun x => f x) (disjUnion s\u2081 s\u2082 h)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b9 : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf g : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nh : Disjoint s\u2081 s\u2082\n\u22a2 \u220f x in disjUnion s\u2081 s\u2082 h, f x = fold (fun x x_1 => x * x_1) 1 (fun x => f x) (disjUnion s\u2081 s\u2082 h)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Algebra/Operations.lean", "full_name": "Submodule.mul_comm", "start": [597, 11], "end": [599, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Alternating/Basic.lean", "full_name": "AlternatingMap.curryLeft_smul", "start": [1031, 1], "end": [1033, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/RingedSpace/OpenImmersion.lean", "full_name": "AlgebraicGeometry.SheafedSpace.IsOpenImmersion.of_stalk_iso", "start": [836, 1], "end": [851, 73], "traced_tactics": [{"tactic": "have h := TopCat.Presheaf.app_isIso_of_stalkFunctor_map_iso\n    (show Y.sheaf \u27f6 (TopCat.Sheaf.pushforward _ f.base).obj X.sheaf from \u27e8f.c\u27e9)", "annotated_tactic": ["have h := <a>TopCat.Presheaf.app_isIso_of_stalkFunctor_map_iso</a>\n          (show Y.sheaf \u27f6 (<a>TopCat.Sheaf.pushforward</a> _ f.base).<a>obj</a> X.sheaf from \u27e8f.c\u27e9)", [{"full_name": "TopCat.Presheaf.app_isIso_of_stalkFunctor_map_iso", "def_path": "Mathlib/Topology/Sheaves/Stalks.lean", "def_pos": [598, 9], "def_end_pos": [598, 42]}, {"full_name": "TopCat.Sheaf.pushforward", "def_path": "Mathlib/Topology/Sheaves/Functors.lean", "def_pos": [63, 5], "def_end_pos": [63, 16]}, {"full_name": "Prefunctor.obj", "def_path": "Mathlib/Combinatorics/Quiver/Basic.lean", "def_pos": [62, 3], "def_end_pos": [62, 6]}]], "state_before": "C : Type u_1\ninst\u271d\u2076 : Category.{u_2, u_1} C\ninst\u271d\u2075 : HasLimits C\ninst\u271d\u2074 : HasColimits C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C)\ninst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C)\ninst\u271d : PreservesFilteredColimits (CategoryTheory.forget C)\nX Y : SheafedSpace C\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f.base\nH : \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f x)\nU : Opens \u2191\u2191X.toPresheafedSpace\n\u22a2 IsIso (f.c.app (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f.base)).obj U)))", "state_after": "C : Type u_1\ninst\u271d\u2076 : Category.{u_2, u_1} C\ninst\u271d\u2075 : HasLimits C\ninst\u271d\u2074 : HasColimits C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C)\ninst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C)\ninst\u271d : PreservesFilteredColimits (CategoryTheory.forget C)\nX Y : SheafedSpace C\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f.base\nH : \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f x)\nU : Opens \u2191\u2191X.toPresheafedSpace\nh :\n  \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace)\n    [inst :\n      \u2200 (x : { x // x \u2208 U }),\n        IsIso\n          ((TopCat.Presheaf.stalkFunctor C \u2191x).map\n            (let_fun this := { val := f.c };\n              this).val)],\n    IsIso\n      ((let_fun this := { val := f.c };\n            this).val.app\n        (op U))\n\u22a2 IsIso (f.c.app (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f.base)).obj U)))"}, {"tactic": "refine @h _ ?_", "annotated_tactic": ["refine @h _ ?_", []], "state_before": "C : Type u_1\ninst\u271d\u2076 : Category.{u_2, u_1} C\ninst\u271d\u2075 : HasLimits C\ninst\u271d\u2074 : HasColimits C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C)\ninst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C)\ninst\u271d : PreservesFilteredColimits (CategoryTheory.forget C)\nX Y : SheafedSpace C\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f.base\nH : \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f x)\nU : Opens \u2191\u2191X.toPresheafedSpace\nh :\n  \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace)\n    [inst :\n      \u2200 (x : { x // x \u2208 U }),\n        IsIso\n          ((TopCat.Presheaf.stalkFunctor C \u2191x).map\n            (let_fun this := { val := f.c };\n              this).val)],\n    IsIso\n      ((let_fun this := { val := f.c };\n            this).val.app\n        (op U))\n\u22a2 IsIso (f.c.app (op ((IsOpenMap.functor (_ : IsOpenMap \u2191f.base)).obj U)))", "state_after": "C : Type u_1\ninst\u271d\u2076 : Category.{u_2, u_1} C\ninst\u271d\u2075 : HasLimits C\ninst\u271d\u2074 : HasColimits C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C)\ninst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C)\ninst\u271d : PreservesFilteredColimits (CategoryTheory.forget C)\nX Y : SheafedSpace C\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f.base\nH : \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f x)\nU : Opens \u2191\u2191X.toPresheafedSpace\nh :\n  \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace)\n    [inst :\n      \u2200 (x : { x // x \u2208 U }),\n        IsIso\n          ((TopCat.Presheaf.stalkFunctor C \u2191x).map\n            (let_fun this := { val := f.c };\n              this).val)],\n    IsIso\n      ((let_fun this := { val := f.c };\n            this).val.app\n        (op U))\n\u22a2 \u2200 (x : { x // x \u2208 (IsOpenMap.functor (_ : IsOpenMap \u2191f.base)).obj U }),\n    IsIso\n      ((TopCat.Presheaf.stalkFunctor C \u2191x).map\n        (let_fun this := { val := f.c };\n          this).val)"}, {"tactic": "rintro \u27e8_, y, hy, rfl\u27e9", "annotated_tactic": ["rintro \u27e8_, y, hy, rfl\u27e9", []], "state_before": "C : Type u_1\ninst\u271d\u2076 : Category.{u_2, u_1} C\ninst\u271d\u2075 : HasLimits C\ninst\u271d\u2074 : HasColimits C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C)\ninst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C)\ninst\u271d : PreservesFilteredColimits (CategoryTheory.forget C)\nX Y : SheafedSpace C\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f.base\nH : \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f x)\nU : Opens \u2191\u2191X.toPresheafedSpace\nh :\n  \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace)\n    [inst :\n      \u2200 (x : { x // x \u2208 U }),\n        IsIso\n          ((TopCat.Presheaf.stalkFunctor C \u2191x).map\n            (let_fun this := { val := f.c };\n              this).val)],\n    IsIso\n      ((let_fun this := { val := f.c };\n            this).val.app\n        (op U))\n\u22a2 \u2200 (x : { x // x \u2208 (IsOpenMap.functor (_ : IsOpenMap \u2191f.base)).obj U }),\n    IsIso\n      ((TopCat.Presheaf.stalkFunctor C \u2191x).map\n        (let_fun this := { val := f.c };\n          this).val)", "state_after": "case mk.intro.intro\nC : Type u_1\ninst\u271d\u2076 : Category.{u_2, u_1} C\ninst\u271d\u2075 : HasLimits C\ninst\u271d\u2074 : HasColimits C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C)\ninst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C)\ninst\u271d : PreservesFilteredColimits (CategoryTheory.forget C)\nX Y : SheafedSpace C\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f.base\nH : \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f x)\nU : Opens \u2191\u2191X.toPresheafedSpace\nh :\n  \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace)\n    [inst :\n      \u2200 (x : { x // x \u2208 U }),\n        IsIso\n          ((TopCat.Presheaf.stalkFunctor C \u2191x).map\n            (let_fun this := { val := f.c };\n              this).val)],\n    IsIso\n      ((let_fun this := { val := f.c };\n            this).val.app\n        (op U))\ny : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhy : y \u2208 \u2191U\n\u22a2 IsIso\n    ((TopCat.Presheaf.stalkFunctor C \u2191{ val := \u2191f.base y, property := (_ : \u2203 a, a \u2208 \u2191U \u2227 \u2191f.base a = \u2191f.base y) }).map\n      (let_fun this := { val := f.c };\n        this).val)"}, {"tactic": "specialize H y", "annotated_tactic": ["specialize H y", []], "state_before": "case mk.intro.intro\nC : Type u_1\ninst\u271d\u2076 : Category.{u_2, u_1} C\ninst\u271d\u2075 : HasLimits C\ninst\u271d\u2074 : HasColimits C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C)\ninst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C)\ninst\u271d : PreservesFilteredColimits (CategoryTheory.forget C)\nX Y : SheafedSpace C\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f.base\nH : \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f x)\nU : Opens \u2191\u2191X.toPresheafedSpace\nh :\n  \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace)\n    [inst :\n      \u2200 (x : { x // x \u2208 U }),\n        IsIso\n          ((TopCat.Presheaf.stalkFunctor C \u2191x).map\n            (let_fun this := { val := f.c };\n              this).val)],\n    IsIso\n      ((let_fun this := { val := f.c };\n            this).val.app\n        (op U))\ny : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhy : y \u2208 \u2191U\n\u22a2 IsIso\n    ((TopCat.Presheaf.stalkFunctor C \u2191{ val := \u2191f.base y, property := (_ : \u2203 a, a \u2208 \u2191U \u2227 \u2191f.base a = \u2191f.base y) }).map\n      (let_fun this := { val := f.c };\n        this).val)", "state_after": "case mk.intro.intro\nC : Type u_1\ninst\u271d\u2076 : Category.{u_2, u_1} C\ninst\u271d\u2075 : HasLimits C\ninst\u271d\u2074 : HasColimits C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C)\ninst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C)\ninst\u271d : PreservesFilteredColimits (CategoryTheory.forget C)\nX Y : SheafedSpace C\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f.base\nU : Opens \u2191\u2191X.toPresheafedSpace\nh :\n  \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace)\n    [inst :\n      \u2200 (x : { x // x \u2208 U }),\n        IsIso\n          ((TopCat.Presheaf.stalkFunctor C \u2191x).map\n            (let_fun this := { val := f.c };\n              this).val)],\n    IsIso\n      ((let_fun this := { val := f.c };\n            this).val.app\n        (op U))\ny : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhy : y \u2208 \u2191U\nH : IsIso (PresheafedSpace.stalkMap f y)\n\u22a2 IsIso\n    ((TopCat.Presheaf.stalkFunctor C \u2191{ val := \u2191f.base y, property := (_ : \u2203 a, a \u2208 \u2191U \u2227 \u2191f.base a = \u2191f.base y) }).map\n      (let_fun this := { val := f.c };\n        this).val)"}, {"tactic": "delta PresheafedSpace.stalkMap at H", "annotated_tactic": ["delta <a>PresheafedSpace.stalkMap</a> at H", [{"full_name": "AlgebraicGeometry.PresheafedSpace.stalkMap", "def_path": "Mathlib/Geometry/RingedSpace/Stalks.lean", "def_pos": [49, 5], "def_end_pos": [49, 13]}]], "state_before": "case mk.intro.intro\nC : Type u_1\ninst\u271d\u2076 : Category.{u_2, u_1} C\ninst\u271d\u2075 : HasLimits C\ninst\u271d\u2074 : HasColimits C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C)\ninst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C)\ninst\u271d : PreservesFilteredColimits (CategoryTheory.forget C)\nX Y : SheafedSpace C\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f.base\nU : Opens \u2191\u2191X.toPresheafedSpace\nh :\n  \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace)\n    [inst :\n      \u2200 (x : { x // x \u2208 U }),\n        IsIso\n          ((TopCat.Presheaf.stalkFunctor C \u2191x).map\n            (let_fun this := { val := f.c };\n              this).val)],\n    IsIso\n      ((let_fun this := { val := f.c };\n            this).val.app\n        (op U))\ny : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhy : y \u2208 \u2191U\nH : IsIso (PresheafedSpace.stalkMap f y)\n\u22a2 IsIso\n    ((TopCat.Presheaf.stalkFunctor C \u2191{ val := \u2191f.base y, property := (_ : \u2203 a, a \u2208 \u2191U \u2227 \u2191f.base a = \u2191f.base y) }).map\n      (let_fun this := { val := f.c };\n        this).val)", "state_after": "case mk.intro.intro\nC : Type u_1\ninst\u271d\u2076 : Category.{u_2, u_1} C\ninst\u271d\u2075 : HasLimits C\ninst\u271d\u2074 : HasColimits C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C)\ninst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C)\ninst\u271d : PreservesFilteredColimits (CategoryTheory.forget C)\nX Y : SheafedSpace C\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f.base\nU : Opens \u2191\u2191X.toPresheafedSpace\nh :\n  \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace)\n    [inst :\n      \u2200 (x : { x // x \u2208 U }),\n        IsIso\n          ((TopCat.Presheaf.stalkFunctor C \u2191x).map\n            (let_fun this := { val := f.c };\n              this).val)],\n    IsIso\n      ((let_fun this := { val := f.c };\n            this).val.app\n        (op U))\ny : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhy : y \u2208 \u2191U\nH :\n  IsIso ((TopCat.Presheaf.stalkFunctor C (\u2191f.base y)).map f.c \u226b TopCat.Presheaf.stalkPushforward C f.base X.presheaf y)\n\u22a2 IsIso\n    ((TopCat.Presheaf.stalkFunctor C \u2191{ val := \u2191f.base y, property := (_ : \u2203 a, a \u2208 \u2191U \u2227 \u2191f.base a = \u2191f.base y) }).map\n      (let_fun this := { val := f.c };\n        this).val)"}, {"tactic": "haveI H' :=\n  TopCat.Presheaf.stalkPushforward.stalkPushforward_iso_of_openEmbedding C hf X.presheaf y", "annotated_tactic": ["haveI H' :=\n        <a>TopCat.Presheaf.stalkPushforward.stalkPushforward_iso_of_openEmbedding</a> C hf X.presheaf y", [{"full_name": "TopCat.Presheaf.stalkPushforward.stalkPushforward_iso_of_openEmbedding", "def_path": "Mathlib/Topology/Sheaves/Stalks.lean", "def_pos": [203, 9], "def_end_pos": [203, 46]}]], "state_before": "case mk.intro.intro\nC : Type u_1\ninst\u271d\u2076 : Category.{u_2, u_1} C\ninst\u271d\u2075 : HasLimits C\ninst\u271d\u2074 : HasColimits C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C)\ninst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C)\ninst\u271d : PreservesFilteredColimits (CategoryTheory.forget C)\nX Y : SheafedSpace C\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f.base\nU : Opens \u2191\u2191X.toPresheafedSpace\nh :\n  \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace)\n    [inst :\n      \u2200 (x : { x // x \u2208 U }),\n        IsIso\n          ((TopCat.Presheaf.stalkFunctor C \u2191x).map\n            (let_fun this := { val := f.c };\n              this).val)],\n    IsIso\n      ((let_fun this := { val := f.c };\n            this).val.app\n        (op U))\ny : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhy : y \u2208 \u2191U\nH :\n  IsIso ((TopCat.Presheaf.stalkFunctor C (\u2191f.base y)).map f.c \u226b TopCat.Presheaf.stalkPushforward C f.base X.presheaf y)\n\u22a2 IsIso\n    ((TopCat.Presheaf.stalkFunctor C \u2191{ val := \u2191f.base y, property := (_ : \u2203 a, a \u2208 \u2191U \u2227 \u2191f.base a = \u2191f.base y) }).map\n      (let_fun this := { val := f.c };\n        this).val)", "state_after": "case mk.intro.intro\nC : Type u_1\ninst\u271d\u2076 : Category.{u_2, u_1} C\ninst\u271d\u2075 : HasLimits C\ninst\u271d\u2074 : HasColimits C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C)\ninst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C)\ninst\u271d : PreservesFilteredColimits (CategoryTheory.forget C)\nX Y : SheafedSpace C\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f.base\nU : Opens \u2191\u2191X.toPresheafedSpace\nh :\n  \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace)\n    [inst :\n      \u2200 (x : { x // x \u2208 U }),\n        IsIso\n          ((TopCat.Presheaf.stalkFunctor C \u2191x).map\n            (let_fun this := { val := f.c };\n              this).val)],\n    IsIso\n      ((let_fun this := { val := f.c };\n            this).val.app\n        (op U))\ny : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhy : y \u2208 \u2191U\nH :\n  IsIso ((TopCat.Presheaf.stalkFunctor C (\u2191f.base y)).map f.c \u226b TopCat.Presheaf.stalkPushforward C f.base X.presheaf y)\nH' : IsIso (TopCat.Presheaf.stalkPushforward C f.base X.presheaf y)\n\u22a2 IsIso\n    ((TopCat.Presheaf.stalkFunctor C \u2191{ val := \u2191f.base y, property := (_ : \u2203 a, a \u2208 \u2191U \u2227 \u2191f.base a = \u2191f.base y) }).map\n      (let_fun this := { val := f.c };\n        this).val)"}, {"tactic": "have := @IsIso.comp_isIso _ _ _ _ _ _ _ H (@IsIso.inv_isIso _ _ _ _ _ H')", "annotated_tactic": ["have := @<a>IsIso.comp_isIso</a> _ _ _ _ _ _ _ H (@<a>IsIso.inv_isIso</a> _ _ _ _ _ H')", [{"full_name": "CategoryTheory.IsIso.comp_isIso", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [387, 28], "def_end_pos": [387, 38]}, {"full_name": "CategoryTheory.IsIso.inv_isIso", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [379, 10], "def_end_pos": [379, 19]}]], "state_before": "case mk.intro.intro\nC : Type u_1\ninst\u271d\u2076 : Category.{u_2, u_1} C\ninst\u271d\u2075 : HasLimits C\ninst\u271d\u2074 : HasColimits C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C)\ninst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C)\ninst\u271d : PreservesFilteredColimits (CategoryTheory.forget C)\nX Y : SheafedSpace C\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f.base\nU : Opens \u2191\u2191X.toPresheafedSpace\nh :\n  \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace)\n    [inst :\n      \u2200 (x : { x // x \u2208 U }),\n        IsIso\n          ((TopCat.Presheaf.stalkFunctor C \u2191x).map\n            (let_fun this := { val := f.c };\n              this).val)],\n    IsIso\n      ((let_fun this := { val := f.c };\n            this).val.app\n        (op U))\ny : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhy : y \u2208 \u2191U\nH :\n  IsIso ((TopCat.Presheaf.stalkFunctor C (\u2191f.base y)).map f.c \u226b TopCat.Presheaf.stalkPushforward C f.base X.presheaf y)\nH' : IsIso (TopCat.Presheaf.stalkPushforward C f.base X.presheaf y)\n\u22a2 IsIso\n    ((TopCat.Presheaf.stalkFunctor C \u2191{ val := \u2191f.base y, property := (_ : \u2203 a, a \u2208 \u2191U \u2227 \u2191f.base a = \u2191f.base y) }).map\n      (let_fun this := { val := f.c };\n        this).val)", "state_after": "case mk.intro.intro\nC : Type u_1\ninst\u271d\u2076 : Category.{u_2, u_1} C\ninst\u271d\u2075 : HasLimits C\ninst\u271d\u2074 : HasColimits C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C)\ninst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C)\ninst\u271d : PreservesFilteredColimits (CategoryTheory.forget C)\nX Y : SheafedSpace C\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f.base\nU : Opens \u2191\u2191X.toPresheafedSpace\nh :\n  \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace)\n    [inst :\n      \u2200 (x : { x // x \u2208 U }),\n        IsIso\n          ((TopCat.Presheaf.stalkFunctor C \u2191x).map\n            (let_fun this := { val := f.c };\n              this).val)],\n    IsIso\n      ((let_fun this := { val := f.c };\n            this).val.app\n        (op U))\ny : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhy : y \u2208 \u2191U\nH :\n  IsIso ((TopCat.Presheaf.stalkFunctor C (\u2191f.base y)).map f.c \u226b TopCat.Presheaf.stalkPushforward C f.base X.presheaf y)\nH' : IsIso (TopCat.Presheaf.stalkPushforward C f.base X.presheaf y)\nthis :\n  IsIso\n    (((TopCat.Presheaf.stalkFunctor C (\u2191f.base y)).map f.c \u226b TopCat.Presheaf.stalkPushforward C f.base X.presheaf y) \u226b\n      inv (TopCat.Presheaf.stalkPushforward C f.base X.presheaf y))\n\u22a2 IsIso\n    ((TopCat.Presheaf.stalkFunctor C \u2191{ val := \u2191f.base y, property := (_ : \u2203 a, a \u2208 \u2191U \u2227 \u2191f.base a = \u2191f.base y) }).map\n      (let_fun this := { val := f.c };\n        this).val)"}, {"tactic": "rwa [Category.assoc, IsIso.hom_inv_id, Category.comp_id] at this", "annotated_tactic": ["rwa [<a>Category.assoc</a>, <a>IsIso.hom_inv_id</a>, <a>Category.comp_id</a>] at this", [{"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [159, 3], "def_end_pos": [159, 8]}, {"full_name": "CategoryTheory.IsIso.hom_inv_id", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [273, 9], "def_end_pos": [273, 19]}, {"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [157, 3], "def_end_pos": [157, 10]}]], "state_before": "case mk.intro.intro\nC : Type u_1\ninst\u271d\u2076 : Category.{u_2, u_1} C\ninst\u271d\u2075 : HasLimits C\ninst\u271d\u2074 : HasColimits C\ninst\u271d\u00b3 : ConcreteCategory C\ninst\u271d\u00b2 : ReflectsIsomorphisms (CategoryTheory.forget C)\ninst\u271d\u00b9 : PreservesLimits (CategoryTheory.forget C)\ninst\u271d : PreservesFilteredColimits (CategoryTheory.forget C)\nX Y : SheafedSpace C\nf : X \u27f6 Y\nhf : OpenEmbedding \u2191f.base\nU : Opens \u2191\u2191X.toPresheafedSpace\nh :\n  \u2200 (U : Opens \u2191\u2191Y.toPresheafedSpace)\n    [inst :\n      \u2200 (x : { x // x \u2208 U }),\n        IsIso\n          ((TopCat.Presheaf.stalkFunctor C \u2191x).map\n            (let_fun this := { val := f.c };\n              this).val)],\n    IsIso\n      ((let_fun this := { val := f.c };\n            this).val.app\n        (op U))\ny : (CategoryTheory.forget TopCat).obj \u2191X.toPresheafedSpace\nhy : y \u2208 \u2191U\nH :\n  IsIso ((TopCat.Presheaf.stalkFunctor C (\u2191f.base y)).map f.c \u226b TopCat.Presheaf.stalkPushforward C f.base X.presheaf y)\nH' : IsIso (TopCat.Presheaf.stalkPushforward C f.base X.presheaf y)\nthis :\n  IsIso\n    (((TopCat.Presheaf.stalkFunctor C (\u2191f.base y)).map f.c \u226b TopCat.Presheaf.stalkPushforward C f.base X.presheaf y) \u226b\n      inv (TopCat.Presheaf.stalkPushforward C f.base X.presheaf y))\n\u22a2 IsIso\n    ((TopCat.Presheaf.stalkFunctor C \u2191{ val := \u2191f.base y, property := (_ : \u2203 a, a \u2208 \u2191U \u2227 \u2191f.base a = \u2191f.base y) }).map\n      (let_fun this := { val := f.c };\n        this).val)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/SMul.lean", "full_name": "bddAbove_smul_iff_of_pos", "start": [306, 1], "end": [307, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Basic.lean", "full_name": "OrderDual.Preorder.dual_dual", "start": [749, 1], "end": [750, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Real/Irrational.lean", "full_name": "Int.not_irrational", "start": [185, 1], "end": [185, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.filterMap_eq_map", "start": [1290, 1], "end": [1291, 37], "traced_tactics": [{"tactic": "funext l", "annotated_tactic": ["funext l", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\n\u22a2 filterMap (some \u2218 f) = map f", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\n\u22a2 filterMap (some \u2218 f) l = map f l"}, {"tactic": "induction l <;> simp [*]", "annotated_tactic": ["induction l <;> simp [*]", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nl : List \u03b1\n\u22a2 filterMap (some \u2218 f) l = map f l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Control/Fix.lean", "full_name": "Part.fix_def", "start": [74, 11], "end": [108, 55], "traced_tactics": [{"tactic": "let p := fun i : \u2115 => (Fix.approx f i x).Dom", "annotated_tactic": ["let p := fun i : \u2115 => (<a>Fix.approx</a> f i x).<a>Dom</a>", [{"full_name": "Part.Fix.approx", "def_path": "Mathlib/Control/Fix.lean", "def_pos": [50, 5], "def_end_pos": [50, 15]}, {"full_name": "Part.Dom", "def_path": "Mathlib/Data/Part.lean", "def_pos": [54, 3], "def_end_pos": [54, 6]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\n\u22a2 Part.fix f x = Fix.approx f (Nat.succ (Nat.find h')) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\n\u22a2 Part.fix f x = Fix.approx f (Nat.succ (Nat.find h')) x"}, {"tactic": "have : p (Nat.find h') := Nat.find_spec h'", "annotated_tactic": ["have : p (<a>Nat.find</a> h') := <a>Nat.find_spec</a> h'", [{"full_name": "Nat.find", "def_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "def_pos": [713, 15], "def_end_pos": [713, 19]}, {"full_name": "Nat.find_spec", "def_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "def_pos": [717, 19], "def_end_pos": [717, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\n\u22a2 Part.fix f x = Fix.approx f (Nat.succ (Nat.find h')) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nthis : p (Nat.find h')\n\u22a2 Part.fix f x = Fix.approx f (Nat.succ (Nat.find h')) x"}, {"tactic": "generalize hk : Nat.find h' = k", "annotated_tactic": ["generalize hk : <a>Nat.find</a> h' = k", [{"full_name": "Nat.find", "def_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "def_pos": [713, 15], "def_end_pos": [713, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nthis : p (Nat.find h')\n\u22a2 Part.fix f x = Fix.approx f (Nat.succ (Nat.find h')) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nthis : p (Nat.find h')\nk : \u2115\nhk : Nat.find h' = k\n\u22a2 Part.fix f x = Fix.approx f (Nat.succ k) x"}, {"tactic": "replace hk : Nat.find h' = k + (@Upto.zero p).val := hk", "annotated_tactic": ["replace hk : <a>Nat.find</a> h' = k + (@<a>Upto.zero</a> p).<a>val</a> := hk", [{"full_name": "Nat.find", "def_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "def_pos": [713, 15], "def_end_pos": [713, 19]}, {"full_name": "Nat.Upto.zero", "def_path": "Mathlib/Data/Nat/Upto.lean", "def_pos": [64, 5], "def_end_pos": [64, 9]}, {"full_name": "Subtype.val", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [564, 3], "def_end_pos": [564, 6]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nthis : p (Nat.find h')\nk : \u2115\nhk : Nat.find h' = k\n\u22a2 Part.fix f x = Fix.approx f (Nat.succ k) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nthis : p (Nat.find h')\nk : \u2115\nhk : Nat.find h' = k + \u2191Upto.zero\n\u22a2 Part.fix f x = Fix.approx f (Nat.succ k) x"}, {"tactic": "rw [hk] at this", "annotated_tactic": ["rw [hk] at this", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nthis : p (Nat.find h')\nk : \u2115\nhk : Nat.find h' = k + \u2191Upto.zero\n\u22a2 Part.fix f x = Fix.approx f (Nat.succ k) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nthis : p (k + \u2191Upto.zero)\nhk : Nat.find h' = k + \u2191Upto.zero\n\u22a2 Part.fix f x = Fix.approx f (Nat.succ k) x"}, {"tactic": "revert hk", "annotated_tactic": ["revert hk", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nthis : p (k + \u2191Upto.zero)\nhk : Nat.find h' = k + \u2191Upto.zero\n\u22a2 Part.fix f x = Fix.approx f (Nat.succ k) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nthis : p (k + \u2191Upto.zero)\n\u22a2 Nat.find h' = k + \u2191Upto.zero \u2192 Part.fix f x = Fix.approx f (Nat.succ k) x"}, {"tactic": "dsimp [Part.fix]", "annotated_tactic": ["dsimp [<a>Part.fix</a>]", [{"full_name": "Part.fix", "def_path": "Mathlib/Control/Fix.lean", "def_pos": [69, 15], "def_end_pos": [69, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nthis : p (k + \u2191Upto.zero)\n\u22a2 Nat.find h' = k + \u2191Upto.zero \u2192 Part.fix f x = Fix.approx f (Nat.succ k) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nthis : p (k + \u2191Upto.zero)\n\u22a2 Nat.find h' = k + \u2191Upto.zero \u2192\n    (assert (\u2203 i, (Fix.approx f i x).Dom) fun h =>\n        WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) Upto.zero x) =\n      Fix.approx f (Nat.succ k) x"}, {"tactic": "rw [assert_pos h']", "annotated_tactic": ["rw [<a>assert_pos</a> h']", [{"full_name": "Part.assert_pos", "def_path": "Mathlib/Data/Part.lean", "def_pos": [472, 9], "def_end_pos": [472, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nthis : p (k + \u2191Upto.zero)\n\u22a2 Nat.find h' = k + \u2191Upto.zero \u2192\n    (assert (\u2203 i, (Fix.approx f i x).Dom) fun h =>\n        WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) Upto.zero x) =\n      Fix.approx f (Nat.succ k) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nthis : p (k + \u2191Upto.zero)\n\u22a2 Nat.find h' = k + \u2191Upto.zero \u2192\n    WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) Upto.zero x =\n      Fix.approx f (Nat.succ k) x"}, {"tactic": "revert this", "annotated_tactic": ["revert this", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nthis : p (k + \u2191Upto.zero)\n\u22a2 Nat.find h' = k + \u2191Upto.zero \u2192\n    WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) Upto.zero x =\n      Fix.approx f (Nat.succ k) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\n\u22a2 p (k + \u2191Upto.zero) \u2192\n    Nat.find h' = k + \u2191Upto.zero \u2192\n      WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) Upto.zero x =\n        Fix.approx f (Nat.succ k) x"}, {"tactic": "generalize Upto.zero = z", "annotated_tactic": ["generalize <a>Upto.zero</a> = z", [{"full_name": "Nat.Upto.zero", "def_path": "Mathlib/Data/Nat/Upto.lean", "def_pos": [64, 5], "def_end_pos": [64, 9]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\n\u22a2 p (k + \u2191Upto.zero) \u2192\n    Nat.find h' = k + \u2191Upto.zero \u2192\n      WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) Upto.zero x =\n        Fix.approx f (Nat.succ k) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nz : Upto p\n\u22a2 p (k + \u2191z) \u2192\n    Nat.find h' = k + \u2191z \u2192\n      WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x =\n        Fix.approx f (Nat.succ k) x"}, {"tactic": "intro _this hk", "annotated_tactic": ["intro _this hk", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nz : Upto p\n\u22a2 p (k + \u2191z) \u2192\n    Nat.find h' = k + \u2191z \u2192\n      WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x =\n        Fix.approx f (Nat.succ k) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nz : Upto p\n_this : p (k + \u2191z)\nhk : Nat.find h' = k + \u2191z\n\u22a2 WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x =\n    Fix.approx f (Nat.succ k) x"}, {"tactic": "suffices \u2200 x',\n  WellFounded.fix (Part.fix.proof_1 f x h') (fixAux f) z x' = Fix.approx f (succ k) x'\n  from this _", "annotated_tactic": ["suffices \u2200 x',\n    <a>WellFounded.fix</a> (Part.fix.proof_1 f x h') (<a>fixAux</a> f) z x' = <a>Fix.approx</a> f (<a>succ</a> k) x'\n    from this _", [{"full_name": "WellFounded.fix", "def_path": "lake-packages/lean4/src/lean/Init/WF.lean", "def_pos": [71, 19], "def_end_pos": [71, 22]}, {"full_name": "Part.fixAux", "def_path": "Mathlib/Control/Fix.lean", "def_pos": [56, 5], "def_end_pos": [56, 11]}, {"full_name": "Part.Fix.approx", "def_path": "Mathlib/Control/Fix.lean", "def_pos": [50, 5], "def_end_pos": [50, 15]}, {"full_name": "Nat.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nz : Upto p\n_this : p (k + \u2191z)\nhk : Nat.find h' = k + \u2191z\n\u22a2 WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x =\n    Fix.approx f (Nat.succ k) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nz : Upto p\n_this : p (k + \u2191z)\nhk : Nat.find h' = k + \u2191z\n\u22a2 \u2200 (x' : \u03b1),\n    WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n      Fix.approx f (Nat.succ k) x'"}, {"tactic": "induction k generalizing z with\n| zero =>\n  intro x'\n  rw [Fix.approx, WellFounded.fix_eq, fixAux]\n  congr\n  ext x: 1\n  rw [assert_neg]\n  rfl\n  rw [Nat.zero_add] at _this\n  simpa only [not_not, Coe]\n| succ n n_ih =>\n  intro x'\n  rw [Fix.approx, WellFounded.fix_eq, fixAux]\n  congr\n  ext : 1\n  have hh : \u00ac(Fix.approx f z.val x).Dom := by\n    apply Nat.find_min h'\n    rw [hk, Nat.succ_add, \u2190 Nat.add_succ]\n    apply Nat.lt_of_succ_le\n    apply Nat.le_add_left\n  rw [succ_add_eq_succ_add] at _this hk\n  rw [assert_pos hh, n_ih (Upto.succ z hh) _this hk]", "annotated_tactic": ["induction k generalizing z with\n  | <a>zero</a> =>\n    intro x'\n    rw [<a>Fix.approx</a>, <a>WellFounded.fix_eq</a>, <a>fixAux</a>]\n    congr\n    ext x: 1\n    rw [<a>assert_neg</a>]\n    rfl\n    rw [<a>Nat.zero_add</a>] at _this\n    simpa only [<a>not_not</a>, <a>Coe</a>]\n  | <a>succ</a> n n_ih =>\n    intro x'\n    rw [<a>Fix.approx</a>, <a>WellFounded.fix_eq</a>, <a>fixAux</a>]\n    congr\n    ext : 1\n    have hh : \u00ac(<a>Fix.approx</a> f z.val x).<a>Dom</a> := by\n      apply <a>Nat.find_min</a> h'\n      rw [hk, <a>Nat.succ_add</a>, \u2190 <a>Nat.add_succ</a>]\n      apply <a>Nat.lt_of_succ_le</a>\n      apply <a>Nat.le_add_left</a>\n    rw [<a>succ_add_eq_succ_add</a>] at _this hk\n    rw [<a>assert_pos</a> hh, n_ih (<a>Upto.succ</a> z hh) _this hk]", [{"full_name": "Nat.zero", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1041, 5], "def_end_pos": [1041, 9]}, {"full_name": "Part.Fix.approx", "def_path": "Mathlib/Control/Fix.lean", "def_pos": [50, 5], "def_end_pos": [50, 15]}, {"full_name": "WellFounded.fix_eq", "def_path": "lake-packages/lean4/src/lean/Init/WF.lean", "def_pos": [75, 9], "def_end_pos": [75, 15]}, {"full_name": "Part.fixAux", "def_path": "Mathlib/Control/Fix.lean", "def_pos": [56, 5], "def_end_pos": [56, 11]}, {"full_name": "Part.assert_neg", "def_path": "Mathlib/Data/Part.lean", "def_pos": [481, 9], "def_end_pos": [481, 19]}, {"full_name": "Nat.zero_add", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [114, 27], "def_end_pos": [114, 35]}, {"full_name": "Classical.not_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [683, 24], "def_end_pos": [683, 31]}, {"full_name": "Coe", "def_path": "lake-packages/lean4/src/lean/Init/Coe.lean", "def_pos": [126, 7], "def_end_pos": [126, 10]}, {"full_name": "Nat.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}, {"full_name": "Part.Fix.approx", "def_path": "Mathlib/Control/Fix.lean", "def_pos": [50, 5], "def_end_pos": [50, 15]}, {"full_name": "WellFounded.fix_eq", "def_path": "lake-packages/lean4/src/lean/Init/WF.lean", "def_pos": [75, 9], "def_end_pos": [75, 15]}, {"full_name": "Part.fixAux", "def_path": "Mathlib/Control/Fix.lean", "def_pos": [56, 5], "def_end_pos": [56, 11]}, {"full_name": "Part.Fix.approx", "def_path": "Mathlib/Control/Fix.lean", "def_pos": [50, 5], "def_end_pos": [50, 15]}, {"full_name": "Part.Dom", "def_path": "Mathlib/Data/Part.lean", "def_pos": [54, 3], "def_end_pos": [54, 6]}, {"full_name": "Nat.find_min", "def_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "def_pos": [721, 19], "def_end_pos": [721, 27]}, {"full_name": "Nat.succ_add", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 17]}, {"full_name": "Nat.add_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 17]}, {"full_name": "Nat.lt_of_succ_le", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [310, 9], "def_end_pos": [310, 22]}, {"full_name": "Nat.le_add_left", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [344, 9], "def_end_pos": [344, 20]}, {"full_name": "Nat.succ_add_eq_succ_add", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [282, 9], "def_end_pos": [282, 29]}, {"full_name": "Part.assert_pos", "def_path": "Mathlib/Data/Part.lean", "def_pos": [472, 9], "def_end_pos": [472, 19]}, {"full_name": "Nat.Upto.succ", "def_path": "Mathlib/Data/Nat/Upto.lean", "def_pos": [69, 5], "def_end_pos": [69, 9]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nk : \u2115\nz : Upto p\n_this : p (k + \u2191z)\nhk : Nat.find h' = k + \u2191z\n\u22a2 \u2200 (x' : \u03b1),\n    WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n      Fix.approx f (Nat.succ k) x'", "state_after": "no goals"}, {"tactic": "intro x'", "annotated_tactic": ["intro x'", []], "state_before": "case zero\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\n\u22a2 \u2200 (x' : \u03b1),\n    WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n      Fix.approx f (Nat.succ Nat.zero) x'", "state_after": "case zero\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\nx' : \u03b1\n\u22a2 WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n    Fix.approx f (Nat.succ Nat.zero) x'"}, {"tactic": "rw [Fix.approx, WellFounded.fix_eq, fixAux]", "annotated_tactic": ["rw [<a>Fix.approx</a>, <a>WellFounded.fix_eq</a>, <a>fixAux</a>]", [{"full_name": "Part.Fix.approx", "def_path": "Mathlib/Control/Fix.lean", "def_pos": [50, 5], "def_end_pos": [50, 15]}, {"full_name": "WellFounded.fix_eq", "def_path": "lake-packages/lean4/src/lean/Init/WF.lean", "def_pos": [75, 9], "def_end_pos": [75, 15]}, {"full_name": "Part.fixAux", "def_path": "Mathlib/Control/Fix.lean", "def_pos": [56, 5], "def_end_pos": [56, 11]}]], "state_before": "case zero\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\nx' : \u03b1\n\u22a2 WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n    Fix.approx f (Nat.succ Nat.zero) x'", "state_after": "case zero\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\nx' : \u03b1\n\u22a2 f\n      (fun x_1 =>\n        assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h =>\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_2 => (Fix.approx f x_2 x).Dom)) (fixAux f) (Upto.succ z h)\n            x_1)\n      x' =\n    f (Fix.approx f Nat.zero) x'"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case zero\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\nx' : \u03b1\n\u22a2 f\n      (fun x_1 =>\n        assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h =>\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_2 => (Fix.approx f x_2 x).Dom)) (fixAux f) (Upto.succ z h)\n            x_1)\n      x' =\n    f (Fix.approx f Nat.zero) x'", "state_after": "case zero.e_a\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\nx' : \u03b1\n\u22a2 (fun x_1 =>\n      assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h =>\n        WellFounded.fix (_ : WellFounded (Upto.GT fun x_2 => (Fix.approx f x_2 x).Dom)) (fixAux f) (Upto.succ z h)\n          x_1) =\n    Fix.approx f Nat.zero"}, {"tactic": "ext x: 1", "annotated_tactic": ["ext x: 1", []], "state_before": "case zero.e_a\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\nx' : \u03b1\n\u22a2 (fun x_1 =>\n      assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h =>\n        WellFounded.fix (_ : WellFounded (Upto.GT fun x_2 => (Fix.approx f x_2 x).Dom)) (fixAux f) (Upto.succ z h)\n          x_1) =\n    Fix.approx f Nat.zero", "state_after": "case zero.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx\u271d : \u03b1\nh' : \u2203 i, (Fix.approx f i x\u271d).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x\u271d).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\nx' x : \u03b1\n\u22a2 (assert (\u00ac(Fix.approx f (\u2191z) x\u271d).Dom) fun h =>\n      WellFounded.fix (_ : WellFounded (Upto.GT fun x => (Fix.approx f x x\u271d).Dom)) (fixAux f) (Upto.succ z h) x) =\n    Fix.approx f Nat.zero x"}, {"tactic": "rw [assert_neg]", "annotated_tactic": ["rw [<a>assert_neg</a>]", [{"full_name": "Part.assert_neg", "def_path": "Mathlib/Data/Part.lean", "def_pos": [481, 9], "def_end_pos": [481, 19]}]], "state_before": "case zero.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx\u271d : \u03b1\nh' : \u2203 i, (Fix.approx f i x\u271d).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x\u271d).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\nx' x : \u03b1\n\u22a2 (assert (\u00ac(Fix.approx f (\u2191z) x\u271d).Dom) fun h =>\n      WellFounded.fix (_ : WellFounded (Upto.GT fun x => (Fix.approx f x x\u271d).Dom)) (fixAux f) (Upto.succ z h) x) =\n    Fix.approx f Nat.zero x", "state_after": "case zero.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx\u271d : \u03b1\nh' : \u2203 i, (Fix.approx f i x\u271d).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x\u271d).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\nx' x : \u03b1\n\u22a2 none = Fix.approx f Nat.zero x\n\ncase zero.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx\u271d : \u03b1\nh' : \u2203 i, (Fix.approx f i x\u271d).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x\u271d).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\nx' x : \u03b1\n\u22a2 \u00ac\u00ac(Fix.approx f (\u2191z) x\u271d).Dom"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case zero.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx\u271d : \u03b1\nh' : \u2203 i, (Fix.approx f i x\u271d).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x\u271d).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\nx' x : \u03b1\n\u22a2 none = Fix.approx f Nat.zero x\n\ncase zero.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx\u271d : \u03b1\nh' : \u2203 i, (Fix.approx f i x\u271d).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x\u271d).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\nx' x : \u03b1\n\u22a2 \u00ac\u00ac(Fix.approx f (\u2191z) x\u271d).Dom", "state_after": "case zero.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx\u271d : \u03b1\nh' : \u2203 i, (Fix.approx f i x\u271d).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x\u271d).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\nx' x : \u03b1\n\u22a2 \u00ac\u00ac(Fix.approx f (\u2191z) x\u271d).Dom"}, {"tactic": "rw [Nat.zero_add] at _this", "annotated_tactic": ["rw [<a>Nat.zero_add</a>] at _this", [{"full_name": "Nat.zero_add", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [114, 27], "def_end_pos": [114, 35]}]], "state_before": "case zero.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx\u271d : \u03b1\nh' : \u2203 i, (Fix.approx f i x\u271d).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x\u271d).Dom\nz : Upto p\n_this : p (Nat.zero + \u2191z)\nhk : Nat.find h' = Nat.zero + \u2191z\nx' x : \u03b1\n\u22a2 \u00ac\u00ac(Fix.approx f (\u2191z) x\u271d).Dom", "state_after": "case zero.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx\u271d : \u03b1\nh' : \u2203 i, (Fix.approx f i x\u271d).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x\u271d).Dom\nz : Upto p\n_this : p \u2191z\nhk : Nat.find h' = Nat.zero + \u2191z\nx' x : \u03b1\n\u22a2 \u00ac\u00ac(Fix.approx f (\u2191z) x\u271d).Dom"}, {"tactic": "simpa only [not_not, Coe]", "annotated_tactic": ["simpa only [<a>not_not</a>, <a>Coe</a>]", [{"full_name": "Classical.not_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [683, 24], "def_end_pos": [683, 31]}, {"full_name": "Coe", "def_path": "lake-packages/lean4/src/lean/Init/Coe.lean", "def_pos": [126, 7], "def_end_pos": [126, 10]}]], "state_before": "case zero.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx\u271d : \u03b1\nh' : \u2203 i, (Fix.approx f i x\u271d).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x\u271d).Dom\nz : Upto p\n_this : p \u2191z\nhk : Nat.find h' = Nat.zero + \u2191z\nx' x : \u03b1\n\u22a2 \u00ac\u00ac(Fix.approx f (\u2191z) x\u271d).Dom", "state_after": "no goals"}, {"tactic": "intro x'", "annotated_tactic": ["intro x'", []], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n  \u2200 (z : Upto p),\n    p (n + \u2191z) \u2192\n      Nat.find h' = n + \u2191z \u2192\n        \u2200 (x' : \u03b1),\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n            Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\n\u22a2 \u2200 (x' : \u03b1),\n    WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n      Fix.approx f (Nat.succ (Nat.succ n)) x'", "state_after": "case succ\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n  \u2200 (z : Upto p),\n    p (n + \u2191z) \u2192\n      Nat.find h' = n + \u2191z \u2192\n        \u2200 (x' : \u03b1),\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n            Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' : \u03b1\n\u22a2 WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n    Fix.approx f (Nat.succ (Nat.succ n)) x'"}, {"tactic": "rw [Fix.approx, WellFounded.fix_eq, fixAux]", "annotated_tactic": ["rw [<a>Fix.approx</a>, <a>WellFounded.fix_eq</a>, <a>fixAux</a>]", [{"full_name": "Part.Fix.approx", "def_path": "Mathlib/Control/Fix.lean", "def_pos": [50, 5], "def_end_pos": [50, 15]}, {"full_name": "WellFounded.fix_eq", "def_path": "lake-packages/lean4/src/lean/Init/WF.lean", "def_pos": [75, 9], "def_end_pos": [75, 15]}, {"full_name": "Part.fixAux", "def_path": "Mathlib/Control/Fix.lean", "def_pos": [56, 5], "def_end_pos": [56, 11]}]], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n  \u2200 (z : Upto p),\n    p (n + \u2191z) \u2192\n      Nat.find h' = n + \u2191z \u2192\n        \u2200 (x' : \u03b1),\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n            Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' : \u03b1\n\u22a2 WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n    Fix.approx f (Nat.succ (Nat.succ n)) x'", "state_after": "case succ\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n  \u2200 (z : Upto p),\n    p (n + \u2191z) \u2192\n      Nat.find h' = n + \u2191z \u2192\n        \u2200 (x' : \u03b1),\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n            Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' : \u03b1\n\u22a2 f\n      (fun x_1 =>\n        assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h =>\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_2 => (Fix.approx f x_2 x).Dom)) (fixAux f) (Upto.succ z h)\n            x_1)\n      x' =\n    f (Fix.approx f (n + 1)) x'"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n  \u2200 (z : Upto p),\n    p (n + \u2191z) \u2192\n      Nat.find h' = n + \u2191z \u2192\n        \u2200 (x' : \u03b1),\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n            Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' : \u03b1\n\u22a2 f\n      (fun x_1 =>\n        assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h =>\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_2 => (Fix.approx f x_2 x).Dom)) (fixAux f) (Upto.succ z h)\n            x_1)\n      x' =\n    f (Fix.approx f (n + 1)) x'", "state_after": "case succ.e_a\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n  \u2200 (z : Upto p),\n    p (n + \u2191z) \u2192\n      Nat.find h' = n + \u2191z \u2192\n        \u2200 (x' : \u03b1),\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n            Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' : \u03b1\n\u22a2 (fun x_1 =>\n      assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h =>\n        WellFounded.fix (_ : WellFounded (Upto.GT fun x_2 => (Fix.approx f x_2 x).Dom)) (fixAux f) (Upto.succ z h)\n          x_1) =\n    Fix.approx f (n + 1)"}, {"tactic": "ext : 1", "annotated_tactic": ["ext : 1", []], "state_before": "case succ.e_a\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n  \u2200 (z : Upto p),\n    p (n + \u2191z) \u2192\n      Nat.find h' = n + \u2191z \u2192\n        \u2200 (x' : \u03b1),\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n            Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' : \u03b1\n\u22a2 (fun x_1 =>\n      assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h =>\n        WellFounded.fix (_ : WellFounded (Upto.GT fun x_2 => (Fix.approx f x_2 x).Dom)) (fixAux f) (Upto.succ z h)\n          x_1) =\n    Fix.approx f (n + 1)", "state_after": "case succ.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n  \u2200 (z : Upto p),\n    p (n + \u2191z) \u2192\n      Nat.find h' = n + \u2191z \u2192\n        \u2200 (x' : \u03b1),\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n            Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' x\u271d : \u03b1\n\u22a2 (assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h =>\n      WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) (Upto.succ z h) x\u271d) =\n    Fix.approx f (n + 1) x\u271d"}, {"tactic": "have hh : \u00ac(Fix.approx f z.val x).Dom := by\n  apply Nat.find_min h'\n  rw [hk, Nat.succ_add, \u2190 Nat.add_succ]\n  apply Nat.lt_of_succ_le\n  apply Nat.le_add_left", "annotated_tactic": ["have hh : \u00ac(<a>Fix.approx</a> f z.val x).<a>Dom</a> := by\n      apply <a>Nat.find_min</a> h'\n      rw [hk, <a>Nat.succ_add</a>, \u2190 <a>Nat.add_succ</a>]\n      apply <a>Nat.lt_of_succ_le</a>\n      apply <a>Nat.le_add_left</a>", [{"full_name": "Part.Fix.approx", "def_path": "Mathlib/Control/Fix.lean", "def_pos": [50, 5], "def_end_pos": [50, 15]}, {"full_name": "Part.Dom", "def_path": "Mathlib/Data/Part.lean", "def_pos": [54, 3], "def_end_pos": [54, 6]}, {"full_name": "Nat.find_min", "def_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "def_pos": [721, 19], "def_end_pos": [721, 27]}, {"full_name": "Nat.succ_add", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 17]}, {"full_name": "Nat.add_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 17]}, {"full_name": "Nat.lt_of_succ_le", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [310, 9], "def_end_pos": [310, 22]}, {"full_name": "Nat.le_add_left", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [344, 9], "def_end_pos": [344, 20]}]], "state_before": "case succ.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n  \u2200 (z : Upto p),\n    p (n + \u2191z) \u2192\n      Nat.find h' = n + \u2191z \u2192\n        \u2200 (x' : \u03b1),\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n            Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' x\u271d : \u03b1\n\u22a2 (assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h =>\n      WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) (Upto.succ z h) x\u271d) =\n    Fix.approx f (n + 1) x\u271d", "state_after": "case succ.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n  \u2200 (z : Upto p),\n    p (n + \u2191z) \u2192\n      Nat.find h' = n + \u2191z \u2192\n        \u2200 (x' : \u03b1),\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n            Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' x\u271d : \u03b1\nhh : \u00ac(Fix.approx f (\u2191z) x).Dom\n\u22a2 (assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h =>\n      WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) (Upto.succ z h) x\u271d) =\n    Fix.approx f (n + 1) x\u271d"}, {"tactic": "rw [succ_add_eq_succ_add] at _this hk", "annotated_tactic": ["rw [<a>succ_add_eq_succ_add</a>] at _this hk", [{"full_name": "Nat.succ_add_eq_succ_add", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [282, 9], "def_end_pos": [282, 29]}]], "state_before": "case succ.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n  \u2200 (z : Upto p),\n    p (n + \u2191z) \u2192\n      Nat.find h' = n + \u2191z \u2192\n        \u2200 (x' : \u03b1),\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n            Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' x\u271d : \u03b1\nhh : \u00ac(Fix.approx f (\u2191z) x).Dom\n\u22a2 (assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h =>\n      WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) (Upto.succ z h) x\u271d) =\n    Fix.approx f (n + 1) x\u271d", "state_after": "case succ.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n  \u2200 (z : Upto p),\n    p (n + \u2191z) \u2192\n      Nat.find h' = n + \u2191z \u2192\n        \u2200 (x' : \u03b1),\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n            Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (n + Nat.succ \u2191z)\nhk : Nat.find h' = n + Nat.succ \u2191z\nx' x\u271d : \u03b1\nhh : \u00ac(Fix.approx f (\u2191z) x).Dom\n\u22a2 (assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h =>\n      WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) (Upto.succ z h) x\u271d) =\n    Fix.approx f (n + 1) x\u271d"}, {"tactic": "rw [assert_pos hh, n_ih (Upto.succ z hh) _this hk]", "annotated_tactic": ["rw [<a>assert_pos</a> hh, n_ih (<a>Upto.succ</a> z hh) _this hk]", [{"full_name": "Part.assert_pos", "def_path": "Mathlib/Data/Part.lean", "def_pos": [472, 9], "def_end_pos": [472, 19]}, {"full_name": "Nat.Upto.succ", "def_path": "Mathlib/Data/Nat/Upto.lean", "def_pos": [69, 5], "def_end_pos": [69, 9]}]], "state_before": "case succ.e_a.h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n  \u2200 (z : Upto p),\n    p (n + \u2191z) \u2192\n      Nat.find h' = n + \u2191z \u2192\n        \u2200 (x' : \u03b1),\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n            Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (n + Nat.succ \u2191z)\nhk : Nat.find h' = n + Nat.succ \u2191z\nx' x\u271d : \u03b1\nhh : \u00ac(Fix.approx f (\u2191z) x).Dom\n\u22a2 (assert (\u00ac(Fix.approx f (\u2191z) x).Dom) fun h =>\n      WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) (Upto.succ z h) x\u271d) =\n    Fix.approx f (n + 1) x\u271d", "state_after": "no goals"}, {"tactic": "apply Nat.find_min h'", "annotated_tactic": ["apply <a>Nat.find_min</a> h'", [{"full_name": "Nat.find_min", "def_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "def_pos": [721, 19], "def_end_pos": [721, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n  \u2200 (z : Upto p),\n    p (n + \u2191z) \u2192\n      Nat.find h' = n + \u2191z \u2192\n        \u2200 (x' : \u03b1),\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n            Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' x\u271d : \u03b1\n\u22a2 \u00ac(Fix.approx f (\u2191z) x).Dom", "state_after": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n  \u2200 (z : Upto p),\n    p (n + \u2191z) \u2192\n      Nat.find h' = n + \u2191z \u2192\n        \u2200 (x' : \u03b1),\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n            Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' x\u271d : \u03b1\n\u22a2 \u2191z < Nat.find h'"}, {"tactic": "rw [hk, Nat.succ_add, \u2190 Nat.add_succ]", "annotated_tactic": ["rw [hk, <a>Nat.succ_add</a>, \u2190 <a>Nat.add_succ</a>]", [{"full_name": "Nat.succ_add", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 17]}, {"full_name": "Nat.add_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n  \u2200 (z : Upto p),\n    p (n + \u2191z) \u2192\n      Nat.find h' = n + \u2191z \u2192\n        \u2200 (x' : \u03b1),\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n            Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' x\u271d : \u03b1\n\u22a2 \u2191z < Nat.find h'", "state_after": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n  \u2200 (z : Upto p),\n    p (n + \u2191z) \u2192\n      Nat.find h' = n + \u2191z \u2192\n        \u2200 (x' : \u03b1),\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n            Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' x\u271d : \u03b1\n\u22a2 \u2191z < n + Nat.succ \u2191z"}, {"tactic": "apply Nat.lt_of_succ_le", "annotated_tactic": ["apply <a>Nat.lt_of_succ_le</a>", [{"full_name": "Nat.lt_of_succ_le", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [310, 9], "def_end_pos": [310, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n  \u2200 (z : Upto p),\n    p (n + \u2191z) \u2192\n      Nat.find h' = n + \u2191z \u2192\n        \u2200 (x' : \u03b1),\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n            Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' x\u271d : \u03b1\n\u22a2 \u2191z < n + Nat.succ \u2191z", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n  \u2200 (z : Upto p),\n    p (n + \u2191z) \u2192\n      Nat.find h' = n + \u2191z \u2192\n        \u2200 (x' : \u03b1),\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n            Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' x\u271d : \u03b1\n\u22a2 Nat.succ \u2191z \u2264 n + Nat.succ \u2191z"}, {"tactic": "apply Nat.le_add_left", "annotated_tactic": ["apply <a>Nat.le_add_left</a>", [{"full_name": "Nat.le_add_left", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [344, 9], "def_end_pos": [344, 20]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\nf : ((a : \u03b1) \u2192 Part (\u03b2 a)) \u2192 (a : \u03b1) \u2192 Part (\u03b2 a)\nx : \u03b1\nh' : \u2203 i, (Fix.approx f i x).Dom\np : \u2115 \u2192 Prop := fun i => (Fix.approx f i x).Dom\nn : \u2115\nn_ih :\n  \u2200 (z : Upto p),\n    p (n + \u2191z) \u2192\n      Nat.find h' = n + \u2191z \u2192\n        \u2200 (x' : \u03b1),\n          WellFounded.fix (_ : WellFounded (Upto.GT fun x_1 => (Fix.approx f x_1 x).Dom)) (fixAux f) z x' =\n            Fix.approx f (Nat.succ n) x'\nz : Upto p\n_this : p (Nat.succ n + \u2191z)\nhk : Nat.find h' = Nat.succ n + \u2191z\nx' x\u271d : \u03b1\n\u22a2 Nat.succ \u2191z \u2264 n + Nat.succ \u2191z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "full_name": "left_mem_affineSpan_pair", "start": [1311, 1], "end": [1312, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "full_name": "Rat.normalize_eq", "start": [25, 1], "end": [31, 73], "traced_tactics": [{"tactic": "simp only [normalize, maybeNormalize_eq,\n  Int.div_eq_ediv_of_dvd (Int.ofNat_dvd_left.2 (Nat.gcd_dvd_left ..))]", "annotated_tactic": ["simp only [<a>normalize</a>, <a>maybeNormalize_eq</a>,\n    <a>Int.div_eq_ediv_of_dvd</a> (<a>Int.ofNat_dvd_left</a>.2 (<a>Nat.gcd_dvd_left</a> ..))]", [{"full_name": "Rat.normalize", "def_path": "lake-packages/std/Std/Data/Rat/Basic.lean", "def_pos": [72, 15], "def_end_pos": [72, 28]}, {"full_name": "Rat.maybeNormalize_eq", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [13, 17], "def_end_pos": [13, 34]}, {"full_name": "Int.div_eq_ediv_of_dvd", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [794, 9], "def_end_pos": [794, 27]}, {"full_name": "Int.ofNat_dvd_left", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [662, 9], "def_end_pos": [662, 23]}, {"full_name": "Nat.gcd_dvd_left", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [42, 9], "def_end_pos": [42, 21]}]], "state_before": "num : Int\nden : Nat\nden_nz : den \u2260 0\n\u22a2 normalize num den = mk' (num / \u2191(Nat.gcd (Int.natAbs num) den)) (den / Nat.gcd (Int.natAbs num) den)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean", "full_name": "PrimeSpectrum.isClosed_range_comap_of_surjective", "start": [733, 1], "end": [736, 29], "traced_tactics": [{"tactic": "rw [range_comap_of_surjective _ f hf]", "annotated_tactic": ["rw [<a>range_comap_of_surjective</a> _ f 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"full_name": "Continuous.exp", "start": [158, 1], "end": [159, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "full_name": "MeasureTheory.condexp_bot", "start": [276, 1], "end": [277, 93], "traced_tactics": [{"tactic": "refine' (condexp_bot' f).trans _", "annotated_tactic": ["refine' (<a>condexp_bot'</a> f).<a>trans</a> _", [{"full_name": "MeasureTheory.condexp_bot'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [248, 9], "def_end_pos": [248, 21]}, {"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}]], "state_before": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\ninst\u271d : IsProbabilityMeasure \u03bc\nf : \u03b1 \u2192 F'\n\u22a2 \u03bc[f|\u22a5] = fun x => \u222b (x : \u03b1), f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\ninst\u271d : IsProbabilityMeasure \u03bc\nf : \u03b1 \u2192 F'\n\u22a2 (fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc) = fun x => \u222b (x : \u03b1), f x \u2202\u03bc"}, {"tactic": "rw [measure_univ, ENNReal.one_toReal, inv_one, one_smul]", "annotated_tactic": ["rw [<a>measure_univ</a>, <a>ENNReal.one_toReal</a>, <a>inv_one</a>, <a>one_smul</a>]", [{"full_name": "MeasureTheory.IsProbabilityMeasure.measure_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3027, 3], "def_end_pos": [3027, 15]}, {"full_name": "ENNReal.one_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [230, 17], "def_end_pos": [230, 27]}, {"full_name": "inv_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1015, 9], "def_end_pos": [1015, 16]}, {"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}]], "state_before": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b2 : NormedSpace \u211d F'\ninst\u271d\u00b9 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\ninst\u271d : IsProbabilityMeasure \u03bc\nf : \u03b1 \u2192 F'\n\u22a2 (fun x => (ENNReal.toReal (\u2191\u2191\u03bc Set.univ))\u207b\u00b9 \u2022 \u222b (x : \u03b1), f x \u2202\u03bc) = fun x => \u222b (x : \u03b1), f x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Prod.lean", "full_name": "MeasureTheory.quasiMeasurePreserving_div", "start": [474, 1], "end": [477, 94], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Order/IntermediateValue.lean", "full_name": "isConnected_Ico", "start": [490, 1], "end": [491, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "full_name": "Int.nneg_mul_add_sq_of_abs_le_one", "start": [185, 1], "end": [196, 49], "traced_tactics": [{"tactic": "have hnx : 0 < n \u2192 0 \u2264 x + n := fun hn => by\n  have := _root_.add_le_add (neg_le_of_abs_le hx) (cast_one_le_of_pos hn)\n  rwa [add_left_neg] at this", "annotated_tactic": ["have hnx : 0 < n \u2192 0 \u2264 x + n := fun hn => by\n    have := <a>_root_.add_le_add</a> (<a>neg_le_of_abs_le</a> hx) (<a>cast_one_le_of_pos</a> hn)\n    rwa [<a>add_left_neg</a>] at this", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "neg_le_of_abs_le", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [240, 9], "def_end_pos": [240, 25]}, {"full_name": "Int.cast_one_le_of_pos", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [169, 9], "def_end_pos": [169, 27]}, {"full_name": "add_left_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1118, 3], "def_end_pos": [1118, 14]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\n\u22a2 0 \u2264 \u2191n * x + \u2191n * \u2191n", "state_after": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\n\u22a2 0 \u2264 \u2191n * x + \u2191n * \u2191n"}, {"tactic": "have hnx' : n < 0 \u2192 x + n \u2264 0 := fun hn => by\n  have := _root_.add_le_add (le_of_abs_le hx) (cast_le_neg_one_of_neg hn)\n  rwa [add_right_neg] at this", "annotated_tactic": ["have hnx' : n < 0 \u2192 x + n \u2264 0 := fun hn => by\n    have := <a>_root_.add_le_add</a> (<a>le_of_abs_le</a> hx) (<a>cast_le_neg_one_of_neg</a> hn)\n    rwa [<a>add_right_neg</a>] at this", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "le_of_abs_le", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [244, 9], "def_end_pos": [244, 21]}, {"full_name": "Int.cast_le_neg_one_of_neg", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [172, 9], "def_end_pos": [172, 31]}, {"full_name": "add_right_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1134, 3], "def_end_pos": [1134, 14]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\n\u22a2 0 \u2264 \u2191n * x + \u2191n * \u2191n", "state_after": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\nhnx' : n < 0 \u2192 x + \u2191n \u2264 0\n\u22a2 0 \u2264 \u2191n * x + \u2191n * \u2191n"}, {"tactic": "rw [\u2190 mul_add, mul_nonneg_iff]", "annotated_tactic": ["rw [\u2190 <a>mul_add</a>, <a>mul_nonneg_iff</a>]", [{"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}, {"full_name": "mul_nonneg_iff", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [1061, 9], "def_end_pos": [1061, 23]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\nhnx' : n < 0 \u2192 x + \u2191n \u2264 0\n\u22a2 0 \u2264 \u2191n * x + \u2191n * \u2191n", "state_after": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\nhnx' : n < 0 \u2192 x + \u2191n \u2264 0\n\u22a2 0 \u2264 \u2191n \u2227 0 \u2264 x + \u2191n \u2228 \u2191n \u2264 0 \u2227 x + \u2191n \u2264 0"}, {"tactic": "rcases lt_trichotomy n 0 with (h | rfl | h)", "annotated_tactic": ["rcases <a>lt_trichotomy</a> n 0 with (h | rfl | h)", [{"full_name": "lt_trichotomy", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [311, 9], "def_end_pos": [311, 22]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\nhnx' : n < 0 \u2192 x + \u2191n \u2264 0\n\u22a2 0 \u2264 \u2191n \u2227 0 \u2264 x + \u2191n \u2228 \u2191n \u2264 0 \u2227 x + \u2191n \u2264 0", "state_after": "case inl\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\nhnx' : n < 0 \u2192 x + \u2191n \u2264 0\nh : n < 0\n\u22a2 0 \u2264 \u2191n \u2227 0 \u2264 x + \u2191n \u2228 \u2191n \u2264 0 \u2227 x + \u2191n \u2264 0\n\ncase inr.inl\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < 0 \u2192 0 \u2264 x + \u21910\nhnx' : 0 < 0 \u2192 x + \u21910 \u2264 0\n\u22a2 0 \u2264 \u21910 \u2227 0 \u2264 x + \u21910 \u2228 \u21910 \u2264 0 \u2227 x + \u21910 \u2264 0\n\ncase inr.inr\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\nhnx' : n < 0 \u2192 x + \u2191n \u2264 0\nh : 0 < n\n\u22a2 0 \u2264 \u2191n \u2227 0 \u2264 x + \u2191n \u2228 \u2191n \u2264 0 \u2227 x + \u2191n \u2264 0"}, {"tactic": "have := _root_.add_le_add (neg_le_of_abs_le hx) (cast_one_le_of_pos hn)", "annotated_tactic": ["have := <a>_root_.add_le_add</a> (<a>neg_le_of_abs_le</a> hx) (<a>cast_one_le_of_pos</a> hn)", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "neg_le_of_abs_le", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [240, 9], "def_end_pos": [240, 25]}, {"full_name": "Int.cast_one_le_of_pos", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [169, 9], "def_end_pos": [169, 27]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhn : 0 < n\n\u22a2 0 \u2264 x + \u2191n", "state_after": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhn : 0 < n\nthis : -1 + 1 \u2264 x + \u2191n\n\u22a2 0 \u2264 x + \u2191n"}, {"tactic": "rwa [add_left_neg] at this", "annotated_tactic": ["rwa [<a>add_left_neg</a>] at this", [{"full_name": "add_left_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1118, 3], "def_end_pos": [1118, 14]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhn : 0 < n\nthis : -1 + 1 \u2264 x + \u2191n\n\u22a2 0 \u2264 x + \u2191n", "state_after": "no goals"}, {"tactic": "have := _root_.add_le_add (le_of_abs_le hx) (cast_le_neg_one_of_neg hn)", "annotated_tactic": ["have := <a>_root_.add_le_add</a> (<a>le_of_abs_le</a> hx) (<a>cast_le_neg_one_of_neg</a> hn)", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "le_of_abs_le", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [244, 9], "def_end_pos": [244, 21]}, {"full_name": "Int.cast_le_neg_one_of_neg", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [172, 9], "def_end_pos": [172, 31]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\nhn : n < 0\n\u22a2 x + \u2191n \u2264 0", "state_after": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\nhn : n < 0\nthis : x + \u2191n \u2264 1 + -1\n\u22a2 x + \u2191n \u2264 0"}, {"tactic": "rwa [add_right_neg] at this", "annotated_tactic": ["rwa [<a>add_right_neg</a>] at this", [{"full_name": "add_right_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1134, 3], "def_end_pos": [1134, 14]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\nhn : n < 0\nthis : x + \u2191n \u2264 1 + -1\n\u22a2 x + \u2191n \u2264 0", "state_after": "no goals"}, {"tactic": "exact Or.inr \u27e8by exact_mod_cast h.le, hnx' h\u27e9", "annotated_tactic": ["exact <a>Or.inr</a> \u27e8by exact_mod_cast h.le, hnx' h\u27e9", [{"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}]], "state_before": "case inl\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\nhnx' : n < 0 \u2192 x + \u2191n \u2264 0\nh : n < 0\n\u22a2 0 \u2264 \u2191n \u2227 0 \u2264 x + \u2191n \u2228 \u2191n \u2264 0 \u2227 x + \u2191n \u2264 0", "state_after": "no goals"}, {"tactic": "exact_mod_cast h.le", "annotated_tactic": ["exact_mod_cast h.le", []], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\nhnx' : n < 0 \u2192 x + \u2191n \u2264 0\nh : n < 0\n\u22a2 \u2191n \u2264 0", "state_after": "no goals"}, {"tactic": "simp [le_total 0 x]", "annotated_tactic": ["simp [<a>le_total</a> 0 x]", [{"full_name": "le_total", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [295, 9], "def_end_pos": [295, 17]}]], "state_before": "case inr.inl\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < 0 \u2192 0 \u2264 x + \u21910\nhnx' : 0 < 0 \u2192 x + \u21910 \u2264 0\n\u22a2 0 \u2264 \u21910 \u2227 0 \u2264 x + \u21910 \u2228 \u21910 \u2264 0 \u2227 x + \u21910 \u2264 0", "state_after": "no goals"}, {"tactic": "exact Or.inl \u27e8by exact_mod_cast h.le, hnx h\u27e9", "annotated_tactic": ["exact <a>Or.inl</a> \u27e8by exact_mod_cast h.le, hnx h\u27e9", [{"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}]], "state_before": "case inr.inr\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\nhnx' : n < 0 \u2192 x + \u2191n \u2264 0\nh : 0 < n\n\u22a2 0 \u2264 \u2191n \u2227 0 \u2264 x + \u2191n \u2228 \u2191n \u2264 0 \u2227 x + \u2191n \u2264 0", "state_after": "no goals"}, {"tactic": "exact_mod_cast h.le", "annotated_tactic": ["exact_mod_cast h.le", []], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\nhnx' : n < 0 \u2192 x + \u2191n \u2264 0\nh : 0 < n\n\u22a2 0 \u2264 \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Normed/Group/Hom.lean", "full_name": "NormedAddGroupHom.coe_injective", "start": [105, 1], "end": [106, 16], "traced_tactics": [{"tactic": "apply coe_inj", "annotated_tactic": ["apply <a>coe_inj</a>", [{"full_name": "NormedAddGroupHom.coe_inj", "def_path": "Mathlib/Analysis/Normed/Group/Hom.lean", "def_pos": [101, 9], "def_end_pos": [101, 16]}]], "state_before": "V : Type u_1\nV\u2081 : Type u_2\nV\u2082 : Type u_3\nV\u2083 : Type u_4\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : SeminormedAddCommGroup V\u2081\ninst\u271d\u00b9 : SeminormedAddCommGroup V\u2082\ninst\u271d : SeminormedAddCommGroup V\u2083\nf g : NormedAddGroupHom V\u2081 V\u2082\n\u22a2 Function.Injective toFun", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/LocallyConvex/Polar.lean", "full_name": "LinearMap.polar_eq_iInter", "start": [74, 1], "end": [76, 62], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NormedCommRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ns : Set E\n\u22a2 polar B s = \u22c2 x \u2208 s, {y | \u2016\u2191(\u2191B x) y\u2016 \u2264 1}", "state_after": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NormedCommRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ns : Set E\nx\u271d : F\n\u22a2 x\u271d \u2208 polar B s \u2194 x\u271d \u2208 \u22c2 x \u2208 s, {y | \u2016\u2191(\u2191B x) y\u2016 \u2264 1}"}, {"tactic": "simp only [polar_mem_iff, Set.mem_iInter, Set.mem_setOf_eq]", "annotated_tactic": ["simp only [<a>polar_mem_iff</a>, <a>Set.mem_iInter</a>, <a>Set.mem_setOf_eq</a>]", [{"full_name": "LinearMap.polar_mem_iff", "def_path": "Mathlib/Analysis/LocallyConvex/Polar.lean", "def_pos": [61, 9], "def_end_pos": [61, 22]}, {"full_name": "Set.mem_iInter", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [207, 9], "def_end_pos": [207, 19]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : NormedCommRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nB : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ns : Set E\nx\u271d : F\n\u22a2 x\u271d \u2208 polar B s \u2194 x\u271d \u2208 \u22c2 x \u2208 s, {y | \u2016\u2191(\u2191B x) y\u2016 \u2264 1}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "full_name": "Matrix.det_updateRow_add_self", "start": [445, 1], "end": [448, 80], "traced_tactics": [{"tactic": "simp [det_updateRow_add,\n  det_zero_of_row_eq hij (updateRow_self.trans (updateRow_ne hij.symm).symm)]", "annotated_tactic": ["simp [<a>det_updateRow_add</a>,\n    <a>det_zero_of_row_eq</a> hij (updateRow_self.trans (<a>updateRow_ne</a> hij.symm).<a>symm</a>)]", [{"full_name": "Matrix.det_updateRow_add", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [388, 9], "def_end_pos": [388, 26]}, {"full_name": "Matrix.det_zero_of_row_eq", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [376, 9], "def_end_pos": [376, 27]}, {"full_name": "Matrix.updateRow_ne", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [2809, 9], "def_end_pos": [2809, 21]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "m : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni j : n\nhij : i \u2260 j\n\u22a2 det (updateRow A i (A i + A j)) = det A", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Localization/Basic.lean", "full_name": "IsLocalization.isLocalization_of_base_ringEquiv", "start": [786, 1], "end": [807, 98], "traced_tactics": [{"tactic": "letI : Algebra P S := ((algebraMap R S).comp h.symm.toRingHom).toAlgebra", "annotated_tactic": ["letI : <a>Algebra</a> P S := ((<a>algebraMap</a> R S).<a>comp</a> h.symm.toRingHom).<a>toAlgebra</a>", [{"full_name": "Algebra", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [116, 7], "def_end_pos": [116, 14]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "RingHom.comp", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [657, 5], "def_end_pos": [657, 9]}, {"full_name": "RingHom.toAlgebra", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [267, 5], "def_end_pos": [267, 22]}]], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\n\u22a2 IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S", "state_after": "R : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n\u22a2 IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n\u22a2 IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S", "state_after": "case map_units'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n\u22a2 \u2200 (y : { x // x \u2208 Submonoid.map (RingEquiv.toMonoidHom h) M }), IsUnit (\u2191(algebraMap P S) \u2191y)\n\ncase surj'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n\u22a2 \u2200 (z : S), \u2203 x, z * \u2191(algebraMap P S) \u2191x.2 = \u2191(algebraMap P S) x.1\n\ncase eq_iff_exists'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n\u22a2 \u2200 {x y : P}, \u2191(algebraMap P S) x = \u2191(algebraMap P S) y \u2194 \u2203 c, \u2191c * x = \u2191c * y"}, {"tactic": "rintro \u27e8_, \u27e8y, hy, rfl\u27e9\u27e9", "annotated_tactic": ["rintro \u27e8_, \u27e8y, hy, rfl\u27e9\u27e9", []], "state_before": "case map_units'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n\u22a2 \u2200 (y : { x // x \u2208 Submonoid.map (RingEquiv.toMonoidHom h) M }), IsUnit (\u2191(algebraMap P S) \u2191y)", "state_after": "case map_units'.mk.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : R\nhy : y \u2208 \u2191M\n\u22a2 IsUnit\n    (\u2191(algebraMap P S)\n      \u2191{ val := \u2191(RingEquiv.toMonoidHom h) y,\n          property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191(RingEquiv.toMonoidHom h) a = \u2191(RingEquiv.toMonoidHom h) y) })"}, {"tactic": "convert IsLocalization.map_units S \u27e8y, hy\u27e9", "annotated_tactic": ["convert <a>IsLocalization.map_units</a> S \u27e8y, hy\u27e9", [{"full_name": "IsLocalization.map_units", "def_path": "Mathlib/RingTheory/Localization/Basic.lean", "def_pos": [120, 9], "def_end_pos": [120, 18]}]], "state_before": "case map_units'.mk.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : R\nhy : y \u2208 \u2191M\n\u22a2 IsUnit\n    (\u2191(algebraMap P S)\n      \u2191{ val := \u2191(RingEquiv.toMonoidHom h) y,\n          property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191(RingEquiv.toMonoidHom h) a = \u2191(RingEquiv.toMonoidHom h) y) })", "state_after": "case h.e'_3\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : R\nhy : y \u2208 \u2191M\n\u22a2 \u2191(algebraMap P S)\n      \u2191{ val := \u2191(RingEquiv.toMonoidHom h) y,\n          property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191(RingEquiv.toMonoidHom h) a = \u2191(RingEquiv.toMonoidHom h) y) } =\n    \u2191(algebraMap R S) \u2191{ val := y, property := hy }"}, {"tactic": "dsimp only [RingHom.algebraMap_toAlgebra, RingHom.comp_apply]", "annotated_tactic": ["dsimp only [<a>RingHom.algebraMap_toAlgebra</a>, <a>RingHom.comp_apply</a>]", [{"full_name": "RingHom.algebraMap_toAlgebra", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [271, 9], "def_end_pos": [271, 37]}, {"full_name": "RingHom.comp_apply", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [672, 9], "def_end_pos": [672, 19]}]], "state_before": "case h.e'_3\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : R\nhy : y \u2208 \u2191M\n\u22a2 \u2191(algebraMap P S)\n      \u2191{ val := \u2191(RingEquiv.toMonoidHom h) y,\n          property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191(RingEquiv.toMonoidHom h) a = \u2191(RingEquiv.toMonoidHom h) y) } =\n    \u2191(algebraMap R S) \u2191{ val := y, property := hy }", "state_after": "case h.e'_3\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : R\nhy : y \u2208 \u2191M\n\u22a2 \u2191(algebraMap R S) (\u2191(RingEquiv.toRingHom (RingEquiv.symm h)) (\u2191(RingEquiv.toMonoidHom h) y)) = \u2191(algebraMap R S) y"}, {"tactic": "exact congr_arg _ (h.symm_apply_apply _)", "annotated_tactic": ["exact <a>congr_arg</a> _ (h.symm_apply_apply _)", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}]], "state_before": "case h.e'_3\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : R\nhy : y \u2208 \u2191M\n\u22a2 \u2191(algebraMap R S) (\u2191(RingEquiv.toRingHom (RingEquiv.symm h)) (\u2191(RingEquiv.toMonoidHom h) y)) = \u2191(algebraMap R S) y", "state_after": "no goals"}, {"tactic": "intro y", "annotated_tactic": ["intro y", []], "state_before": "case surj'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n\u22a2 \u2200 (z : S), \u2203 x, z * \u2191(algebraMap P S) \u2191x.2 = \u2191(algebraMap P S) x.1", "state_after": "case surj'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : S\n\u22a2 \u2203 x, y * \u2191(algebraMap P S) \u2191x.2 = \u2191(algebraMap P S) x.1"}, {"tactic": "obtain \u27e8\u27e8x, s\u27e9, e\u27e9 := IsLocalization.surj M y", "annotated_tactic": ["obtain \u27e8\u27e8x, s\u27e9, e\u27e9 := <a>IsLocalization.surj</a> M y", [{"full_name": "IsLocalization.surj", "def_path": "Mathlib/RingTheory/Localization/Basic.lean", "def_pos": [125, 9], "def_end_pos": [125, 13]}]], "state_before": "case surj'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : S\n\u22a2 \u2203 x, y * \u2191(algebraMap P S) \u2191x.2 = \u2191(algebraMap P S) x.1", "state_after": "case surj'.intro.mk\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : S\nx : R\ns : { x // x \u2208 M }\ne : y * \u2191(algebraMap R S) \u2191(x, s).2 = \u2191(algebraMap R S) (x, s).1\n\u22a2 \u2203 x, y * \u2191(algebraMap P S) \u2191x.2 = \u2191(algebraMap P S) x.1"}, {"tactic": "refine' \u27e8\u27e8h x, _, _, s.prop, rfl\u27e9, _\u27e9", "annotated_tactic": ["refine' \u27e8\u27e8h x, _, _, s.prop, <a>rfl</a>\u27e9, _\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case surj'.intro.mk\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : S\nx : R\ns : { x // x \u2208 M }\ne : y * \u2191(algebraMap R S) \u2191(x, s).2 = \u2191(algebraMap R S) (x, s).1\n\u22a2 \u2203 x, y * \u2191(algebraMap P S) \u2191x.2 = \u2191(algebraMap P S) x.1", "state_after": "case surj'.intro.mk\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : S\nx : R\ns : { x // x \u2208 M }\ne : y * \u2191(algebraMap R S) \u2191(x, s).2 = \u2191(algebraMap R S) (x, s).1\n\u22a2 y *\n      \u2191(algebraMap P S)\n        \u2191(\u2191h x,\n              { val := \u2191(RingEquiv.toMonoidHom h) \u2191s,\n                property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191(RingEquiv.toMonoidHom h) a = \u2191(RingEquiv.toMonoidHom h) \u2191s) }).2 =\n    \u2191(algebraMap P S)\n      (\u2191h x,\n          { val := \u2191(RingEquiv.toMonoidHom h) \u2191s,\n            property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191(RingEquiv.toMonoidHom h) a = \u2191(RingEquiv.toMonoidHom h) \u2191s) }).1"}, {"tactic": "dsimp only [RingHom.algebraMap_toAlgebra, RingHom.comp_apply] at e \u22a2", "annotated_tactic": ["dsimp only [<a>RingHom.algebraMap_toAlgebra</a>, <a>RingHom.comp_apply</a>] at e \u22a2", [{"full_name": "RingHom.algebraMap_toAlgebra", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [271, 9], "def_end_pos": [271, 37]}, {"full_name": "RingHom.comp_apply", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [672, 9], "def_end_pos": [672, 19]}]], "state_before": "case surj'.intro.mk\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : S\nx : R\ns : { x // x \u2208 M }\ne : y * \u2191(algebraMap R S) \u2191(x, s).2 = \u2191(algebraMap R S) (x, s).1\n\u22a2 y *\n      \u2191(algebraMap P S)\n        \u2191(\u2191h x,\n              { val := \u2191(RingEquiv.toMonoidHom h) \u2191s,\n                property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191(RingEquiv.toMonoidHom h) a = \u2191(RingEquiv.toMonoidHom h) \u2191s) }).2 =\n    \u2191(algebraMap P S)\n      (\u2191h x,\n          { val := \u2191(RingEquiv.toMonoidHom h) \u2191s,\n            property := (_ : \u2203 a, a \u2208 \u2191M \u2227 \u2191(RingEquiv.toMonoidHom h) a = \u2191(RingEquiv.toMonoidHom h) \u2191s) }).1", "state_after": "case surj'.intro.mk\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : S\nx : R\ns : { x // x \u2208 M }\ne : y * \u2191(algebraMap R S) \u2191s = \u2191(algebraMap R S) x\n\u22a2 y * \u2191(algebraMap R S) (\u2191(RingEquiv.toRingHom (RingEquiv.symm h)) (\u2191(RingEquiv.toMonoidHom h) \u2191s)) =\n    \u2191(algebraMap R S) (\u2191(RingEquiv.toRingHom (RingEquiv.symm h)) (\u2191h x))"}, {"tactic": "convert e <;> exact h.symm_apply_apply _", "annotated_tactic": ["convert e <;> exact h.symm_apply_apply _", []], "state_before": "case surj'.intro.mk\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\ny : S\nx : R\ns : { x // x \u2208 M }\ne : y * \u2191(algebraMap R S) \u2191s = \u2191(algebraMap R S) x\n\u22a2 y * \u2191(algebraMap R S) (\u2191(RingEquiv.toRingHom (RingEquiv.symm h)) (\u2191(RingEquiv.toMonoidHom h) \u2191s)) =\n    \u2191(algebraMap R S) (\u2191(RingEquiv.toRingHom (RingEquiv.symm h)) (\u2191h x))", "state_after": "no goals"}, {"tactic": "intro x y", "annotated_tactic": ["intro x y", []], "state_before": "case eq_iff_exists'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n\u22a2 \u2200 {x y : P}, \u2191(algebraMap P S) x = \u2191(algebraMap P S) y \u2194 \u2203 c, \u2191c * x = \u2191c * y", "state_after": "case eq_iff_exists'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n\u22a2 \u2191(algebraMap P S) x = \u2191(algebraMap P S) y \u2194 \u2203 c, \u2191c * x = \u2191c * y"}, {"tactic": "rw [RingHom.algebraMap_toAlgebra, RingHom.comp_apply, RingHom.comp_apply,\n  IsLocalization.eq_iff_exists M S]", "annotated_tactic": ["rw [<a>RingHom.algebraMap_toAlgebra</a>, <a>RingHom.comp_apply</a>, <a>RingHom.comp_apply</a>,\n      <a>IsLocalization.eq_iff_exists</a> M S]", [{"full_name": "RingHom.algebraMap_toAlgebra", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [271, 9], "def_end_pos": [271, 37]}, {"full_name": "RingHom.comp_apply", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [672, 9], "def_end_pos": [672, 19]}, {"full_name": "RingHom.comp_apply", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [672, 9], "def_end_pos": [672, 19]}, {"full_name": "IsLocalization.eq_iff_exists", "def_path": "Mathlib/RingTheory/Localization/Basic.lean", "def_pos": [130, 9], "def_end_pos": [130, 22]}]], "state_before": "case eq_iff_exists'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n\u22a2 \u2191(algebraMap P S) x = \u2191(algebraMap P S) y \u2194 \u2203 c, \u2191c * x = \u2191c * y", "state_after": "case eq_iff_exists'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n\u22a2 (\u2203 c, \u2191c * \u2191(RingEquiv.toRingHom (RingEquiv.symm h)) x = \u2191c * \u2191(RingEquiv.toRingHom (RingEquiv.symm h)) y) \u2194\n    \u2203 c, \u2191c * x = \u2191c * y"}, {"tactic": "simp_rw [\u2190 h.toEquiv.apply_eq_iff_eq]", "annotated_tactic": ["simp_rw [\u2190 h.toEquiv.apply_eq_iff_eq]", []], "state_before": "case eq_iff_exists'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n\u22a2 (\u2203 c, \u2191c * \u2191(RingEquiv.toRingHom (RingEquiv.symm h)) x = \u2191c * \u2191(RingEquiv.toRingHom (RingEquiv.symm h)) y) \u2194\n    \u2203 c, \u2191c * x = \u2191c * y", "state_after": "case eq_iff_exists'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n\u22a2 (\u2203 c,\n      \u2191h.toEquiv (\u2191c * \u2191(RingEquiv.toRingHom (RingEquiv.symm h)) x) =\n        \u2191h.toEquiv (\u2191c * \u2191(RingEquiv.toRingHom (RingEquiv.symm h)) y)) \u2194\n    \u2203 c, \u2191c * x = \u2191c * y"}, {"tactic": "change (\u2203 c : M, h (c * h.symm x) = h (c * h.symm y)) \u2194 _", "annotated_tactic": ["change (\u2203 c : M, h (c * h.symm x) = h (c * h.symm y)) \u2194 _", []], "state_before": "case eq_iff_exists'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n\u22a2 (\u2203 c,\n      \u2191h.toEquiv (\u2191c * \u2191(RingEquiv.toRingHom (RingEquiv.symm h)) x) =\n        \u2191h.toEquiv (\u2191c * \u2191(RingEquiv.toRingHom (RingEquiv.symm h)) y)) \u2194\n    \u2203 c, \u2191c * x = \u2191c * y", "state_after": "case eq_iff_exists'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n\u22a2 (\u2203 c, \u2191h (\u2191c * \u2191(RingEquiv.symm h) x) = \u2191h (\u2191c * \u2191(RingEquiv.symm h) y)) \u2194 \u2203 c, \u2191c * x = \u2191c * y"}, {"tactic": "simp only [RingEquiv.apply_symm_apply, RingEquiv.map_mul]", "annotated_tactic": ["simp only [<a>RingEquiv.apply_symm_apply</a>, <a>RingEquiv.map_mul</a>]", [{"full_name": "RingEquiv.apply_symm_apply", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [355, 9], "def_end_pos": [355, 25]}, {"full_name": "RingEquiv.map_mul", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [166, 19], "def_end_pos": [166, 26]}]], "state_before": "case eq_iff_exists'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n\u22a2 (\u2203 c, \u2191h (\u2191c * \u2191(RingEquiv.symm h) x) = \u2191h (\u2191c * \u2191(RingEquiv.symm h) y)) \u2194 \u2203 c, \u2191c * x = \u2191c * y", "state_after": "case eq_iff_exists'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n\u22a2 (\u2203 c, \u2191h \u2191c * x = \u2191h \u2191c * y) \u2194 \u2203 c, \u2191c * x = \u2191c * y"}, {"tactic": "exact\n  \u27e8fun \u27e8c, e\u27e9 => \u27e8\u27e8_, _, c.prop, rfl\u27e9, e\u27e9, fun \u27e8\u27e8_, c, h, e\u2081\u27e9, e\u2082\u27e9 => \u27e8\u27e8_, h\u27e9, e\u2081.symm \u25b8 e\u2082\u27e9\u27e9", "annotated_tactic": ["exact\n      \u27e8fun \u27e8c, e\u27e9 => \u27e8\u27e8_, _, c.prop, <a>rfl</a>\u27e9, e\u27e9, fun \u27e8\u27e8_, c, h, e\u2081\u27e9, e\u2082\u27e9 => \u27e8\u27e8_, h\u27e9, e\u2081.symm \u25b8 e\u2082\u27e9\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case eq_iff_exists'\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra R S\nP : Type u_3\ninst\u271d\u00b9 : CommSemiring P\ninst\u271d : IsLocalization M S\nh : R \u2243+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nx y : P\n\u22a2 (\u2203 c, \u2191h \u2191c * x = \u2191h \u2191c * y) \u2194 \u2203 c, \u2191c * x = \u2191c * y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Lattice.lean", "full_name": "Subtype.coe_sup", "start": [1428, 1], "end": [1431, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/Basic.lean", "full_name": "Nat.self_add_sub_one", "start": [330, 1], "end": [334, 51], "traced_tactics": [{"tactic": "cases n", "annotated_tactic": ["cases n", []], "state_before": "m n\u271d k n : \u2115\n\u22a2 n + (n - 1) = 2 * n - 1", "state_after": "case zero\nm n k : \u2115\n\u22a2 zero + (zero - 1) = 2 * zero - 1\n\ncase succ\nm n k n\u271d : \u2115\n\u22a2 succ n\u271d + (succ n\u271d - 1) = 2 * succ n\u271d - 1"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case zero\nm n k : \u2115\n\u22a2 zero + (zero - 1) = 2 * zero - 1", "state_after": "no goals"}, {"tactic": "rw [two_mul]", "annotated_tactic": ["rw [<a>two_mul</a>]", [{"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}]], "state_before": "case succ\nm n k n\u271d : \u2115\n\u22a2 succ n\u271d + (succ n\u271d - 1) = 2 * succ n\u271d - 1", "state_after": "case succ\nm n k n\u271d : \u2115\n\u22a2 succ n\u271d + (succ n\u271d - 1) = succ n\u271d + succ n\u271d - 1"}, {"tactic": "convert (add_succ_sub_one (Nat.succ _) _).symm", "annotated_tactic": ["convert (<a>add_succ_sub_one</a> (<a>Nat.succ</a> _) _).<a>symm</a>", [{"full_name": "Nat.add_succ_sub_one", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [298, 9], "def_end_pos": [298, 25]}, {"full_name": "Nat.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case succ\nm n k n\u271d : \u2115\n\u22a2 succ n\u271d + (succ n\u271d - 1) = succ n\u271d + succ n\u271d - 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/SimpleGraph/Density.lean", "full_name": "SimpleGraph.swap_mem_interedges_iff", "start": [419, 1], "end": [420, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/IsROrC/Basic.lean", "full_name": "IsROrC.I_mul_I", "start": [339, 1], "end": [340, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.SimpleFunc.mem\u2112p_iff", "start": [351, 1], "end": [354, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Instances/ENNReal.lean", "full_name": "ENNReal.inv_limsup", "start": [520, 1], "end": [522, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/TensorProduct.lean", "full_name": "Algebra.TensorProduct.map_comp", "start": [842, 1], "end": [844, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean", "full_name": "GaussianInt.norm_le_norm_mul_left", "start": [256, 1], "end": [261, 49], "traced_tactics": [{"tactic": "rw [Zsqrtd.norm_mul, Int.natAbs_mul]", "annotated_tactic": ["rw [<a>Zsqrtd.norm_mul</a>, <a>Int.natAbs_mul</a>]", [{"full_name": "Zsqrtd.norm_mul", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [541, 9], "def_end_pos": [541, 17]}, {"full_name": "Int.natAbs_mul", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [171, 9], "def_end_pos": [171, 19]}]], "state_before": "x y : \u2124[i]\nhy : y \u2260 0\n\u22a2 Int.natAbs (norm x) \u2264 Int.natAbs (norm (x * y))", "state_after": "x y : \u2124[i]\nhy : y \u2260 0\n\u22a2 Int.natAbs (norm x) \u2264 Int.natAbs (norm x) * Int.natAbs (norm y)"}, {"tactic": "exact le_mul_of_one_le_right (Nat.zero_le _) (Int.ofNat_le.1 (by\n  rw [abs_coe_nat_norm]\n  exact Int.add_one_le_of_lt (norm_pos.2 hy)))", "annotated_tactic": ["exact <a>le_mul_of_one_le_right</a> (<a>Nat.zero_le</a> _) (<a>Int.ofNat_le</a>.1 (by\n    rw [<a>abs_coe_nat_norm</a>]\n    exact <a>Int.add_one_le_of_lt</a> (<a>norm_pos</a>.2 hy)))", [{"full_name": "le_mul_of_one_le_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [678, 9], "def_end_pos": [678, 31]}, {"full_name": "Nat.zero_le", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 20]}, {"full_name": "Int.ofNat_le", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [574, 28], "def_end_pos": [574, 36]}, {"full_name": "GaussianInt.abs_coe_nat_norm", "def_path": "Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean", "def_pos": [172, 9], "def_end_pos": [172, 25]}, {"full_name": "Int.add_one_le_of_lt", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [1255, 9], "def_end_pos": [1255, 25]}, {"full_name": "GaussianInt.norm_pos", "def_path": "Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean", "def_pos": [168, 9], "def_end_pos": [168, 17]}]], "state_before": "x y : \u2124[i]\nhy : y \u2260 0\n\u22a2 Int.natAbs (norm x) \u2264 Int.natAbs (norm x) * Int.natAbs (norm y)", "state_after": "no goals"}, {"tactic": "rw [abs_coe_nat_norm]", "annotated_tactic": ["rw [<a>abs_coe_nat_norm</a>]", [{"full_name": "GaussianInt.abs_coe_nat_norm", "def_path": "Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean", "def_pos": [172, 9], "def_end_pos": [172, 25]}]], "state_before": "x y : \u2124[i]\nhy : y \u2260 0\n\u22a2 \u21911 \u2264 \u2191(Int.natAbs (norm y))", "state_after": "x y : \u2124[i]\nhy : y \u2260 0\n\u22a2 \u21911 \u2264 norm y"}, {"tactic": "exact Int.add_one_le_of_lt (norm_pos.2 hy)", "annotated_tactic": ["exact <a>Int.add_one_le_of_lt</a> (<a>norm_pos</a>.2 hy)", [{"full_name": "Int.add_one_le_of_lt", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [1255, 9], "def_end_pos": [1255, 25]}, {"full_name": "GaussianInt.norm_pos", "def_path": "Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean", "def_pos": [168, 9], "def_end_pos": [168, 17]}]], "state_before": "x y : \u2124[i]\nhy : y \u2260 0\n\u22a2 \u21911 \u2264 norm y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "full_name": "div_mem_comm_iff", "start": [164, 1], "end": [165, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Multiset/Nodup.lean", "full_name": "Multiset.count_eq_of_nodup", "start": [94, 1], "end": [98, 37], "traced_tactics": [{"tactic": "split_ifs with h", "annotated_tactic": ["split_ifs with h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d t : Multiset \u03b1\na\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\nd : Nodup s\n\u22a2 count a s = if a \u2208 s then 1 else 0", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d t : Multiset \u03b1\na\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\nd : Nodup s\nh : a \u2208 s\n\u22a2 count a s = 1\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d t : Multiset \u03b1\na\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\nd : Nodup s\nh : \u00aca \u2208 s\n\u22a2 count a s = 0"}, {"tactic": "exact count_eq_one_of_mem d h", "annotated_tactic": ["exact <a>count_eq_one_of_mem</a> d h", [{"full_name": "Multiset.count_eq_one_of_mem", "def_path": "Mathlib/Data/Multiset/Nodup.lean", "def_pos": [89, 9], "def_end_pos": [89, 28]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d t : Multiset \u03b1\na\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\nd : Nodup s\nh : a \u2208 s\n\u22a2 count a s = 1", "state_after": "no goals"}, {"tactic": "exact count_eq_zero_of_not_mem h", "annotated_tactic": ["exact <a>count_eq_zero_of_not_mem</a> h", [{"full_name": "Multiset.count_eq_zero_of_not_mem", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2421, 9], "def_end_pos": [2421, 33]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d t : Multiset \u03b1\na\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\ns : Multiset \u03b1\nd : Nodup s\nh : \u00aca \u2208 s\n\u22a2 count a s = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Parity.lean", "full_name": "Int.even_add", "start": [101, 1], "end": [104, 40], "traced_tactics": [{"tactic": "cases' emod_two_eq_zero_or_one m with h\u2081 h\u2081 <;>\ncases' emod_two_eq_zero_or_one n with h\u2082 h\u2082 <;>\nsimp [even_iff, h\u2081, h\u2082, Int.add_emod]", "annotated_tactic": ["cases' <a>emod_two_eq_zero_or_one</a> m with h\u2081 h\u2081 <;>\n  cases' <a>emod_two_eq_zero_or_one</a> n with h\u2082 h\u2082 <;>\n  simp [<a>even_iff</a>, h\u2081, h\u2082, <a>Int.add_emod</a>]", [{"full_name": "Int.emod_two_eq_zero_or_one", "def_path": "Mathlib/Data/Int/Order/Basic.lean", "def_pos": [315, 9], "def_end_pos": [315, 32]}, {"full_name": "Int.emod_two_eq_zero_or_one", "def_path": "Mathlib/Data/Int/Order/Basic.lean", "def_pos": [315, 9], "def_end_pos": [315, 32]}, {"full_name": "Int.even_iff", "def_path": "Mathlib/Data/Int/Parity.lean", "def_pos": [36, 9], "def_end_pos": [36, 17]}, {"full_name": "Int.add_emod", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [428, 9], "def_end_pos": [428, 17]}]], "state_before": "m n : \u2124\n\u22a2 Even (m + n) \u2194 (Even m \u2194 Even n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.sub_neg", "start": [370, 11], "end": [370, 87], "traced_tactics": [{"tactic": "simp [Int.sub_eq_add_neg]", "annotated_tactic": ["simp [<a>Int.sub_eq_add_neg</a>]", [{"full_name": "Int.sub_eq_add_neg", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [94, 19], "def_end_pos": [94, 33]}]], "state_before": "a b : Int\n\u22a2 a - -b = a + b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "full_name": "SimpleGraph.Walk.cons_copy", "start": [158, 1], "end": [161, 6], "traced_tactics": [{"tactic": "subst_vars", "annotated_tactic": ["subst_vars", []], "state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w v' w' : V\nh : Adj G u v\np : Walk G v' w'\nhv : v' = v\nhw : w' = w\n\u22a2 cons h (Walk.copy p hv hw) = Walk.copy (cons (_ : Adj G u v') p) (_ : u = u) hw", "state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v' w' : V\np : Walk G v' w'\nh : Adj G u v'\n\u22a2 cons h (Walk.copy p (_ : v' = v') (_ : w' = w')) = Walk.copy (cons (_ : Adj G u v') p) (_ : u = u) (_ : w' = w')"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v' w' : V\np : Walk G v' w'\nh : Adj G u v'\n\u22a2 cons h (Walk.copy p (_ : v' = v') (_ : w' = w')) = Walk.copy (cons (_ : Adj G u v') p) (_ : u = u) (_ : w' = w')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.exists_image_iff", "start": [602, 1], "end": [605, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/Digits.lean", "full_name": "Nat.digits_zero_zero", "start": [94, 1], "end": [95, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean", "full_name": "NonUnitalAlgebra.inf_toNonUnitalSubsemiring", "start": [669, 1], "end": [671, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Pointwise.lean", "full_name": "sSup_inv", "start": [61, 1], "end": [63, 43], "traced_tactics": [{"tactic": "rw [\u2190 image_inv, sSup_image]", "annotated_tactic": ["rw [\u2190 <a>image_inv</a>, <a>sSup_image</a>]", [{"full_name": "Set.image_inv", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 18]}, {"full_name": "sSup_image", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1452, 9], "def_end_pos": [1452, 19]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : CompleteLattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ns\u271d t s : Set \u03b1\n\u22a2 sSup s\u207b\u00b9 = (sInf s)\u207b\u00b9", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b3 : CompleteLattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ns\u271d t s : Set \u03b1\n\u22a2 \u2a06 a \u2208 s, a\u207b\u00b9 = (sInf s)\u207b\u00b9"}, {"tactic": "exact ((OrderIso.inv \u03b1).map_sInf _).symm", "annotated_tactic": ["exact ((<a>OrderIso.inv</a> \u03b1).<a>map_sInf</a> _).<a>symm</a>", [{"full_name": "OrderIso.inv", "def_path": "Mathlib/Algebra/Order/Group/OrderIso.lean", "def_pos": [37, 5], "def_end_pos": [37, 17]}, {"full_name": "OrderIso.map_sInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1042, 9], "def_end_pos": [1042, 26]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : CompleteLattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ns\u271d t s : Set \u03b1\n\u22a2 \u2a06 a \u2208 s, a\u207b\u00b9 = (sInf s)\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "iInf_ne_top_subtype", "start": [1683, 1], "end": [1684, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Basic.lean", "full_name": "LinearMap.range_comp_of_range_eq_top", "start": [1639, 1], "end": [1641, 92], "traced_tactics": [{"tactic": "rw [range_comp, hf, Submodule.map_top]", "annotated_tactic": ["rw [<a>range_comp</a>, hf, <a>Submodule.map_top</a>]", [{"full_name": "LinearMap.range_comp", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [1112, 9], "def_end_pos": [1112, 19]}, {"full_name": "Submodule.map_top", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [1530, 9], "def_end_pos": [1530, 16]}]], "state_before": "R : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : Semiring R\u2082\ninst\u271d\u00b9\u2070 : Semiring R\u2083\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2083\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\ninst\u271d\u00b2 : RingHomSurjective \u03c4\u2081\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c4\u2082\u2083\ninst\u271d : RingHomSurjective \u03c4\u2081\u2083\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c4\u2082\u2083] M\u2083\nhf : range f = \u22a4\n\u22a2 range (comp g f) = range g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean", "full_name": "Real.arccos_one", "start": [384, 1], "end": [384, 54], "traced_tactics": [{"tactic": "simp [arccos]", "annotated_tactic": ["simp [<a>arccos</a>]", [{"full_name": "Real.arccos", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean", "def_pos": [335, 19], "def_end_pos": [335, 25]}]], "state_before": "\u22a2 arccos 1 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Complex/Order.lean", "full_name": "Complex.zero_lt_real", "start": [79, 1], "end": [80, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Subgroup/Pointwise.lean", "full_name": "AddSubgroup.pointwise_smul_le_pointwise_smul_iff", "start": [504, 1], "end": [506, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Ideal/Operations.lean", "full_name": "RingHom.ker_ne_top", "start": [2095, 1], "end": [2096, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/UniformGroup.lean", "full_name": "UniformCauchySeqOn.mul", "start": [518, 1], "end": [520, 82], "traced_tactics": [{"tactic": "simpa using (uniformContinuous_mul.comp_uniformCauchySeqOn (hf.prod' hf')) u hu", "annotated_tactic": ["simpa using (uniformContinuous_mul.comp_uniformCauchySeqOn (hf.prod' hf')) u hu", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\n\u03b9 : Type u_3\nl : Filter \u03b9\nl' : Filter \u03b2\nf f' : \u03b9 \u2192 \u03b2 \u2192 \u03b1\ng g' : \u03b2 \u2192 \u03b1\ns : Set \u03b2\nhf : UniformCauchySeqOn f l s\nhf' : UniformCauchySeqOn f' l s\nu : Set (\u03b1 \u00d7 \u03b1)\nhu : u \u2208 \ud835\udce4 \u03b1\n\u22a2 \u2200\u1da0 (m : \u03b9 \u00d7 \u03b9) in l \u00d7\u02e2 l, \u2200 (x : \u03b2), x \u2208 s \u2192 ((f * f') m.1 x, (f * f') m.2 x) \u2208 u", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/Padics/RingHoms.lean", "full_name": "PadicInt.isCauSeq_nthHom", "start": [505, 1], "end": [513, 54], "traced_tactics": [{"tactic": "intro \u03b5 h\u03b5", "annotated_tactic": ["intro \u03b5 h\u03b5", []], "state_before": "p : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\n\u22a2 IsCauSeq (padicNorm p) fun n => \u2191(nthHom f r n)", "state_after": "p : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 padicNorm p ((fun n => \u2191(nthHom f r n)) j - (fun n => \u2191(nthHom f r n)) i) < \u03b5"}, {"tactic": "obtain \u27e8k, hk\u27e9 : \u2203 k : \u2115, (p : \u211a) ^ (-((k : \u2115) : \u2124)) < \u03b5 := exists_pow_neg_lt_rat p h\u03b5", "annotated_tactic": ["obtain \u27e8k, hk\u27e9 : \u2203 k : \u2115, (p : \u211a) ^ (-((k : \u2115) : \u2124)) < \u03b5 := <a>exists_pow_neg_lt_rat</a> p h\u03b5", [{"full_name": "PadicInt.exists_pow_neg_lt_rat", "def_path": "Mathlib/NumberTheory/Padics/PadicIntegers.lean", "def_pos": [344, 9], "def_end_pos": [344, 30]}]], "state_before": "p : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 padicNorm p ((fun n => \u2191(nthHom f r n)) j - (fun n => \u2191(nthHom f r n)) i) < \u03b5", "state_after": "case intro\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nk : \u2115\nhk : \u2191p ^ (-\u2191k) < \u03b5\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 padicNorm p ((fun n => \u2191(nthHom f r n)) j - (fun n => \u2191(nthHom f r n)) i) < \u03b5"}, {"tactic": "use k", "annotated_tactic": ["use k", []], "state_before": "case intro\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nk : \u2115\nhk : \u2191p ^ (-\u2191k) < \u03b5\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 padicNorm p ((fun n => \u2191(nthHom f r n)) j - (fun n => \u2191(nthHom f r n)) i) < \u03b5", "state_after": "case h\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nk : \u2115\nhk : \u2191p ^ (-\u2191k) < \u03b5\n\u22a2 \u2200 (j : \u2115), j \u2265 k \u2192 padicNorm p ((fun n => \u2191(nthHom f r n)) j - (fun n => \u2191(nthHom f r n)) k) < \u03b5"}, {"tactic": "intro j hj", "annotated_tactic": ["intro j hj", []], "state_before": "case h\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nk : \u2115\nhk : \u2191p ^ (-\u2191k) < \u03b5\n\u22a2 \u2200 (j : \u2115), j \u2265 k \u2192 padicNorm p ((fun n => \u2191(nthHom f r n)) j - (fun n => \u2191(nthHom f r n)) k) < \u03b5", "state_after": "case h\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nk : \u2115\nhk : \u2191p ^ (-\u2191k) < \u03b5\nj : \u2115\nhj : j \u2265 k\n\u22a2 padicNorm p ((fun n => \u2191(nthHom f r n)) j - (fun n => \u2191(nthHom f r n)) k) < \u03b5"}, {"tactic": "refine' lt_of_le_of_lt _ hk", "annotated_tactic": ["refine' <a>lt_of_le_of_lt</a> _ hk", [{"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}]], "state_before": "case h\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nk : \u2115\nhk : \u2191p ^ (-\u2191k) < \u03b5\nj : \u2115\nhj : j \u2265 k\n\u22a2 padicNorm p ((fun n => \u2191(nthHom f r n)) j - (fun n => \u2191(nthHom f r n)) k) < \u03b5", "state_after": "case h\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nk : \u2115\nhk : \u2191p ^ (-\u2191k) < \u03b5\nj : \u2115\nhj : j \u2265 k\n\u22a2 padicNorm p ((fun n => \u2191(nthHom f r n)) j - (fun n => \u2191(nthHom f r n)) k) \u2264 \u2191p ^ (-\u2191k)"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "case h\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nk : \u2115\nhk : \u2191p ^ (-\u2191k) < \u03b5\nj : \u2115\nhj : j \u2265 k\n\u22a2 padicNorm p ((fun n => \u2191(nthHom f r n)) j - (fun n => \u2191(nthHom f r n)) k) \u2264 \u2191p ^ (-\u2191k)", "state_after": "case h\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nk : \u2115\nhk : \u2191p ^ (-\u2191k) < \u03b5\nj : \u2115\nhj : j \u2265 k\n\u22a2 padicNorm p \u2191(nthHom f r j - nthHom f r k) \u2264 \u2191p ^ (-\u2191k)"}, {"tactic": "rw [\u2190 padicNorm.dvd_iff_norm_le]", "annotated_tactic": ["rw [\u2190 <a>padicNorm.dvd_iff_norm_le</a>]", [{"full_name": "padicNorm.dvd_iff_norm_le", "def_path": "Mathlib/NumberTheory/Padics/PadicNorm.lean", "def_pos": [255, 9], "def_end_pos": [255, 24]}]], "state_before": "case h\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nk : \u2115\nhk : \u2191p ^ (-\u2191k) < \u03b5\nj : \u2115\nhj : j \u2265 k\n\u22a2 padicNorm p \u2191(nthHom f r j - nthHom f r k) \u2264 \u2191p ^ (-\u2191k)", "state_after": "case h\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nk : \u2115\nhk : \u2191p ^ (-\u2191k) < \u03b5\nj : \u2115\nhj : j \u2265 k\n\u22a2 \u2191(p ^ k) \u2223 nthHom f r j - nthHom f r k"}, {"tactic": "exact_mod_cast pow_dvd_nthHom_sub f_compat r k j hj", "annotated_tactic": ["exact_mod_cast <a>pow_dvd_nthHom_sub</a> f_compat r k j hj", [{"full_name": "PadicInt.pow_dvd_nthHom_sub", "def_path": "Mathlib/NumberTheory/Padics/RingHoms.lean", "def_pos": [496, 9], "def_end_pos": [496, 27]}]], "state_before": "case h\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\nr : R\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nk : \u2115\nhk : \u2191p ^ (-\u2191k) < \u03b5\nj : \u2115\nhj : j \u2265 k\n\u22a2 \u2191(p ^ k) \u2223 nthHom f r j - nthHom f r k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/Zip.lean", "full_name": "List.map_fst_zip", "start": [128, 1], "end": [135, 35], "traced_tactics": [{"tactic": "simp [succ_le_succ_iff] at h", "annotated_tactic": ["simp [<a>succ_le_succ_iff</a>] at h", [{"full_name": "Nat.succ_le_succ_iff", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [257, 9], "def_end_pos": [257, 25]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhead\u271d\u00b9 : \u03b1\nas : List \u03b1\nhead\u271d : \u03b2\nbs : List \u03b2\nh : length (head\u271d\u00b9 :: as) \u2264 length (head\u271d :: bs)\n\u22a2 map Prod.fst (zip (head\u271d\u00b9 :: as) (head\u271d :: bs)) = head\u271d\u00b9 :: as", "state_after": "\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhead\u271d\u00b9 : \u03b1\nas : List \u03b1\nhead\u271d : \u03b2\nbs : List \u03b2\nh : length as \u2264 length bs\n\u22a2 map Prod.fst (zip (head\u271d\u00b9 :: as) (head\u271d :: bs)) = head\u271d\u00b9 :: as"}, {"tactic": "change _ :: map Prod.fst (zip as bs) = _ :: as", "annotated_tactic": ["change _ :: <a>map</a> <a>Prod.fst</a> (<a>zip</a> as bs) = _ :: as", [{"full_name": "List.map", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [151, 19], "def_end_pos": [151, 22]}, {"full_name": "Prod.fst", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [468, 3], "def_end_pos": [468, 6]}, {"full_name": "List.zip", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [557, 5], "def_end_pos": [557, 8]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhead\u271d\u00b9 : \u03b1\nas : List \u03b1\nhead\u271d : \u03b2\nbs : List \u03b2\nh : length as \u2264 length bs\n\u22a2 map Prod.fst (zip (head\u271d\u00b9 :: as) (head\u271d :: bs)) = head\u271d\u00b9 :: as", "state_after": "\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhead\u271d\u00b9 : \u03b1\nas : List \u03b1\nhead\u271d : \u03b2\nbs : List \u03b2\nh : length as \u2264 length bs\n\u22a2 (head\u271d\u00b9, head\u271d).1 :: map Prod.fst (zip as bs) = head\u271d\u00b9 :: as"}, {"tactic": "rw [map_fst_zip as bs h]", "annotated_tactic": ["rw [map_fst_zip as bs h]", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\nhead\u271d\u00b9 : \u03b1\nas : List \u03b1\nhead\u271d : \u03b2\nbs : List \u03b2\nh : length as \u2264 length bs\n\u22a2 (head\u271d\u00b9, head\u271d).1 :: map Prod.fst (zip as bs) = head\u271d\u00b9 :: as", "state_after": "no goals"}, {"tactic": "simp at h", "annotated_tactic": ["simp at h", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\n\u03b4 : Type u_3\n\u03b5 : Type u_4\na : \u03b1\nas : List \u03b1\nh : length (a :: as) \u2264 length []\n\u22a2 map Prod.fst (zip (a :: as) []) = a :: as", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean", "full_name": "NonUnitalSubalgebra.prod_top", "start": [834, 1], "end": [834, 82], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "R : Type u\nA : Type v\nB : Type w\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : NonUnitalNonAssocSemiring A\ninst\u271d\u2076 : Module R A\ninst\u271d\u2075 : IsScalarTower R A A\ninst\u271d\u2074 : SMulCommClass R A A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : Module R B\ninst\u271d\u00b9 : IsScalarTower R B B\ninst\u271d : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nS\u2081 : NonUnitalSubalgebra R B\n\u22a2 prod \u22a4 \u22a4 = \u22a4", "state_after": "case h\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : NonUnitalNonAssocSemiring A\ninst\u271d\u2076 : Module R A\ninst\u271d\u2075 : IsScalarTower R A A\ninst\u271d\u2074 : SMulCommClass R A A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : Module R B\ninst\u271d\u00b9 : IsScalarTower R B B\ninst\u271d : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nS\u2081 : NonUnitalSubalgebra R B\nx\u271d : A \u00d7 B\n\u22a2 x\u271d \u2208 prod \u22a4 \u22a4 \u2194 x\u271d \u2208 \u22a4"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : NonUnitalNonAssocSemiring A\ninst\u271d\u2076 : Module R A\ninst\u271d\u2075 : IsScalarTower R A A\ninst\u271d\u2074 : SMulCommClass R A A\ninst\u271d\u00b3 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b2 : Module R B\ninst\u271d\u00b9 : IsScalarTower R B B\ninst\u271d : SMulCommClass R B B\nS : NonUnitalSubalgebra R A\nS\u2081 : NonUnitalSubalgebra R B\nx\u271d : A \u00d7 B\n\u22a2 x\u271d \u2208 prod \u22a4 \u22a4 \u2194 x\u271d \u2208 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/GroupAction/Support.lean", "full_name": "MulAction.Supports.mono", "start": [46, 1], "end": [47, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Dynamics/OmegaLimit.lean", "full_name": "omegaLimit_subset_of_tendsto", "start": [74, 1], "end": [78, 76], "traced_tactics": [{"tactic": "refine' iInter\u2082_mono' fun u hu \u21a6 \u27e8m \u207b\u00b9' u, tendsto_def.mp hf _ hu, _\u27e9", "annotated_tactic": ["refine' <a>iInter\u2082_mono'</a> fun u hu \u21a6 \u27e8m \u207b\u00b9' u, tendsto_def.mp hf _ hu, _\u27e9", [{"full_name": "Set.iInter\u2082_mono'", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [529, 9], "def_end_pos": [529, 22]}]], "state_before": "\u03c4 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nf : Filter \u03c4\n\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2\ns s\u2081 s\u2082 : Set \u03b1\nm : \u03c4 \u2192 \u03c4\nf\u2081 f\u2082 : Filter \u03c4\nhf : Tendsto m f\u2081 f\u2082\n\u22a2 \u03c9 f\u2081 (fun t x => \u03d5 (m t) x) s \u2286 \u03c9 f\u2082 \u03d5 s", "state_after": "\u03c4 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nf : Filter \u03c4\n\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2\ns s\u2081 s\u2082 : Set \u03b1\nm : \u03c4 \u2192 \u03c4\nf\u2081 f\u2082 : Filter \u03c4\nhf : Tendsto m f\u2081 f\u2082\nu : Set \u03c4\nhu : u \u2208 f\u2082\n\u22a2 closure (image2 (fun t x => \u03d5 (m t) x) (m \u207b\u00b9' u) s) \u2286 closure (image2 \u03d5 u s)"}, {"tactic": "rw [\u2190 image2_image_left]", "annotated_tactic": ["rw [\u2190 <a>image2_image_left</a>]", [{"full_name": "Set.image2_image_left", "def_path": "Mathlib/Data/Set/NAry.lean", "def_pos": [286, 9], "def_end_pos": [286, 26]}]], "state_before": "\u03c4 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nf : Filter \u03c4\n\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2\ns s\u2081 s\u2082 : Set \u03b1\nm : \u03c4 \u2192 \u03c4\nf\u2081 f\u2082 : Filter \u03c4\nhf : Tendsto m f\u2081 f\u2082\nu : Set \u03c4\nhu : u \u2208 f\u2082\n\u22a2 closure (image2 (fun t x => \u03d5 (m t) x) (m \u207b\u00b9' u) s) \u2286 closure (image2 \u03d5 u s)", "state_after": "\u03c4 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b9 : Type u_4\ninst\u271d : 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"Mathlib/Data/Set/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}]], "state_before": "\u03c4 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nf : Filter \u03c4\n\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2\ns s\u2081 s\u2082 : Set \u03b1\nm : \u03c4 \u2192 \u03c4\nf\u2081 f\u2082 : Filter \u03c4\nhf : Tendsto m f\u2081 f\u2082\nu : Set \u03c4\nhu : u \u2208 f\u2082\n\u22a2 closure (image2 \u03d5 ((fun t => m t) '' (m \u207b\u00b9' u)) s) \u2286 closure (image2 \u03d5 u s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Monad.lean", "full_name": "MvPolynomial.bind\u2082_C_left", "start": [184, 1], "end": [184, 96], "traced_tactics": [{"tactic": "ext : 2 <;> simp", "annotated_tactic": ["ext : 2 <;> simp", []], "state_before": "\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u22a2 bind\u2082 C = RingHom.id (MvPolynomial \u03c3 R)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Functor/Flat.lean", "full_name": "CategoryTheory.flat_iff_lan_flat", "start": [382, 1], "end": [390, 41], "traced_tactics": [{"tactic": "haveI := preservesFiniteLimitsOfFlat (lan F.op : _ \u2964 D\u1d52\u1d56 \u2964 Type u\u2081)", "annotated_tactic": ["haveI := <a>preservesFiniteLimitsOfFlat</a> (<a>lan</a> F.op : _ \u2964 D\u1d52\u1d56 \u2964 Type u\u2081)", [{"full_name": "CategoryTheory.preservesFiniteLimitsOfFlat", "def_path": "Mathlib/CategoryTheory/Functor/Flat.lean", "def_pos": [275, 19], "def_end_pos": [275, 46]}, {"full_name": "CategoryTheory.lan", "def_path": "Mathlib/CategoryTheory/Limits/KanExtension.lean", "def_pos": [337, 5], "def_end_pos": [337, 8]}]], "state_before": "C D : Type u\u2081\ninst\u271d\u2079 : SmallCategory C\ninst\u271d\u2078 : SmallCategory D\nE : Type u\u2082\ninst\u271d\u2077 : Category.{u\u2081, u\u2082} E\ninst\u271d\u2076 : ConcreteCategory E\ninst\u271d\u2075 : HasLimits E\ninst\u271d\u2074 : HasColimits E\ninst\u271d\u00b3 : ReflectsLimits (forget E)\ninst\u271d\u00b2 : PreservesFilteredColimits (forget E)\ninst\u271d\u00b9 : PreservesLimits (forget E)\ninst\u271d : HasFiniteLimits C\nF : C \u2964 D\nH : RepresentablyFlat (lan F.op)\n\u22a2 RepresentablyFlat F", "state_after": "C D : Type u\u2081\ninst\u271d\u2079 : SmallCategory C\ninst\u271d\u2078 : SmallCategory D\nE : Type u\u2082\ninst\u271d\u2077 : Category.{u\u2081, u\u2082} E\ninst\u271d\u2076 : ConcreteCategory E\ninst\u271d\u2075 : HasLimits E\ninst\u271d\u2074 : HasColimits E\ninst\u271d\u00b3 : ReflectsLimits (forget E)\ninst\u271d\u00b2 : PreservesFilteredColimits (forget E)\ninst\u271d\u00b9 : PreservesLimits (forget E)\ninst\u271d : HasFiniteLimits C\nF : C \u2964 D\nH : RepresentablyFlat (lan F.op)\nthis : PreservesFiniteLimits (lan F.op)\n\u22a2 RepresentablyFlat F"}, {"tactic": "haveI : PreservesFiniteLimits F := by\n  apply preservesFiniteLimitsOfPreservesFiniteLimitsOfSize.{u\u2081}\n  intros; skip; apply preservesLimitOfLanPreservesLimit", "annotated_tactic": ["haveI : <a>PreservesFiniteLimits</a> F := by\n      apply <a>preservesFiniteLimitsOfPreservesFiniteLimitsOfSize</a>.{u\u2081}\n      intros; skip; apply <a>preservesLimitOfLanPreservesLimit</a>", [{"full_name": "CategoryTheory.Limits.PreservesFiniteLimits", "def_path": "Mathlib/CategoryTheory/Limits/Preserves/Finite.lean", "def_pos": [45, 7], "def_end_pos": [45, 28]}, {"full_name": "CategoryTheory.Limits.preservesFiniteLimitsOfPreservesFiniteLimitsOfSize", "def_path": "Mathlib/CategoryTheory/Limits/Preserves/Finite.lean", "def_pos": [85, 5], "def_end_pos": [85, 55]}, {"full_name": "CategoryTheory.preservesLimitOfLanPreservesLimit", "def_path": "Mathlib/CategoryTheory/Limits/Preserves/FunctorCategory.lean", "def_pos": [108, 19], "def_end_pos": [108, 52]}]], "state_before": "C D : Type u\u2081\ninst\u271d\u2079 : SmallCategory C\ninst\u271d\u2078 : SmallCategory D\nE : Type u\u2082\ninst\u271d\u2077 : Category.{u\u2081, u\u2082} E\ninst\u271d\u2076 : ConcreteCategory E\ninst\u271d\u2075 : HasLimits E\ninst\u271d\u2074 : HasColimits E\ninst\u271d\u00b3 : ReflectsLimits (forget E)\ninst\u271d\u00b2 : PreservesFilteredColimits (forget E)\ninst\u271d\u00b9 : PreservesLimits (forget E)\ninst\u271d : HasFiniteLimits C\nF : C \u2964 D\nH : RepresentablyFlat (lan F.op)\nthis : PreservesFiniteLimits (lan F.op)\n\u22a2 RepresentablyFlat F", "state_after": "C D : Type u\u2081\ninst\u271d\u2079 : SmallCategory C\ninst\u271d\u2078 : SmallCategory D\nE : Type u\u2082\ninst\u271d\u2077 : Category.{u\u2081, u\u2082} E\ninst\u271d\u2076 : ConcreteCategory E\ninst\u271d\u2075 : HasLimits E\ninst\u271d\u2074 : HasColimits E\ninst\u271d\u00b3 : ReflectsLimits (forget E)\ninst\u271d\u00b2 : PreservesFilteredColimits (forget E)\ninst\u271d\u00b9 : PreservesLimits (forget E)\ninst\u271d : HasFiniteLimits C\nF : C \u2964 D\nH : RepresentablyFlat (lan F.op)\nthis\u271d : PreservesFiniteLimits (lan F.op)\nthis : PreservesFiniteLimits F\n\u22a2 RepresentablyFlat F"}, {"tactic": "apply flat_of_preservesFiniteLimits", "annotated_tactic": ["apply <a>flat_of_preservesFiniteLimits</a>", [{"full_name": "CategoryTheory.flat_of_preservesFiniteLimits", "def_path": "Mathlib/CategoryTheory/Functor/Flat.lean", "def_pos": [148, 9], "def_end_pos": [148, 38]}]], "state_before": "C D : Type u\u2081\ninst\u271d\u2079 : SmallCategory C\ninst\u271d\u2078 : SmallCategory D\nE : Type u\u2082\ninst\u271d\u2077 : Category.{u\u2081, u\u2082} E\ninst\u271d\u2076 : ConcreteCategory E\ninst\u271d\u2075 : HasLimits E\ninst\u271d\u2074 : HasColimits E\ninst\u271d\u00b3 : ReflectsLimits (forget E)\ninst\u271d\u00b2 : PreservesFilteredColimits (forget E)\ninst\u271d\u00b9 : PreservesLimits (forget E)\ninst\u271d : HasFiniteLimits C\nF : C \u2964 D\nH : RepresentablyFlat (lan F.op)\nthis\u271d : PreservesFiniteLimits (lan F.op)\nthis : PreservesFiniteLimits F\n\u22a2 RepresentablyFlat F", "state_after": "no goals"}, {"tactic": "apply preservesFiniteLimitsOfPreservesFiniteLimitsOfSize.{u\u2081}", "annotated_tactic": ["apply <a>preservesFiniteLimitsOfPreservesFiniteLimitsOfSize</a>.{u\u2081}", [{"full_name": "CategoryTheory.Limits.preservesFiniteLimitsOfPreservesFiniteLimitsOfSize", "def_path": "Mathlib/CategoryTheory/Limits/Preserves/Finite.lean", "def_pos": [85, 5], "def_end_pos": [85, 55]}]], "state_before": "C D : Type u\u2081\ninst\u271d\u2079 : SmallCategory C\ninst\u271d\u2078 : SmallCategory D\nE : Type u\u2082\ninst\u271d\u2077 : Category.{u\u2081, u\u2082} E\ninst\u271d\u2076 : ConcreteCategory E\ninst\u271d\u2075 : HasLimits E\ninst\u271d\u2074 : HasColimits E\ninst\u271d\u00b3 : ReflectsLimits (forget E)\ninst\u271d\u00b2 : PreservesFilteredColimits (forget E)\ninst\u271d\u00b9 : PreservesLimits (forget E)\ninst\u271d : HasFiniteLimits C\nF : C \u2964 D\nH : RepresentablyFlat (lan F.op)\nthis : PreservesFiniteLimits (lan F.op)\n\u22a2 PreservesFiniteLimits F", "state_after": "case h\nC D : Type u\u2081\ninst\u271d\u2079 : SmallCategory C\ninst\u271d\u2078 : SmallCategory D\nE : Type u\u2082\ninst\u271d\u2077 : Category.{u\u2081, u\u2082} E\ninst\u271d\u2076 : ConcreteCategory E\ninst\u271d\u2075 : HasLimits E\ninst\u271d\u2074 : HasColimits E\ninst\u271d\u00b3 : ReflectsLimits (forget E)\ninst\u271d\u00b2 : PreservesFilteredColimits (forget E)\ninst\u271d\u00b9 : PreservesLimits (forget E)\ninst\u271d : HasFiniteLimits C\nF : C \u2964 D\nH : RepresentablyFlat (lan F.op)\nthis : PreservesFiniteLimits (lan F.op)\n\u22a2 (J : Type u\u2081) \u2192 {\ud835\udca5 : SmallCategory J} \u2192 FinCategory J \u2192 PreservesLimitsOfShape J F"}, {"tactic": "intros", "annotated_tactic": ["intros", []], "state_before": "case h\nC D : Type u\u2081\ninst\u271d\u2079 : SmallCategory C\ninst\u271d\u2078 : SmallCategory D\nE : Type u\u2082\ninst\u271d\u2077 : Category.{u\u2081, u\u2082} E\ninst\u271d\u2076 : ConcreteCategory E\ninst\u271d\u2075 : HasLimits E\ninst\u271d\u2074 : HasColimits E\ninst\u271d\u00b3 : ReflectsLimits (forget E)\ninst\u271d\u00b2 : PreservesFilteredColimits (forget E)\ninst\u271d\u00b9 : PreservesLimits (forget E)\ninst\u271d : HasFiniteLimits C\nF : C \u2964 D\nH : RepresentablyFlat (lan F.op)\nthis : PreservesFiniteLimits (lan F.op)\n\u22a2 (J : Type u\u2081) \u2192 {\ud835\udca5 : SmallCategory J} \u2192 FinCategory J \u2192 PreservesLimitsOfShape J F", "state_after": "case h\nC D : Type u\u2081\ninst\u271d\u2079 : SmallCategory C\ninst\u271d\u2078 : SmallCategory D\nE : Type u\u2082\ninst\u271d\u2077 : Category.{u\u2081, u\u2082} E\ninst\u271d\u2076 : ConcreteCategory E\ninst\u271d\u2075 : HasLimits E\ninst\u271d\u2074 : HasColimits E\ninst\u271d\u00b3 : ReflectsLimits (forget E)\ninst\u271d\u00b2 : PreservesFilteredColimits (forget E)\ninst\u271d\u00b9 : PreservesLimits (forget E)\ninst\u271d : HasFiniteLimits C\nF : C \u2964 D\nH : RepresentablyFlat (lan F.op)\nthis : PreservesFiniteLimits (lan F.op)\nJ\u271d : Type u\u2081\n\ud835\udca5\u271d : SmallCategory J\u271d\nx\u271d : FinCategory J\u271d\n\u22a2 PreservesLimitsOfShape J\u271d F"}, {"tactic": "apply preservesLimitOfLanPreservesLimit", "annotated_tactic": ["apply <a>preservesLimitOfLanPreservesLimit</a>", [{"full_name": "CategoryTheory.preservesLimitOfLanPreservesLimit", "def_path": "Mathlib/CategoryTheory/Limits/Preserves/FunctorCategory.lean", "def_pos": [108, 19], "def_end_pos": [108, 52]}]], "state_before": "case h\nC D : Type u\u2081\ninst\u271d\u2079 : SmallCategory C\ninst\u271d\u2078 : SmallCategory D\nE : Type u\u2082\ninst\u271d\u2077 : Category.{u\u2081, u\u2082} E\ninst\u271d\u2076 : ConcreteCategory E\ninst\u271d\u2075 : HasLimits E\ninst\u271d\u2074 : HasColimits E\ninst\u271d\u00b3 : ReflectsLimits (forget E)\ninst\u271d\u00b2 : PreservesFilteredColimits (forget E)\ninst\u271d\u00b9 : PreservesLimits (forget E)\ninst\u271d : HasFiniteLimits C\nF : C \u2964 D\nH : RepresentablyFlat (lan F.op)\nthis : PreservesFiniteLimits (lan F.op)\nJ\u271d : Type u\u2081\n\ud835\udca5\u271d : SmallCategory J\u271d\nx\u271d : FinCategory J\u271d\n\u22a2 PreservesLimitsOfShape J\u271d F", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/Prime.lean", "full_name": "Nat.Prime.ne_one", "start": [84, 1], "end": [85, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Ring/Ideal.lean", "full_name": "QuotientRing.isOpenMap_coe", "start": [61, 1], "end": [65, 64], "traced_tactics": [{"tactic": "intro s s_op", "annotated_tactic": ["intro s s_op", []], "state_before": "R : Type u_1\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : CommRing R\nN : Ideal R\ninst\u271d : TopologicalRing R\n\u22a2 IsOpenMap \u2191(mk N)", "state_after": "R : Type u_1\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : CommRing R\nN : Ideal R\ninst\u271d : TopologicalRing R\ns : Set R\ns_op : IsOpen s\n\u22a2 IsOpen (\u2191(mk N) '' s)"}, {"tactic": "change IsOpen (mk N \u207b\u00b9' (mk N '' s))", "annotated_tactic": ["change <a>IsOpen</a> (<a>mk</a> N \u207b\u00b9' (<a>mk</a> N '' s))", [{"full_name": "IsOpen", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [101, 5], "def_end_pos": [101, 11]}, {"full_name": "Ideal.Quotient.mk", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [96, 5], "def_end_pos": [96, 7]}, {"full_name": "Ideal.Quotient.mk", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [96, 5], "def_end_pos": [96, 7]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : CommRing R\nN : Ideal R\ninst\u271d : TopologicalRing R\ns : Set R\ns_op : IsOpen s\n\u22a2 IsOpen (\u2191(mk N) '' s)", "state_after": "R : Type u_1\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : CommRing R\nN : Ideal R\ninst\u271d : TopologicalRing R\ns : Set R\ns_op : IsOpen s\n\u22a2 IsOpen (\u2191(mk N) \u207b\u00b9' (\u2191(mk N) '' s))"}, {"tactic": "rw [quotient_ring_saturate]", "annotated_tactic": ["rw [<a>quotient_ring_saturate</a>]", [{"full_name": "Ideal.Quotient.quotient_ring_saturate", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [169, 9], "def_end_pos": [169, 31]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : CommRing R\nN : Ideal R\ninst\u271d : TopologicalRing R\ns : Set R\ns_op : IsOpen s\n\u22a2 IsOpen (\u2191(mk N) \u207b\u00b9' (\u2191(mk N) '' s))", "state_after": "R : Type u_1\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : CommRing R\nN : Ideal R\ninst\u271d : TopologicalRing R\ns : Set R\ns_op : IsOpen s\n\u22a2 IsOpen (\u22c3 x, (fun y => \u2191x + y) '' s)"}, {"tactic": "exact isOpen_iUnion fun \u27e8n, _\u27e9 => isOpenMap_add_left n s s_op", "annotated_tactic": ["exact <a>isOpen_iUnion</a> fun \u27e8n, _\u27e9 => <a>isOpenMap_add_left</a> n s s_op", [{"full_name": "isOpen_iUnion", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [147, 9], "def_end_pos": [147, 22]}, {"full_name": "isOpenMap_add_left", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [76, 3], "def_end_pos": [76, 14]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : CommRing R\nN : Ideal R\ninst\u271d : TopologicalRing R\ns : Set R\ns_op : IsOpen s\n\u22a2 IsOpen (\u22c3 x, (fun y => \u2191x + y) '' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/QuotientGroup.lean", "full_name": "QuotientGroup.homQuotientZPowOfHom_id", "start": [518, 1], "end": [519, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/ContinuousOn.lean", "full_name": "continuousWithinAt_const", "start": [1029, 1], "end": [1031, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "full_name": "Nat.div2_succ", "start": [117, 1], "end": [124, 9], "traced_tactics": [{"tactic": "simp only [bodd, boddDiv2, div2]", "annotated_tactic": ["simp only [<a>bodd</a>, <a>boddDiv2</a>, <a>div2</a>]", [{"full_name": "Nat.bodd", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [50, 5], "def_end_pos": [50, 9]}, {"full_name": "Nat.boddDiv2", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [36, 5], "def_end_pos": [36, 13]}, {"full_name": "Nat.div2", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [45, 5], "def_end_pos": [45, 9]}]], "state_before": "n : \u2115\n\u22a2 div2 (succ n) = bif bodd n then succ (div2 n) else div2 n", "state_after": "n : \u2115\n\u22a2 (match boddDiv2 n with\n      | (false, m) => (true, m)\n      | (true, m) => (false, succ m)).snd =\n    bif (boddDiv2 n).fst then succ (boddDiv2 n).snd else (boddDiv2 n).snd"}, {"tactic": "cases' boddDiv2 n with fst snd", "annotated_tactic": ["cases' <a>boddDiv2</a> n with fst snd", [{"full_name": "Nat.boddDiv2", "def_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "def_pos": [36, 5], "def_end_pos": [36, 13]}]], "state_before": "n : \u2115\n\u22a2 (match boddDiv2 n with\n      | (false, m) => (true, m)\n      | (true, m) => (false, succ m)).snd =\n    bif (boddDiv2 n).fst then succ (boddDiv2 n).snd else (boddDiv2 n).snd", "state_after": "case mk\nn : \u2115\nfst : Bool\nsnd : \u2115\n\u22a2 (match (fst, snd) with\n      | (false, m) => (true, m)\n      | (true, m) => (false, succ m)).snd =\n    bif (fst, snd).fst then succ (fst, snd).snd else (fst, snd).snd"}, {"tactic": "cases fst", "annotated_tactic": ["cases fst", []], "state_before": "case mk\nn : \u2115\nfst : Bool\nsnd : \u2115\n\u22a2 (match (fst, snd) with\n      | (false, m) => (true, m)\n      | (true, m) => (false, succ m)).snd =\n    bif (fst, snd).fst then succ (fst, snd).snd else (fst, snd).snd", "state_after": "case mk.false\nn snd : \u2115\n\u22a2 (match (false, snd) with\n      | (false, m) => (true, m)\n      | (true, m) => (false, succ m)).snd =\n    bif (false, snd).fst then succ (false, snd).snd else (false, snd).snd\n\ncase mk.true\nn snd : \u2115\n\u22a2 (match (true, snd) with\n      | (false, m) => (true, m)\n      | (true, m) => (false, succ m)).snd =\n    bif (true, snd).fst then succ (true, snd).snd else (true, snd).snd"}, {"tactic": "case mk.false =>\n  simp", "annotated_tactic": ["case mk.false =>\n    simp", []], "state_before": "case mk.false\nn snd : \u2115\n\u22a2 (match (false, snd) with\n      | (false, m) => (true, m)\n      | (true, m) => (false, succ m)).snd =\n    bif (false, snd).fst then succ (false, snd).snd else (false, snd).snd\n\ncase mk.true\nn snd : \u2115\n\u22a2 (match (true, snd) with\n      | (false, m) => (true, m)\n      | (true, m) => (false, succ m)).snd =\n    bif (true, snd).fst then succ (true, snd).snd else (true, snd).snd", "state_after": "case mk.true\nn snd : \u2115\n\u22a2 (match (true, snd) with\n      | (false, m) => (true, m)\n      | (true, m) => (false, succ m)).snd =\n    bif (true, snd).fst then succ (true, snd).snd else (true, snd).snd"}, {"tactic": "case mk.true =>\n  simp", "annotated_tactic": ["case mk.true =>\n    simp", []], "state_before": "case mk.true\nn snd : \u2115\n\u22a2 (match (true, snd) with\n      | (false, m) => (true, m)\n      | (true, m) => (false, succ m)).snd =\n    bif (true, snd).fst then succ (true, snd).snd else (true, snd).snd", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "n snd : \u2115\n\u22a2 (match (false, snd) with\n      | (false, m) => (true, m)\n      | (true, m) => (false, succ m)).snd =\n    bif (false, snd).fst then succ (false, snd).snd else (false, snd).snd", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "n snd : \u2115\n\u22a2 (match (true, snd) with\n      | (false, m) => (true, m)\n      | (true, m) => (false, succ m)).snd =\n    bif (true, snd).fst then succ (true, snd).snd else (true, snd).snd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Category/TopCat/Basic.lean", "full_name": "TopCat.openEmbedding_iff_comp_isIso'", "start": [169, 1], "end": [172, 41], "traced_tactics": [{"tactic": "simp only [\u2190Functor.map_comp]", "annotated_tactic": ["simp only [\u2190<a>Functor.map_comp</a>]", [{"full_name": "CategoryTheory.Functor.map_comp", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [43, 3], "def_end_pos": [43, 11]}]], "state_before": "X Y Z : TopCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d : IsIso g\n\u22a2 OpenEmbedding ((forget TopCat).map f \u226b (forget TopCat).map g) \u2194 OpenEmbedding \u2191f", "state_after": "X Y Z : TopCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d : IsIso g\n\u22a2 OpenEmbedding ((forget TopCat).map (f \u226b g)) \u2194 OpenEmbedding \u2191f"}, {"tactic": "exact openEmbedding_iff_comp_isIso f g", "annotated_tactic": ["exact <a>openEmbedding_iff_comp_isIso</a> f g", [{"full_name": "TopCat.openEmbedding_iff_comp_isIso", "def_path": "Mathlib/Topology/Category/TopCat/Basic.lean", "def_pos": [162, 9], "def_end_pos": [162, 37]}]], "state_before": "X Y Z : TopCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d : IsIso g\n\u22a2 OpenEmbedding ((forget TopCat).map (f \u226b g)) \u2194 OpenEmbedding \u2191f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/Deriv/Pow.lean", "full_name": "derivWithin_pow'", "start": [112, 1], "end": [114, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/Basic.lean", "full_name": "Polynomial.eq_zero_of_eq_zero", "start": [869, 1], "end": [870, 38], "traced_tactics": [{"tactic": "rw [\u2190 one_smul R p, \u2190 h, zero_smul]", "annotated_tactic": ["rw [\u2190 <a>one_smul</a> R p, \u2190 h, <a>zero_smul</a>]", [{"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}, {"full_name": "zero_smul", "def_path": "Mathlib/Algebra/SMulWithZero.lean", "def_pos": [70, 9], "def_end_pos": [70, 18]}]], "state_before": "R : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\nh : 0 = 1\np : R[X]\n\u22a2 p = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Euclidean/MongePoint.lean", "full_name": "Affine.Triangle.orthocenter_mem_altitude", "start": [490, 1], "end": [496, 36], "traced_tactics": [{"tactic": "obtain \u27e8i\u2082, i\u2083, h\u2081\u2082, h\u2082\u2083, h\u2081\u2083\u27e9 : \u2203 i\u2082 i\u2083, i\u2081 \u2260 i\u2082 \u2227 i\u2082 \u2260 i\u2083 \u2227 i\u2081 \u2260 i\u2083 := by\n  fin_cases i\u2081 <;> decide", "annotated_tactic": ["obtain \u27e8i\u2082, i\u2083, h\u2081\u2082, h\u2082\u2083, h\u2081\u2083\u27e9 : \u2203 i\u2082 i\u2083, i\u2081 \u2260 i\u2082 \u2227 i\u2082 \u2260 i\u2083 \u2227 i\u2081 \u2260 i\u2083 := by\n    -- porting note: was `decide!`\n    fin_cases i\u2081 <;> decide", []], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nt : Triangle \u211d P\ni\u2081 : Fin 3\n\u22a2 orthocenter t \u2208 altitude t i\u2081", "state_after": "case intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nt : Triangle \u211d P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\n\u22a2 orthocenter t \u2208 altitude t i\u2081"}, {"tactic": "rw [orthocenter_eq_mongePoint, t.altitude_eq_mongePlane h\u2081\u2082 h\u2081\u2083 h\u2082\u2083]", "annotated_tactic": ["rw [<a>orthocenter_eq_mongePoint</a>, t.altitude_eq_mongePlane h\u2081\u2082 h\u2081\u2083 h\u2082\u2083]", [{"full_name": "Affine.Triangle.orthocenter_eq_mongePoint", "def_path": "Mathlib/Geometry/Euclidean/MongePoint.lean", "def_pos": [445, 9], "def_end_pos": [445, 34]}]], "state_before": "case intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nt : Triangle \u211d P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\n\u22a2 orthocenter t \u2208 altitude t i\u2081", "state_after": "case intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nt : Triangle \u211d P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\n\u22a2 mongePoint t \u2208 mongePlane t i\u2082 i\u2083"}, {"tactic": "exact t.mongePoint_mem_mongePlane", "annotated_tactic": ["exact t.mongePoint_mem_mongePlane", []], "state_before": "case intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nt : Triangle \u211d P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\n\u22a2 mongePoint t \u2208 mongePlane t i\u2082 i\u2083", "state_after": "no goals"}, {"tactic": "fin_cases i\u2081 <;> decide", "annotated_tactic": ["fin_cases i\u2081 <;> decide", []], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nt : Triangle \u211d P\ni\u2081 : Fin 3\n\u22a2 \u2203 i\u2082 i\u2083, i\u2081 \u2260 i\u2082 \u2227 i\u2082 \u2260 i\u2083 \u2227 i\u2081 \u2260 i\u2083", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "full_name": "Set.Nonempty.inv", "start": [251, 1], "end": [252, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.Ico_union_Ico_eq_Ico", "start": [794, 1], "end": [796, 89], "traced_tactics": [{"tactic": "rw [\u2190 coe_inj, coe_union, coe_Ico, coe_Ico, coe_Ico, Set.Ico_union_Ico_eq_Ico hab hbc]", "annotated_tactic": ["rw [\u2190 <a>coe_inj</a>, <a>coe_union</a>, <a>coe_Ico</a>, <a>coe_Ico</a>, <a>coe_Ico</a>, <a>Set.Ico_union_Ico_eq_Ico</a> hab hbc]", [{"full_name": "Finset.coe_inj", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [244, 9], "def_end_pos": [244, 16]}, {"full_name": "Finset.coe_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1399, 9], "def_end_pos": [1399, 18]}, {"full_name": "Finset.coe_Ico", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [351, 9], "def_end_pos": [351, 16]}, {"full_name": "Finset.coe_Ico", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [351, 9], "def_end_pos": [351, 16]}, {"full_name": "Finset.coe_Ico", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [351, 9], "def_end_pos": [351, 16]}, {"full_name": "Set.Ico_union_Ico_eq_Ico", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [1529, 9], "def_end_pos": [1529, 29]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nhab : a \u2264 b\nhbc : b \u2264 c\n\u22a2 Ico a b \u222a Ico b c = Ico a c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.singleton_one", "start": [100, 1], "end": [101, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Matrix/BilinearForm.lean", "full_name": "Matrix.nondegenerate_toBilin_iff", "start": [566, 1], "end": [568, 100], "traced_tactics": [{"tactic": "rw [\u2190 Matrix.nondegenerate_toBilin'_iff_nondegenerate_toBilin, Matrix.nondegenerate_toBilin'_iff]", "annotated_tactic": ["rw [\u2190 <a>Matrix.nondegenerate_toBilin'_iff_nondegenerate_toBilin</a>, <a>Matrix.nondegenerate_toBilin'_iff</a>]", [{"full_name": "Matrix.nondegenerate_toBilin'_iff_nondegenerate_toBilin", "def_path": "Mathlib/LinearAlgebra/Matrix/BilinearForm.lean", "def_pos": [543, 9], "def_end_pos": [543, 71]}, {"full_name": "Matrix.nondegenerate_toBilin'_iff", "def_path": "Mathlib/LinearAlgebra/Matrix/BilinearForm.lean", "def_pos": [555, 9], "def_end_pos": [555, 49]}]], "state_before": "R : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2079 : Semiring R\ninst\u271d\u00b9\u2078 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2077 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2076 : Ring R\u2081\ninst\u271d\u00b9\u2075 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2074 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b9 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2070 : CommRing R\u2083\ninst\u271d\u2079 : AddCommGroup M\u2083\ninst\u271d\u2078 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nA : Type u_11\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : IsDomain A\ninst\u271d\u00b2 : Module A M\u2083\nB\u2083 : BilinForm A M\u2083\n\u03b9 : Type u_12\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nM : Matrix \u03b9 \u03b9 R\u2083\nb : Basis \u03b9 R\u2083 M\u2083\n\u22a2 Nondegenerate (\u2191(toBilin b) M) \u2194 Matrix.Nondegenerate M", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean", "full_name": "Orientation.oangle_sign_add_left", "start": [968, 1], "end": [969, 53], "traced_tactics": [{"tactic": "rw [\u2190 o.oangle_sign_add_smul_left x y 1, one_smul]", "annotated_tactic": ["rw [\u2190 o.oangle_sign_add_smul_left x y 1, <a>one_smul</a>]", [{"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}]], "state_before": "V : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u22a2 Real.Angle.sign (oangle o (x + y) y) = Real.Angle.sign (oangle o x y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Perm/Basic.lean", "full_name": "Equiv.Perm.ofSubtype_apply_coe", "start": [455, 1], "end": [456, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Lie/Basic.lean", "full_name": "LieEquiv.refl_symm", "start": [648, 1], "end": [649, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Game/Birthday.lean", "full_name": "SetTheory.PGame.birthday_add_zero", "start": [176, 1], "end": [176, 69], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "a b x : PGame\n\u22a2 birthday (a + 0) = birthday a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Euclidean/Circumcenter.lean", "full_name": "Affine.Simplex.point_eq_affineCombination_of_pointsWithCircumcenter", "start": [592, 1], "end": [606, 8], "traced_tactics": [{"tactic": "rw [\u2190 pointsWithCircumcenter_point]", "annotated_tactic": ["rw [\u2190 <a>pointsWithCircumcenter_point</a>]", [{"full_name": "Affine.Simplex.pointsWithCircumcenter_point", "def_path": "Mathlib/Geometry/Euclidean/Circumcenter.lean", "def_pos": [562, 9], "def_end_pos": [562, 37]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ns : Simplex \u211d P n\ni : Fin (n + 1)\n\u22a2 points s i = \u2191(affineCombination \u211d univ (pointsWithCircumcenter s)) (pointWeightsWithCircumcenter i)", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ns : Simplex \u211d P n\ni : Fin (n + 1)\n\u22a2 pointsWithCircumcenter s (point_index i) =\n    \u2191(affineCombination \u211d univ (pointsWithCircumcenter s)) (pointWeightsWithCircumcenter i)"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ns : Simplex \u211d P n\ni : Fin (n + 1)\n\u22a2 pointsWithCircumcenter s (point_index i) =\n    \u2191(affineCombination \u211d univ (pointsWithCircumcenter s)) (pointWeightsWithCircumcenter i)", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ns : Simplex \u211d P n\ni : Fin (n + 1)\n\u22a2 \u2191(affineCombination \u211d univ (pointsWithCircumcenter s)) (pointWeightsWithCircumcenter i) =\n    pointsWithCircumcenter s (point_index i)"}, {"tactic": "refine'\n  affineCombination_of_eq_one_of_eq_zero _ _ _ (mem_univ _)\n    (by simp [pointWeightsWithCircumcenter]) _", "annotated_tactic": ["refine'\n    <a>affineCombination_of_eq_one_of_eq_zero</a> _ _ _ (<a>mem_univ</a> _)\n      (by simp [<a>pointWeightsWithCircumcenter</a>]) _", [{"full_name": "Finset.affineCombination_of_eq_one_of_eq_zero", "def_path": "Mathlib/LinearAlgebra/AffineSpace/Combination.lean", "def_pos": [477, 9], "def_end_pos": [477, 47]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Affine.Simplex.pointWeightsWithCircumcenter", "def_path": "Mathlib/Geometry/Euclidean/Circumcenter.lean", "def_pos": [577, 5], "def_end_pos": [577, 33]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ns : Simplex \u211d P n\ni : Fin (n + 1)\n\u22a2 \u2191(affineCombination \u211d univ (pointsWithCircumcenter s)) (pointWeightsWithCircumcenter i) =\n    pointsWithCircumcenter s (point_index i)", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ns : Simplex \u211d P n\ni : Fin (n + 1)\n\u22a2 \u2200 (i2 : PointsWithCircumcenterIndex n), i2 \u2208 univ \u2192 i2 \u2260 point_index i \u2192 pointWeightsWithCircumcenter i i2 = 0"}, {"tactic": "intro i hi hn", "annotated_tactic": ["intro i hi hn", []], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ns : Simplex \u211d P n\ni : Fin (n + 1)\n\u22a2 \u2200 (i2 : PointsWithCircumcenterIndex n), i2 \u2208 univ \u2192 i2 \u2260 point_index i \u2192 pointWeightsWithCircumcenter i i2 = 0", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ns : Simplex \u211d P n\ni\u271d : Fin (n + 1)\ni : PointsWithCircumcenterIndex n\nhi : i \u2208 univ\nhn : i \u2260 point_index i\u271d\n\u22a2 pointWeightsWithCircumcenter i\u271d i = 0"}, {"tactic": "cases i", "annotated_tactic": ["cases i", []], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ns : Simplex \u211d P n\ni\u271d : Fin (n + 1)\ni : PointsWithCircumcenterIndex n\nhi : i \u2208 univ\nhn : i \u2260 point_index i\u271d\n\u22a2 pointWeightsWithCircumcenter i\u271d i = 0", "state_after": "case point_index\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ns : Simplex \u211d P n\ni a\u271d : Fin (n + 1)\nhi : point_index a\u271d \u2208 univ\nhn : point_index a\u271d \u2260 point_index i\n\u22a2 pointWeightsWithCircumcenter i (point_index a\u271d) = 0\n\ncase circumcenter_index\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ns : Simplex \u211d P n\ni : Fin (n + 1)\nhi : circumcenter_index \u2208 univ\nhn : circumcenter_index \u2260 point_index i\n\u22a2 pointWeightsWithCircumcenter i circumcenter_index = 0"}, {"tactic": "simp [pointWeightsWithCircumcenter]", "annotated_tactic": ["simp [<a>pointWeightsWithCircumcenter</a>]", [{"full_name": "Affine.Simplex.pointWeightsWithCircumcenter", "def_path": "Mathlib/Geometry/Euclidean/Circumcenter.lean", "def_pos": [577, 5], "def_end_pos": [577, 33]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ns : Simplex \u211d P n\ni : Fin (n + 1)\n\u22a2 pointWeightsWithCircumcenter i (point_index i) = 1", "state_after": "no goals"}, {"tactic": "have h : _ \u2260 i := fun h => hn (h \u25b8 rfl)", "annotated_tactic": ["have h : _ \u2260 i := fun h => hn (h \u25b8 <a>rfl</a>)", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case point_index\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ns : Simplex \u211d P n\ni a\u271d : Fin (n + 1)\nhi : point_index a\u271d \u2208 univ\nhn : point_index a\u271d \u2260 point_index i\n\u22a2 pointWeightsWithCircumcenter i (point_index a\u271d) = 0", "state_after": "case point_index\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ns : Simplex \u211d P n\ni a\u271d : Fin (n + 1)\nhi : point_index a\u271d \u2208 univ\nhn : point_index a\u271d \u2260 point_index i\nh : a\u271d \u2260 i\n\u22a2 pointWeightsWithCircumcenter i (point_index a\u271d) = 0"}, {"tactic": "simp [pointWeightsWithCircumcenter, h]", "annotated_tactic": ["simp [<a>pointWeightsWithCircumcenter</a>, h]", [{"full_name": "Affine.Simplex.pointWeightsWithCircumcenter", "def_path": "Mathlib/Geometry/Euclidean/Circumcenter.lean", "def_pos": [577, 5], "def_end_pos": [577, 33]}]], "state_before": "case point_index\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ns : Simplex \u211d P n\ni a\u271d : Fin (n + 1)\nhi : point_index a\u271d \u2208 univ\nhn : point_index a\u271d \u2260 point_index i\nh : a\u271d \u2260 i\n\u22a2 pointWeightsWithCircumcenter i (point_index a\u271d) = 0", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case circumcenter_index\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ns : Simplex \u211d P n\ni : Fin (n + 1)\nhi : circumcenter_index \u2208 univ\nhn : circumcenter_index \u2260 point_index i\n\u22a2 pointWeightsWithCircumcenter i circumcenter_index = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean", "full_name": "Complex.GammaSeq_eq_approx_Gamma_integral", "start": [264, 1], "end": [285, 33], "traced_tactics": [{"tactic": "have : \u2200 x : \u211d, x = x / n * n := by intro x; rw [div_mul_cancel]; exact Nat.cast_ne_zero.mpr hn", "annotated_tactic": ["have : \u2200 x : \u211d, x = x / n * n := by intro x; rw [<a>div_mul_cancel</a>]; exact Nat.cast_ne_zero.mpr hn", [{"full_name": "div_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [66, 9], "def_end_pos": [66, 23]}]], "state_before": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\n\u22a2 GammaSeq s n = \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191x ^ (s - 1)", "state_after": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\n\u22a2 GammaSeq s n = \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191x ^ (s - 1)"}, {"tactic": "conv_rhs => enter [1, x, 2, 1]; rw [this x]", "annotated_tactic": ["conv_rhs => enter [1, x, 2, 1]; rw [this x]", []], "state_before": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\n\u22a2 GammaSeq s n = \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191x ^ (s - 1)", "state_after": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\n\u22a2 GammaSeq s n = \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1)"}, {"tactic": "rw [GammaSeq_eq_betaIntegral_of_re_pos hs]", "annotated_tactic": ["rw [<a>GammaSeq_eq_betaIntegral_of_re_pos</a> hs]", [{"full_name": "Complex.GammaSeq_eq_betaIntegral_of_re_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean", "def_pos": [246, 9], "def_end_pos": [246, 43]}]], "state_before": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\n\u22a2 GammaSeq s n = \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1)", "state_after": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\n\u22a2 \u2191n ^ s * betaIntegral s (\u2191n + 1) = \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1)"}, {"tactic": "have := intervalIntegral.integral_comp_div (a := 0) (b := n)\n  (fun x => \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1) : \u211d \u2192 \u2102) (Nat.cast_ne_zero.mpr hn)", "annotated_tactic": ["have := <a>intervalIntegral.integral_comp_div</a> (a := 0) (b := n)\n    (fun x => \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1) : \u211d \u2192 \u2102) (Nat.cast_ne_zero.mpr hn)", [{"full_name": "intervalIntegral.integral_comp_div", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [724, 9], "def_end_pos": [724, 26]}]], "state_before": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\n\u22a2 \u2191n ^ s * betaIntegral s (\u2191n + 1) = \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1)", "state_after": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, (fun x => \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)) (x / \u2191n) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, (fun x => \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)) x\n\u22a2 \u2191n ^ s * betaIntegral s (\u2191n + 1) = \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1)"}, {"tactic": "dsimp only at this", "annotated_tactic": ["dsimp only at this", []], "state_before": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, (fun x => \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)) (x / \u2191n) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, (fun x => \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)) x\n\u22a2 \u2191n ^ s * betaIntegral s (\u2191n + 1) = \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1)", "state_after": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\n\u22a2 \u2191n ^ s * betaIntegral s (\u2191n + 1) = \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1)"}, {"tactic": "rw [betaIntegral, this, real_smul, zero_div, div_self, add_sub_cancel,\n  \u2190 intervalIntegral.integral_const_mul, \u2190 intervalIntegral.integral_const_mul]", "annotated_tactic": ["rw [<a>betaIntegral</a>, this, <a>real_smul</a>, <a>zero_div</a>, <a>div_self</a>, <a>add_sub_cancel</a>,\n    \u2190 <a>intervalIntegral.integral_const_mul</a>, \u2190 <a>intervalIntegral.integral_const_mul</a>]", [{"full_name": "Complex.betaIntegral", "def_path": "Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean", "def_pos": [59, 19], "def_end_pos": [59, 31]}, {"full_name": "Complex.real_smul", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [383, 9], "def_end_pos": [383, 18]}, {"full_name": "zero_div", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [291, 9], "def_end_pos": [291, 17]}, {"full_name": "div_self", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [29, 9], "def_end_pos": [29, 17]}, {"full_name": "add_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [739, 30], "def_end_pos": [739, 44]}, {"full_name": "intervalIntegral.integral_const_mul", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [615, 9], "def_end_pos": [615, 27]}, {"full_name": "intervalIntegral.integral_const_mul", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [615, 9], "def_end_pos": [615, 27]}]], "state_before": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\n\u22a2 \u2191n ^ s * betaIntegral s (\u2191n + 1) = \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1)", "state_after": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\n\u22a2 \u222b (x : \u211d) in 0 ..1, \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) =\n    \u222b (x : \u211d) in 0 ..1, \u2191\u2191n * (\u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1))\n\ns : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\n\u22a2 \u2191n \u2260 0"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\n\u22a2 \u222b (x : \u211d) in 0 ..1, \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) =\n    \u222b (x : \u211d) in 0 ..1, \u2191\u2191n * (\u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1))\n\ns : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\n\u22a2 \u2191n \u2260 0", "state_after": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\n\u22a2 \u2191n \u2260 0\n\ns : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\n\u22a2 \u222b (x : \u211d) in 0 ..1, \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) =\n    \u222b (x : \u211d) in 0 ..1, \u2191\u2191n * (\u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1))"}, {"tactic": "simp_rw [intervalIntegral.integral_of_le zero_le_one]", "annotated_tactic": ["simp_rw [<a>intervalIntegral.integral_of_le</a> <a>zero_le_one</a>]", [{"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [465, 9], "def_end_pos": [465, 23]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}]], "state_before": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\n\u22a2 \u222b (x : \u211d) in 0 ..1, \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) =\n    \u222b (x : \u211d) in 0 ..1, \u2191\u2191n * (\u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1))", "state_after": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\n\u22a2 \u222b (x : \u211d) in Ioc 0 1, \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) =\n    \u222b (x : \u211d) in Ioc 0 1, \u2191\u2191n * (\u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1))"}, {"tactic": "refine' set_integral_congr measurableSet_Ioc fun x hx => _", "annotated_tactic": ["refine' <a>set_integral_congr</a> <a>measurableSet_Ioc</a> fun x hx => _", [{"full_name": "MeasureTheory.set_integral_congr", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [87, 9], "def_end_pos": [87, 27]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 26]}]], "state_before": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\n\u22a2 \u222b (x : \u211d) in Ioc 0 1, \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) =\n    \u222b (x : \u211d) in Ioc 0 1, \u2191\u2191n * (\u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1))", "state_after": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\nx : \u211d\nhx : x \u2208 Ioc 0 1\n\u22a2 \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) = \u2191\u2191n * (\u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1))"}, {"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "state_before": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\nx : \u211d\nhx : x \u2208 Ioc 0 1\n\u22a2 \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) = \u2191\u2191n * (\u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1))", "state_after": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\nx : \u211d\nhx : x \u2208 Ioc 0 1\n\u22a2 \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) = \u2191n * ((1 - \u2191x) ^ n * (\u2191x * \u2191n) ^ (s - 1))"}, {"tactic": "have hn' : (n : \u2102) \u2260 0 := Nat.cast_ne_zero.mpr hn", "annotated_tactic": ["have hn' : (n : \u2102) \u2260 0 := Nat.cast_ne_zero.mpr hn", []], "state_before": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\nx : \u211d\nhx : x \u2208 Ioc 0 1\n\u22a2 \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) = \u2191n * ((1 - \u2191x) ^ n * (\u2191x * \u2191n) ^ (s - 1))", "state_after": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\nx : \u211d\nhx : x \u2208 Ioc 0 1\nhn' : \u2191n \u2260 0\n\u22a2 \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) = \u2191n * ((1 - \u2191x) ^ n * (\u2191x * \u2191n) ^ (s - 1))"}, {"tactic": "have A : (n : \u2102) ^ s = (n : \u2102) ^ (s - 1) * n := by\n  conv_lhs => rw [(by ring : s = s - 1 + 1), cpow_add _ _ hn']\n  simp", "annotated_tactic": ["have A : (n : \u2102) ^ s = (n : \u2102) ^ (s - 1) * n := by\n    conv_lhs => rw [(by ring : s = s - 1 + 1), <a>cpow_add</a> _ _ hn']\n    simp", [{"full_name": "Complex.cpow_add", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean", "def_pos": [92, 9], "def_end_pos": [92, 17]}]], "state_before": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\nx : \u211d\nhx : x \u2208 Ioc 0 1\nhn' : \u2191n \u2260 0\n\u22a2 \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) = \u2191n * ((1 - \u2191x) ^ n * (\u2191x * \u2191n) ^ (s - 1))", "state_after": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\nx : \u211d\nhx : x \u2208 Ioc 0 1\nhn' : \u2191n \u2260 0\nA : \u2191n ^ s = \u2191n ^ (s - 1) * \u2191n\n\u22a2 \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) = \u2191n * ((1 - \u2191x) ^ n * (\u2191x * \u2191n) ^ (s - 1))"}, {"tactic": "have B : ((x : \u2102) * \u2191n) ^ (s - 1) = (x : \u2102) ^ (s - 1) * (n : \u2102) ^ (s - 1) := by\n  rw [\u2190 ofReal_nat_cast,\n    mul_cpow_ofReal_nonneg hx.1.le (Nat.cast_pos.mpr (Nat.pos_of_ne_zero hn)).le]", "annotated_tactic": ["have B : ((x : \u2102) * \u2191n) ^ (s - 1) = (x : \u2102) ^ (s - 1) * (n : \u2102) ^ (s - 1) := by\n    rw [\u2190 <a>ofReal_nat_cast</a>,\n      <a>mul_cpow_ofReal_nonneg</a> hx.1.<a>le</a> (Nat.cast_pos.mpr (<a>Nat.pos_of_ne_zero</a> hn)).<a>le</a>]", [{"full_name": "Complex.ofReal_nat_cast", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [787, 9], "def_end_pos": [787, 24]}, {"full_name": "Complex.mul_cpow_ofReal_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean", "def_pos": [164, 9], "def_end_pos": [164, 31]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "Nat.pos_of_ne_zero", "def_path": "lake-packages/std/Std/Data/Nat/Init/Lemmas.lean", "def_pos": [25, 19], "def_end_pos": [25, 33]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\nx : \u211d\nhx : x \u2208 Ioc 0 1\nhn' : \u2191n \u2260 0\nA : \u2191n ^ s = \u2191n ^ (s - 1) * \u2191n\n\u22a2 \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) = \u2191n * ((1 - \u2191x) ^ n * (\u2191x * \u2191n) ^ (s - 1))", "state_after": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\nx : \u211d\nhx : x \u2208 Ioc 0 1\nhn' : \u2191n \u2260 0\nA : \u2191n ^ s = \u2191n ^ (s - 1) * \u2191n\nB : (\u2191x * \u2191n) ^ (s - 1) = \u2191x ^ (s - 1) * \u2191n ^ (s - 1)\n\u22a2 \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) = \u2191n * ((1 - \u2191x) ^ n * (\u2191x * \u2191n) ^ (s - 1))"}, {"tactic": "rw [A, B, cpow_nat_cast]", "annotated_tactic": ["rw [A, B, <a>cpow_nat_cast</a>]", [{"full_name": "Complex.cpow_nat_cast", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean", "def_pos": [116, 9], "def_end_pos": [116, 22]}]], "state_before": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\nx : \u211d\nhx : x \u2208 Ioc 0 1\nhn' : \u2191n \u2260 0\nA : \u2191n ^ s = \u2191n ^ (s - 1) * \u2191n\nB : (\u2191x * \u2191n) ^ (s - 1) = \u2191x ^ (s - 1) * \u2191n ^ (s - 1)\n\u22a2 \u2191n ^ s * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ \u2191n) = \u2191n * ((1 - \u2191x) ^ n * (\u2191x * \u2191n) ^ (s - 1))", "state_after": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\nx : \u211d\nhx : x \u2208 Ioc 0 1\nhn' : \u2191n \u2260 0\nA : \u2191n ^ s = \u2191n ^ (s - 1) * \u2191n\nB : (\u2191x * \u2191n) ^ (s - 1) = \u2191x ^ (s - 1) * \u2191n ^ (s - 1)\n\u22a2 \u2191n ^ (s - 1) * \u2191n * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ n) = \u2191n * ((1 - \u2191x) ^ n * (\u2191x ^ (s - 1) * \u2191n ^ (s - 1)))"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\nx : \u211d\nhx : x \u2208 Ioc 0 1\nhn' : \u2191n \u2260 0\nA : \u2191n ^ s = \u2191n ^ (s - 1) * \u2191n\nB : (\u2191x * \u2191n) ^ (s - 1) = \u2191x ^ (s - 1) * \u2191n ^ (s - 1)\n\u22a2 \u2191n ^ (s - 1) * \u2191n * (\u2191x ^ (s - 1) * (1 - \u2191x) ^ n) = \u2191n * ((1 - \u2191x) ^ n * (\u2191x ^ (s - 1) * \u2191n ^ (s - 1)))", "state_after": "no goals"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\n\u22a2 \u2200 (x : \u211d), x = x / \u2191n * \u2191n", "state_after": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nx : \u211d\n\u22a2 x = x / \u2191n * \u2191n"}, {"tactic": "rw [div_mul_cancel]", "annotated_tactic": ["rw [<a>div_mul_cancel</a>]", [{"full_name": "div_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [66, 9], "def_end_pos": [66, 23]}]], "state_before": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nx : \u211d\n\u22a2 x = x / \u2191n * \u2191n", "state_after": "case h\ns : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nx : \u211d\n\u22a2 \u2191n \u2260 0"}, {"tactic": "exact Nat.cast_ne_zero.mpr hn", "annotated_tactic": ["exact Nat.cast_ne_zero.mpr hn", []], "state_before": "case h\ns : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nx : \u211d\n\u22a2 \u2191n \u2260 0", "state_after": "no goals"}, {"tactic": "exact Nat.cast_ne_zero.mpr hn", "annotated_tactic": ["exact Nat.cast_ne_zero.mpr hn", []], "state_before": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\n\u22a2 \u2191n \u2260 0", "state_after": "no goals"}, {"tactic": "conv_lhs => rw [(by ring : s = s - 1 + 1), cpow_add _ _ hn']", "annotated_tactic": ["conv_lhs => rw [(by ring : s = s - 1 + 1), <a>cpow_add</a> _ _ hn']", [{"full_name": "Complex.cpow_add", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean", "def_pos": [92, 9], "def_end_pos": [92, 17]}]], "state_before": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\nx : \u211d\nhx : x \u2208 Ioc 0 1\nhn' : \u2191n \u2260 0\n\u22a2 \u2191n ^ s = \u2191n ^ (s - 1) * \u2191n", "state_after": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\nx : \u211d\nhx : x \u2208 Ioc 0 1\nhn' : \u2191n \u2260 0\n\u22a2 \u2191n ^ (s - 1) * \u2191n ^ 1 = \u2191n ^ (s - 1) * \u2191n"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\nx : \u211d\nhx : x \u2208 Ioc 0 1\nhn' : \u2191n \u2260 0\n\u22a2 \u2191n ^ (s - 1) * \u2191n ^ 1 = \u2191n ^ (s - 1) * \u2191n", "state_after": "no goals"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\nx : \u211d\nhx : x \u2208 Ioc 0 1\nhn' : \u2191n \u2260 0\n\u22a2 s = s - 1 + 1", "state_after": "no goals"}, {"tactic": "rw [\u2190 ofReal_nat_cast,\n  mul_cpow_ofReal_nonneg hx.1.le (Nat.cast_pos.mpr (Nat.pos_of_ne_zero hn)).le]", "annotated_tactic": ["rw [\u2190 <a>ofReal_nat_cast</a>,\n      <a>mul_cpow_ofReal_nonneg</a> hx.1.<a>le</a> (Nat.cast_pos.mpr (<a>Nat.pos_of_ne_zero</a> hn)).<a>le</a>]", [{"full_name": "Complex.ofReal_nat_cast", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [787, 9], "def_end_pos": [787, 24]}, {"full_name": "Complex.mul_cpow_ofReal_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean", "def_pos": [164, 9], "def_end_pos": [164, 31]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "Nat.pos_of_ne_zero", "def_path": "lake-packages/std/Std/Data/Nat/Init/Lemmas.lean", "def_pos": [25, 19], "def_end_pos": [25, 33]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "s : \u2102\nhs : 0 < s.re\nn : \u2115\nhn : n \u2260 0\nthis\u271d : \u2200 (x : \u211d), x = x / \u2191n * \u2191n\nthis :\n  \u222b (x : \u211d) in 0 ..\u2191n, \u2191((1 - x / \u2191n) ^ n) * \u2191(x / \u2191n * \u2191n) ^ (s - 1) =\n    \u2191n \u2022 \u222b (x : \u211d) in 0 / \u2191n..\u2191n / \u2191n, \u2191((1 - x) ^ n) * \u2191(x * \u2191n) ^ (s - 1)\nx : \u211d\nhx : x \u2208 Ioc 0 1\nhn' : \u2191n \u2260 0\nA : \u2191n ^ s = \u2191n ^ (s - 1) * \u2191n\n\u22a2 (\u2191x * \u2191n) ^ (s - 1) = \u2191x ^ (s - 1) * \u2191n ^ (s - 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Pi.lean", "full_name": "Function.one_lt_const", "start": [131, 1], "end": [132, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "full_name": "add_tsub_add_eq_tsub_left", "start": [396, 1], "end": [397, 62], "traced_tactics": [{"tactic": "rw [add_comm a b, add_comm a c, add_tsub_add_eq_tsub_right]", "annotated_tactic": ["rw [<a>add_comm</a> a b, <a>add_comm</a> a c, <a>add_tsub_add_eq_tsub_right</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "add_tsub_add_eq_tsub_right", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [389, 9], "def_end_pos": [389, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : AddCommSemigroup \u03b1\ninst\u271d\u00b3 : Sub \u03b1\ninst\u271d\u00b2 : OrderedSub \u03b1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\n\u22a2 a + b - (a + c) = b - c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/ModelTheory/Semantics.lean", "full_name": "FirstOrder.Language.BoundedFormula.realize_sup", "start": [325, 1], "end": [327, 8], "traced_tactics": [{"tactic": "simp only [realize, Sup.sup, realize_not, eq_iff_iff]", "annotated_tactic": ["simp only [<a>realize</a>, <a>Sup.sup</a>, <a>realize_not</a>, <a>eq_iff_iff</a>]", [{"full_name": "FirstOrder.Language.Term.realize", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [72, 5], "def_end_pos": [72, 12]}, {"full_name": "Sup.sup", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [993, 3], "def_end_pos": [993, 6]}, {"full_name": "FirstOrder.Language.BoundedFormula.realize_not", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [268, 9], "def_end_pos": [268, 20]}, {"full_name": "eq_iff_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [53, 17], "def_end_pos": [53, 27]}]], "state_before": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u22a2 Realize (\u03c6 \u2294 \u03c8) v xs \u2194 Realize \u03c6 v xs \u2228 Realize \u03c8 v xs", "state_after": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u22a2 Realize (\u223c\u03c6 \u27f9 \u03c8) v xs \u2194 Realize \u03c6 v xs \u2228 Realize \u03c8 v xs"}, {"tactic": "tauto", "annotated_tactic": ["tauto", []], "state_before": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u22a2 Realize (\u223c\u03c6 \u27f9 \u03c8) v xs \u2194 Realize \u03c6 v xs \u2228 Realize \u03c8 v xs", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "measure_Ico_lt_top", "start": [4415, 1], "end": [4416, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Abelian/RightDerived.lean", "full_name": "CategoryTheory.Abelian.Functor.exact_of_map_injectiveResolution", "start": [198, 1], "end": [205, 66], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\ninst\u271d\u2075 : Category.{w, u} C\nD : Type u\ninst\u271d\u2074 : Category.{w, u} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b3 : Abelian C\ninst\u271d\u00b2 : Abelian D\ninst\u271d\u00b9 : Functor.Additive F\nP : InjectiveResolution X\ninst\u271d : PreservesFiniteLimits F\n\u22a2 (Iso.refl (F.obj (HomologicalComplex.X ((CochainComplex.single\u2080 C).obj X) 0))).hom \u226b\n      F.map (HomologicalComplex.Hom.f P.\u03b9 0) =\n    F.map (HomologicalComplex.Hom.f P.\u03b9 0) \u226b (Iso.refl (F.obj (HomologicalComplex.X P.cocomplex 0))).hom", "state_after": "no goals"}, {"tactic": "rw [Iso.refl_hom, Category.id_comp, Iso.symm_hom, HomologicalComplex.dFrom_eq] <;> congr", "annotated_tactic": ["rw [<a>Iso.refl_hom</a>, <a>Category.id_comp</a>, <a>Iso.symm_hom</a>, <a>HomologicalComplex.dFrom_eq</a>] <;> congr", [{"full_name": "CategoryTheory.Iso.refl_hom", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [133, 9], "def_end_pos": [133, 14]}, {"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [155, 3], "def_end_pos": [155, 10]}, {"full_name": "CategoryTheory.Iso.symm_hom", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [103, 9], "def_end_pos": [103, 17]}, {"full_name": "HomologicalComplex.dFrom_eq", "def_path": "Mathlib/Algebra/Homology/HomologicalComplex.lean", "def_pos": [477, 9], "def_end_pos": [477, 17]}]], "state_before": "C : Type u\ninst\u271d\u2075 : Category.{w, u} C\nD : Type u\ninst\u271d\u2074 : Category.{w, u} D\nF : C \u2964 D\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ninst\u271d\u00b3 : Abelian C\ninst\u271d\u00b2 : Abelian D\ninst\u271d\u00b9 : Functor.Additive F\nP : InjectiveResolution X\ninst\u271d : PreservesFiniteLimits F\n\u22a2 (Iso.refl (F.obj (HomologicalComplex.X P.cocomplex 0))).hom \u226b\n      HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.up \u2115)).obj P.cocomplex) 0 =\n    F.map (HomologicalComplex.d P.cocomplex 0 1) \u226b\n      (HomologicalComplex.xNextIso ((mapHomologicalComplex F (ComplexShape.up \u2115)).obj P.cocomplex)\n            (_ : 0 + 1 = 0 + 1)).symm.hom", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/HasseDeriv.lean", "full_name": "Polynomial.hasseDeriv_C", "start": [129, 1], "end": [131, 35], "traced_tactics": [{"tactic": "rw [\u2190 monomial_zero_left, hasseDeriv_monomial, Nat.choose_eq_zero_of_lt hk, Nat.cast_zero,\n  zero_mul, monomial_zero_right]", "annotated_tactic": ["rw [\u2190 <a>monomial_zero_left</a>, <a>hasseDeriv_monomial</a>, <a>Nat.choose_eq_zero_of_lt</a> hk, <a>Nat.cast_zero</a>,\n    <a>zero_mul</a>, <a>monomial_zero_right</a>]", [{"full_name": "Polynomial.monomial_zero_left", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [506, 9], "def_end_pos": [506, 27]}, {"full_name": "Polynomial.hasseDeriv_monomial", "def_path": "Mathlib/Data/Polynomial/HasseDeriv.lean", "def_pos": [113, 9], "def_end_pos": [113, 28]}, {"full_name": "Nat.choose_eq_zero_of_lt", "def_path": "Mathlib/Data/Nat/Choose/Basic.lean", "def_pos": [68, 9], "def_end_pos": [68, 29]}, {"full_name": "Nat.cast_zero", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [114, 9], "def_end_pos": [114, 18]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "Polynomial.monomial_zero_right", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [448, 9], "def_end_pos": [448, 28]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\nk : \u2115\nf : R[X]\nr : R\nhk : 0 < k\n\u22a2 \u2191(hasseDeriv k) (\u2191C r) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "full_name": "MvPowerSeries.coeff_monomial_mul", "start": [231, 1], "end": [239, 74], "traced_tactics": [{"tactic": "classical\nhave :\n  \u2200 p \u2208 antidiagonal m,\n    coeff R (p : (\u03c3 \u2192\u2080 \u2115) \u00d7 (\u03c3 \u2192\u2080 \u2115)).1 (monomial R n a) * coeff R p.2 \u03c6 \u2260 0 \u2192 p.1 = n :=\n  fun p _ hp => eq_of_coeff_monomial_ne_zero (left_ne_zero_of_mul hp)\nrw [coeff_mul, \u2190 Finset.sum_filter_of_ne this, antidiagonal_filter_fst_eq, Finset.sum_ite_index]\nsimp only [Finset.sum_singleton, coeff_monomial_same, Finset.sum_empty]", "annotated_tactic": ["classical\n  have :\n    \u2200 p \u2208 <a>antidiagonal</a> m,\n      <a>coeff</a> R (p : (\u03c3 \u2192\u2080 \u2115) \u00d7 (\u03c3 \u2192\u2080 \u2115)).1 (<a>monomial</a> R n a) * <a>coeff</a> R p.2 \u03c6 \u2260 0 \u2192 p.1 = n :=\n    fun p _ hp => <a>eq_of_coeff_monomial_ne_zero</a> (<a>left_ne_zero_of_mul</a> hp)\n  rw [<a>coeff_mul</a>, \u2190 <a>Finset.sum_filter_of_ne</a> this, <a>antidiagonal_filter_fst_eq</a>, <a>Finset.sum_ite_index</a>]\n  simp only [<a>Finset.sum_singleton</a>, <a>coeff_monomial_same</a>, <a>Finset.sum_empty</a>]", [{"full_name": "Finsupp.antidiagonal", "def_path": "Mathlib/Data/Finsupp/Antidiagonal.lean", "def_pos": [42, 5], "def_end_pos": [42, 17]}, {"full_name": "MvPowerSeries.coeff", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [126, 5], "def_end_pos": [126, 10]}, {"full_name": "MvPowerSeries.monomial", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [120, 5], "def_end_pos": [120, 13]}, {"full_name": "MvPowerSeries.coeff", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [126, 5], "def_end_pos": [126, 10]}, {"full_name": "MvPowerSeries.eq_of_coeff_monomial_ne_zero", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [173, 9], "def_end_pos": [173, 37]}, {"full_name": "left_ne_zero_of_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [51, 9], "def_end_pos": [51, 28]}, {"full_name": "MvPowerSeries.coeff_mul", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [217, 9], "def_end_pos": [217, 18]}, {"full_name": "Finset.sum_filter_of_ne", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [752, 3], "def_end_pos": [752, 14]}, {"full_name": "Finsupp.antidiagonal_filter_fst_eq", "def_path": "Mathlib/Data/Finsupp/Antidiagonal.lean", "def_pos": [58, 9], "def_end_pos": [58, 35]}, {"full_name": "Finset.sum_ite_index", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1105, 3], "def_end_pos": [1105, 14]}, {"full_name": "Finset.sum_singleton", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [343, 3], "def_end_pos": [343, 14]}, {"full_name": "MvPowerSeries.coeff_monomial_same", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [161, 9], "def_end_pos": [161, 28]}, {"full_name": "Finset.sum_empty", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [298, 3], "def_end_pos": [298, 14]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nm n : \u03c3 \u2192\u2080 \u2115\n\u03c6 \u03c8 : MvPowerSeries \u03c3 R\na : R\n\u22a2 \u2191(coeff R m) (\u2191(monomial R n) a * \u03c6) = if n \u2264 m then a * \u2191(coeff R (m - n)) \u03c6 else 0", "state_after": "no goals"}, {"tactic": "have :\n  \u2200 p \u2208 antidiagonal m,\n    coeff R (p : (\u03c3 \u2192\u2080 \u2115) \u00d7 (\u03c3 \u2192\u2080 \u2115)).1 (monomial R n a) * coeff R p.2 \u03c6 \u2260 0 \u2192 p.1 = n :=\n  fun p _ hp => eq_of_coeff_monomial_ne_zero (left_ne_zero_of_mul hp)", "annotated_tactic": ["have :\n    \u2200 p \u2208 <a>antidiagonal</a> m,\n      <a>coeff</a> R (p : (\u03c3 \u2192\u2080 \u2115) \u00d7 (\u03c3 \u2192\u2080 \u2115)).1 (<a>monomial</a> R n a) * <a>coeff</a> R p.2 \u03c6 \u2260 0 \u2192 p.1 = n :=\n    fun p _ hp => <a>eq_of_coeff_monomial_ne_zero</a> (<a>left_ne_zero_of_mul</a> hp)", [{"full_name": "Finsupp.antidiagonal", "def_path": "Mathlib/Data/Finsupp/Antidiagonal.lean", "def_pos": [42, 5], "def_end_pos": [42, 17]}, {"full_name": "MvPowerSeries.coeff", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [126, 5], "def_end_pos": [126, 10]}, {"full_name": "MvPowerSeries.monomial", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [120, 5], "def_end_pos": [120, 13]}, {"full_name": "MvPowerSeries.coeff", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [126, 5], "def_end_pos": [126, 10]}, {"full_name": "MvPowerSeries.eq_of_coeff_monomial_ne_zero", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [173, 9], "def_end_pos": [173, 37]}, {"full_name": "left_ne_zero_of_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [51, 9], "def_end_pos": [51, 28]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nm n : \u03c3 \u2192\u2080 \u2115\n\u03c6 \u03c8 : MvPowerSeries \u03c3 R\na : R\n\u22a2 \u2191(coeff R m) (\u2191(monomial R n) a * \u03c6) = if n \u2264 m then a * \u2191(coeff R (m - n)) \u03c6 else 0", "state_after": "\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nm n : \u03c3 \u2192\u2080 \u2115\n\u03c6 \u03c8 : MvPowerSeries \u03c3 R\na : R\nthis :\n  \u2200 (p : (\u03c3 \u2192\u2080 \u2115) \u00d7 (\u03c3 \u2192\u2080 \u2115)), p \u2208 antidiagonal m \u2192 \u2191(coeff R p.1) (\u2191(monomial R n) a) * \u2191(coeff R p.2) \u03c6 \u2260 0 \u2192 p.1 = n\n\u22a2 \u2191(coeff R m) (\u2191(monomial R n) a * \u03c6) = if n \u2264 m then a * \u2191(coeff R (m - n)) \u03c6 else 0"}, {"tactic": "rw [coeff_mul, \u2190 Finset.sum_filter_of_ne this, antidiagonal_filter_fst_eq, Finset.sum_ite_index]", "annotated_tactic": ["rw [<a>coeff_mul</a>, \u2190 <a>Finset.sum_filter_of_ne</a> this, <a>antidiagonal_filter_fst_eq</a>, <a>Finset.sum_ite_index</a>]", [{"full_name": "MvPowerSeries.coeff_mul", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [217, 9], "def_end_pos": [217, 18]}, {"full_name": "Finset.sum_filter_of_ne", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [752, 3], "def_end_pos": [752, 14]}, {"full_name": "Finsupp.antidiagonal_filter_fst_eq", "def_path": "Mathlib/Data/Finsupp/Antidiagonal.lean", "def_pos": [58, 9], "def_end_pos": [58, 35]}, {"full_name": "Finset.sum_ite_index", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1105, 3], "def_end_pos": [1105, 14]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nm n : \u03c3 \u2192\u2080 \u2115\n\u03c6 \u03c8 : MvPowerSeries \u03c3 R\na : R\nthis :\n  \u2200 (p : (\u03c3 \u2192\u2080 \u2115) \u00d7 (\u03c3 \u2192\u2080 \u2115)), p \u2208 antidiagonal m \u2192 \u2191(coeff R p.1) (\u2191(monomial R n) a) * \u2191(coeff R p.2) \u03c6 \u2260 0 \u2192 p.1 = n\n\u22a2 \u2191(coeff R m) (\u2191(monomial R n) a * \u03c6) = if n \u2264 m then a * \u2191(coeff R (m - n)) \u03c6 else 0", "state_after": "\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nm n : \u03c3 \u2192\u2080 \u2115\n\u03c6 \u03c8 : MvPowerSeries \u03c3 R\na : R\nthis :\n  \u2200 (p : (\u03c3 \u2192\u2080 \u2115) \u00d7 (\u03c3 \u2192\u2080 \u2115)), p \u2208 antidiagonal m \u2192 \u2191(coeff R p.1) (\u2191(monomial R n) a) * \u2191(coeff R p.2) \u03c6 \u2260 0 \u2192 p.1 = n\n\u22a2 (if n \u2264 m then \u2211 x in {(n, m - n)}, \u2191(coeff R x.1) (\u2191(monomial R n) a) * \u2191(coeff R x.2) \u03c6\n    else \u2211 x in \u2205, \u2191(coeff R x.1) (\u2191(monomial R n) a) * \u2191(coeff R x.2) \u03c6) =\n    if n \u2264 m then a * \u2191(coeff R (m - n)) \u03c6 else 0"}, {"tactic": "simp only [Finset.sum_singleton, coeff_monomial_same, Finset.sum_empty]", "annotated_tactic": ["simp only [<a>Finset.sum_singleton</a>, <a>coeff_monomial_same</a>, <a>Finset.sum_empty</a>]", [{"full_name": "Finset.sum_singleton", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [343, 3], "def_end_pos": [343, 14]}, {"full_name": "MvPowerSeries.coeff_monomial_same", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [161, 9], "def_end_pos": [161, 28]}, {"full_name": "Finset.sum_empty", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [298, 3], "def_end_pos": [298, 14]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\ninst\u271d : Semiring R\nm n : \u03c3 \u2192\u2080 \u2115\n\u03c6 \u03c8 : MvPowerSeries \u03c3 R\na : R\nthis :\n  \u2200 (p : (\u03c3 \u2192\u2080 \u2115) \u00d7 (\u03c3 \u2192\u2080 \u2115)), p \u2208 antidiagonal m \u2192 \u2191(coeff R p.1) (\u2191(monomial R n) a) * \u2191(coeff R p.2) \u03c6 \u2260 0 \u2192 p.1 = n\n\u22a2 (if n \u2264 m then \u2211 x in {(n, m - n)}, \u2191(coeff R x.1) (\u2191(monomial R n) a) * \u2191(coeff R x.2) \u03c6\n    else \u2211 x in \u2205, \u2191(coeff R x.1) (\u2191(monomial R n) a) * \u2191(coeff R x.2) \u03c6) =\n    if n \u2264 m then a * \u2191(coeff R (m - n)) \u03c6 else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Limits/Final.lean", "full_name": "CategoryTheory.Functor.initial_natIso_iff", "start": [169, 1], "end": [170, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "full_name": "Array.get?_push_eq", "start": [82, 1], "end": [83, 33], "traced_tactics": [{"tactic": "rw [getElem?_pos, get_push_eq]", "annotated_tactic": ["rw [<a>getElem?_pos</a>, <a>get_push_eq</a>]", [{"full_name": "getElem?_pos", "def_path": "lake-packages/std/Std/Data/Array/Lemmas.lean", "def_pos": [23, 9], "def_end_pos": [23, 21]}, {"full_name": "Array.get_push_eq", "def_path": "lake-packages/std/Std/Data/Array/Init/Lemmas.lean", "def_pos": [131, 17], "def_end_pos": [131, 28]}]], "state_before": "\u03b1 : Type u_1\na : Array \u03b1\nx : \u03b1\n\u22a2 (push a x)[size a]? = some x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Surreal/Basic.lean", "full_name": "SetTheory.PGame.Numeric.moveLeft", "start": [89, 1], "end": [90, 25], "traced_tactics": [{"tactic": "cases x", "annotated_tactic": ["cases x", []], "state_before": "x : PGame\no : Numeric x\ni : LeftMoves x\n\u22a2 Numeric (PGame.moveLeft x i)", "state_after": "case mk\n\u03b1\u271d \u03b2\u271d : Type u_1\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\no : Numeric (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\ni : LeftMoves (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\n\u22a2 Numeric (PGame.moveLeft (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) i)"}, {"tactic": "exact o.2.1 i", "annotated_tactic": ["exact o.2.1 i", []], "state_before": "case mk\n\u03b1\u271d \u03b2\u271d : Type u_1\na\u271d\u00b9 : \u03b1\u271d \u2192 PGame\na\u271d : \u03b2\u271d \u2192 PGame\no : Numeric (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\ni : LeftMoves (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d)\n\u22a2 Numeric (PGame.moveLeft (PGame.mk \u03b1\u271d \u03b2\u271d a\u271d\u00b9 a\u271d) i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Real/ENNReal.lean", "full_name": "ENNReal.top_ne_zero", "start": [337, 9], "end": [337, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "full_name": "MeasureTheory.snorm'_measure_zero_of_pos", "start": [225, 1], "end": [226, 27], "traced_tactics": [{"tactic": "simp [snorm', hq_pos]", "annotated_tactic": ["simp [<a>snorm'</a>, hq_pos]", [{"full_name": "MeasureTheory.snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [73, 5], "def_end_pos": [73, 11]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 F\nhq_pos : 0 < q\n\u22a2 snorm' f q 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "frontier_Iic_subset", "start": [668, 1], "end": [669, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/JordanHolder.lean", "full_name": "CompositionSeries.ext_fun", "start": [206, 1], "end": [212, 38], "traced_tactics": [{"tactic": "cases s\u2081", "annotated_tactic": ["cases s\u2081", []], "state_before": "X : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\ns\u2081 s\u2082 : CompositionSeries X\nhl : s\u2081.length = s\u2082.length\nh : \u2200 (i : Fin (s\u2081.length + 1)), series s\u2081 i = series s\u2082 (Fin.cast (_ : Nat.succ s\u2081.length = Nat.succ s\u2082.length) i)\n\u22a2 s\u2081 = s\u2082", "state_after": "case mk\nX : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\ns\u2082 : CompositionSeries X\nlength\u271d : \u2115\nseries\u271d : Fin (length\u271d + 1) \u2192 X\nstep'\u271d : \u2200 (i : Fin length\u271d), IsMaximal (series\u271d (Fin.castSucc i)) (series\u271d (Fin.succ i))\nhl : { length := length\u271d, series := series\u271d, step' := step'\u271d }.length = s\u2082.length\nh :\n  \u2200 (i : Fin ({ length := length\u271d, series := series\u271d, step' := step'\u271d }.length + 1)),\n    series { length := length\u271d, series := series\u271d, step' := step'\u271d } i =\n      series s\u2082\n        (Fin.cast (_ : Nat.succ { length := length\u271d, series := series\u271d, step' := step'\u271d }.length = Nat.succ s\u2082.length)\n          i)\n\u22a2 { length := length\u271d, series := series\u271d, step' := step'\u271d } = s\u2082"}, {"tactic": "cases s\u2082", "annotated_tactic": ["cases s\u2082", []], "state_before": "case mk\nX : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\ns\u2082 : CompositionSeries X\nlength\u271d : \u2115\nseries\u271d : Fin (length\u271d + 1) \u2192 X\nstep'\u271d : \u2200 (i : Fin length\u271d), IsMaximal (series\u271d (Fin.castSucc i)) (series\u271d (Fin.succ i))\nhl : { length := length\u271d, series := series\u271d, step' := step'\u271d }.length = s\u2082.length\nh :\n  \u2200 (i : Fin ({ length := length\u271d, series := series\u271d, step' := step'\u271d }.length + 1)),\n    series { length := length\u271d, series := series\u271d, step' := step'\u271d } i =\n      series s\u2082\n        (Fin.cast (_ : Nat.succ { length := length\u271d, series := series\u271d, step' := step'\u271d }.length = Nat.succ s\u2082.length)\n          i)\n\u22a2 { length := length\u271d, series := series\u271d, step' := step'\u271d } = s\u2082", "state_after": "case mk.mk\nX : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\nlength\u271d\u00b9 : \u2115\nseries\u271d\u00b9 : Fin (length\u271d\u00b9 + 1) \u2192 X\nstep'\u271d\u00b9 : \u2200 (i : Fin length\u271d\u00b9), IsMaximal (series\u271d\u00b9 (Fin.castSucc i)) (series\u271d\u00b9 (Fin.succ i))\nlength\u271d : \u2115\nseries\u271d : Fin (length\u271d + 1) \u2192 X\nstep'\u271d : \u2200 (i : Fin length\u271d), IsMaximal (series\u271d (Fin.castSucc i)) (series\u271d (Fin.succ i))\nhl :\n  { length := length\u271d\u00b9, series := series\u271d\u00b9, step' := step'\u271d\u00b9 }.length =\n    { length := length\u271d, series := series\u271d, step' := step'\u271d }.length\nh :\n  \u2200 (i : Fin ({ length := length\u271d\u00b9, series := series\u271d\u00b9, step' := step'\u271d\u00b9 }.length + 1)),\n    series { length := length\u271d\u00b9, series := series\u271d\u00b9, step' := step'\u271d\u00b9 } i =\n      series { length := length\u271d, series := series\u271d, step' := step'\u271d }\n        (Fin.cast\n          (_ :\n            Nat.succ { length := length\u271d\u00b9, series := series\u271d\u00b9, step' := step'\u271d\u00b9 }.length =\n              Nat.succ { length := length\u271d, series := series\u271d, step' := step'\u271d }.length)\n          i)\n\u22a2 { length := length\u271d\u00b9, series := series\u271d\u00b9, step' := step'\u271d\u00b9 } =\n    { length := length\u271d, series := series\u271d, step' := step'\u271d }"}, {"tactic": "dsimp at hl h", "annotated_tactic": ["dsimp at hl h", []], "state_before": "case mk.mk\nX : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\nlength\u271d\u00b9 : \u2115\nseries\u271d\u00b9 : Fin (length\u271d\u00b9 + 1) \u2192 X\nstep'\u271d\u00b9 : \u2200 (i : Fin length\u271d\u00b9), IsMaximal (series\u271d\u00b9 (Fin.castSucc i)) (series\u271d\u00b9 (Fin.succ i))\nlength\u271d : \u2115\nseries\u271d : Fin (length\u271d + 1) \u2192 X\nstep'\u271d : \u2200 (i : Fin length\u271d), IsMaximal (series\u271d (Fin.castSucc i)) (series\u271d (Fin.succ i))\nhl :\n  { length := length\u271d\u00b9, series := series\u271d\u00b9, step' := step'\u271d\u00b9 }.length =\n    { length := length\u271d, series := series\u271d, step' := step'\u271d }.length\nh :\n  \u2200 (i : Fin ({ length := length\u271d\u00b9, series := series\u271d\u00b9, step' := step'\u271d\u00b9 }.length + 1)),\n    series { length := length\u271d\u00b9, series := series\u271d\u00b9, step' := step'\u271d\u00b9 } i =\n      series { length := length\u271d, series := series\u271d, step' := step'\u271d }\n        (Fin.cast\n          (_ :\n            Nat.succ { length := length\u271d\u00b9, series := series\u271d\u00b9, step' := step'\u271d\u00b9 }.length =\n              Nat.succ { length := length\u271d, series := series\u271d, step' := step'\u271d }.length)\n          i)\n\u22a2 { length := length\u271d\u00b9, series := series\u271d\u00b9, step' := step'\u271d\u00b9 } =\n    { length := length\u271d, series := series\u271d, step' := step'\u271d }", "state_after": "case mk.mk\nX : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\nlength\u271d\u00b9 : \u2115\nseries\u271d\u00b9 : Fin (length\u271d\u00b9 + 1) \u2192 X\nstep'\u271d\u00b9 : \u2200 (i : Fin length\u271d\u00b9), IsMaximal (series\u271d\u00b9 (Fin.castSucc i)) (series\u271d\u00b9 (Fin.succ i))\nlength\u271d : \u2115\nseries\u271d : Fin (length\u271d + 1) \u2192 X\nstep'\u271d : \u2200 (i : Fin length\u271d), IsMaximal (series\u271d (Fin.castSucc i)) (series\u271d (Fin.succ i))\nhl : length\u271d\u00b9 = length\u271d\nh : \u2200 (i : Fin (length\u271d\u00b9 + 1)), series\u271d\u00b9 i = series\u271d (Fin.cast (_ : Nat.succ length\u271d\u00b9 = Nat.succ length\u271d) i)\n\u22a2 { length := length\u271d\u00b9, series := series\u271d\u00b9, step' := step'\u271d\u00b9 } =\n    { length := length\u271d, series := series\u271d, step' := step'\u271d }"}, {"tactic": "subst hl", "annotated_tactic": ["subst hl", []], "state_before": "case mk.mk\nX : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\nlength\u271d\u00b9 : \u2115\nseries\u271d\u00b9 : Fin (length\u271d\u00b9 + 1) \u2192 X\nstep'\u271d\u00b9 : \u2200 (i : Fin length\u271d\u00b9), IsMaximal (series\u271d\u00b9 (Fin.castSucc i)) (series\u271d\u00b9 (Fin.succ i))\nlength\u271d : \u2115\nseries\u271d : Fin (length\u271d + 1) \u2192 X\nstep'\u271d : \u2200 (i : Fin length\u271d), IsMaximal (series\u271d (Fin.castSucc i)) (series\u271d (Fin.succ i))\nhl : length\u271d\u00b9 = length\u271d\nh : \u2200 (i : Fin (length\u271d\u00b9 + 1)), series\u271d\u00b9 i = series\u271d (Fin.cast (_ : Nat.succ length\u271d\u00b9 = Nat.succ length\u271d) i)\n\u22a2 { length := length\u271d\u00b9, series := series\u271d\u00b9, step' := step'\u271d\u00b9 } =\n    { length := length\u271d, series := series\u271d, step' := step'\u271d }", "state_after": "case mk.mk\nX : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\nlength\u271d : \u2115\nseries\u271d\u00b9 : Fin (length\u271d + 1) \u2192 X\nstep'\u271d\u00b9 : \u2200 (i : Fin length\u271d), IsMaximal (series\u271d\u00b9 (Fin.castSucc i)) (series\u271d\u00b9 (Fin.succ i))\nseries\u271d : Fin (length\u271d + 1) \u2192 X\nstep'\u271d : \u2200 (i : Fin length\u271d), IsMaximal (series\u271d (Fin.castSucc i)) (series\u271d (Fin.succ i))\nh : \u2200 (i : Fin (length\u271d + 1)), series\u271d\u00b9 i = series\u271d (Fin.cast (_ : Nat.succ length\u271d = Nat.succ length\u271d) i)\n\u22a2 { length := length\u271d, series := series\u271d\u00b9, step' := step'\u271d\u00b9 } =\n    { length := length\u271d, series := series\u271d, step' := step'\u271d }"}, {"tactic": "simpa [Function.funext_iff] using h", "annotated_tactic": ["simpa [<a>Function.funext_iff</a>] using h", [{"full_name": "Function.funext_iff", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [82, 9], "def_end_pos": [82, 19]}]], "state_before": "case mk.mk\nX : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\nlength\u271d : \u2115\nseries\u271d\u00b9 : Fin (length\u271d + 1) \u2192 X\nstep'\u271d\u00b9 : \u2200 (i : Fin length\u271d), IsMaximal (series\u271d\u00b9 (Fin.castSucc i)) (series\u271d\u00b9 (Fin.succ i))\nseries\u271d : Fin (length\u271d + 1) \u2192 X\nstep'\u271d : \u2200 (i : Fin length\u271d), IsMaximal (series\u271d (Fin.castSucc i)) (series\u271d (Fin.succ i))\nh : \u2200 (i : Fin (length\u271d + 1)), series\u271d\u00b9 i = series\u271d (Fin.cast (_ : Nat.succ length\u271d = Nat.succ length\u271d) i)\n\u22a2 { length := length\u271d, series := series\u271d\u00b9, step' := step'\u271d\u00b9 } =\n    { length := length\u271d, series := series\u271d, step' := step'\u271d }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "full_name": "Nat.pos_of_lt_add_left", "start": [327, 11], "end": [328, 61], "traced_tactics": [{"tactic": "rw [Nat.zero_add]", "annotated_tactic": ["rw [<a>Nat.zero_add</a>]", [{"full_name": "Nat.zero_add", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [114, 27], "def_end_pos": [114, 35]}]], "state_before": "n k : Nat\nh : n < k + n\n\u22a2 0 + ?m.18605 h < k + ?m.18605 h", "state_after": "n k : Nat\nh : n < k + n\n\u22a2 ?m.18605 h < k + ?m.18605 h\n\n\n\u22a2 {n k : Nat} \u2192 n < k + n \u2192 Nat"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "n k : Nat\nh : n < k + n\n\u22a2 ?m.18605 h < k + ?m.18605 h\n\n\n\u22a2 {n k : Nat} \u2192 n < k + n \u2192 Nat", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Hom/Basic.lean", "full_name": "OrderIso.ext", "start": [796, 1], "end": [797, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "full_name": "MeasureTheory.Measure.map_prod_map", "start": [699, 1], "end": [706, 38], "traced_tactics": [{"tactic": "haveI := hgc.of_map \u03bcc hg.aemeasurable", "annotated_tactic": ["haveI := hgc.of_map \u03bcc hg.aemeasurable", []], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\n\u03b4 : Type u_7\ninst\u271d : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\n\u03bca : Measure \u03b1\n\u03bcc : Measure \u03b3\nhfa : SigmaFinite (map f \u03bca)\nhgc : SigmaFinite (map g \u03bcc)\nhf : Measurable f\nhg : Measurable g\n\u22a2 Measure.prod (map f \u03bca) (map g \u03bcc) = map (Prod.map f g) (Measure.prod \u03bca \u03bcc)", "state_after": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\n\u03b4 : Type u_7\ninst\u271d : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\n\u03bca : Measure \u03b1\n\u03bcc : Measure \u03b3\nhfa : SigmaFinite (map f \u03bca)\nhgc : SigmaFinite (map g \u03bcc)\nhf : Measurable f\nhg : Measurable g\nthis : SigmaFinite \u03bcc\n\u22a2 Measure.prod (map f \u03bca) (map g \u03bcc) = map (Prod.map f g) (Measure.prod \u03bca \u03bcc)"}, {"tactic": "refine' prod_eq fun s t hs ht => _", "annotated_tactic": ["refine' <a>prod_eq</a> fun s t hs ht => _", [{"full_name": "MeasureTheory.Measure.prod_eq", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [565, 9], "def_end_pos": [565, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\n\u03b4 : Type u_7\ninst\u271d : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\n\u03bca : Measure \u03b1\n\u03bcc : Measure \u03b3\nhfa : SigmaFinite (map f \u03bca)\nhgc : SigmaFinite (map g \u03bcc)\nhf : Measurable f\nhg : Measurable g\nthis : SigmaFinite \u03bcc\n\u22a2 Measure.prod (map f \u03bca) (map g \u03bcc) = map (Prod.map f g) (Measure.prod \u03bca \u03bcc)", "state_after": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\n\u03b4 : Type u_7\ninst\u271d : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\n\u03bca : Measure \u03b1\n\u03bcc : Measure \u03b3\nhfa : SigmaFinite (map f \u03bca)\nhgc : SigmaFinite (map g \u03bcc)\nhf : Measurable f\nhg : Measurable g\nthis : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191\u2191(map (Prod.map f g) (Measure.prod \u03bca \u03bcc)) (s \u00d7\u02e2 t) = \u2191\u2191(map f \u03bca) s * \u2191\u2191(map g \u03bcc) t"}, {"tactic": "rw [map_apply (hf.prod_map hg) (hs.prod ht), map_apply hf hs, map_apply hg ht]", "annotated_tactic": ["rw [<a>map_apply</a> (hf.prod_map hg) (hs.prod ht), <a>map_apply</a> hf hs, <a>map_apply</a> hg ht]", [{"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\n\u03b4 : Type u_7\ninst\u271d : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\n\u03bca : Measure \u03b1\n\u03bcc : Measure \u03b3\nhfa : SigmaFinite (map f \u03bca)\nhgc : SigmaFinite (map g \u03bcc)\nhf : Measurable f\nhg : Measurable g\nthis : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191\u2191(map (Prod.map f g) (Measure.prod \u03bca \u03bcc)) (s \u00d7\u02e2 t) = \u2191\u2191(map f \u03bca) s * \u2191\u2191(map g \u03bcc) t", "state_after": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\n\u03b4 : Type u_7\ninst\u271d : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\n\u03bca : Measure \u03b1\n\u03bcc : Measure \u03b3\nhfa : SigmaFinite (map f \u03bca)\nhgc : SigmaFinite (map g \u03bcc)\nhf : Measurable f\nhg : Measurable g\nthis : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.prod \u03bca \u03bcc) (Prod.map f g \u207b\u00b9' s \u00d7\u02e2 t) = \u2191\u2191\u03bca (f \u207b\u00b9' s) * \u2191\u2191\u03bcc (g \u207b\u00b9' t)"}, {"tactic": "exact prod_prod (f \u207b\u00b9' s) (g \u207b\u00b9' t)", "annotated_tactic": ["exact <a>prod_prod</a> (f \u207b\u00b9' s) (g \u207b\u00b9' t)", [{"full_name": "MeasureTheory.Measure.prod_prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [316, 9], "def_end_pos": [316, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2078 : MeasurableSpace \u03b1\ninst\u271d\u2077 : MeasurableSpace \u03b1'\ninst\u271d\u2076 : MeasurableSpace \u03b2\ninst\u271d\u2075 : MeasurableSpace \u03b2'\ninst\u271d\u2074 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : SigmaFinite \u03bc\n\u03b4 : Type u_7\ninst\u271d : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\n\u03bca : Measure \u03b1\n\u03bcc : Measure \u03b3\nhfa : SigmaFinite (map f \u03bca)\nhgc : SigmaFinite (map g \u03bcc)\nhf : Measurable f\nhg : Measurable g\nthis : SigmaFinite \u03bcc\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.prod \u03bca \u03bcc) (Prod.map f g \u207b\u00b9' s \u00d7\u02e2 t) = \u2191\u2191\u03bca (f \u207b\u00b9' s) * \u2191\u2191\u03bcc (g \u207b\u00b9' t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.comap_embedding_atBot", "start": [1398, 1], "end": [1400, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Algebra/Tower.lean", "full_name": "AlgEquiv.restrictScalars_apply", "start": [241, 1], "end": [241, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Composition.lean", "full_name": "ProbabilityTheory.kernel.compProd_apply_eq_compProdFun", "start": [216, 1], "end": [227, 6], "traced_tactics": [{"tactic": "rw [compProd, dif_pos]", "annotated_tactic": ["rw [<a>compProd</a>, <a>dif_pos</a>]", [{"full_name": "ProbabilityTheory.kernel.compProd", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [194, 19], "def_end_pos": [194, 27]}, {"full_name": "dif_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [807, 9], "def_end_pos": [807, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s = compProdFun \u03ba \u03b7 a s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191{\n                val := fun a =>\n                  Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)\n                    (_ :\n                      \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),\n                        (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n                          Pairwise (Disjoint on f) \u2192\n                            compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i)),\n                property :=\n                  (_ :\n                    Measurable fun a =>\n                      Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)\n                        (_ :\n                          \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),\n                            (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n                              Pairwise (Disjoint on f) \u2192\n                                compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i))) }\n            a)\n      s =\n    compProdFun \u03ba \u03b7 a s\n\ncase hc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 IsSFiniteKernel \u03ba \u2227 IsSFiniteKernel \u03b7"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191{\n                val := fun a =>\n                  Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)\n                    (_ :\n                      \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),\n                        (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n                          Pairwise (Disjoint on f) \u2192\n                            compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i)),\n                property :=\n                  (_ :\n                    Measurable fun a =>\n                      Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)\n                        (_ :\n                          \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),\n                            (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n                              Pairwise (Disjoint on f) \u2192\n                                compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i))) }\n            a)\n      s =\n    compProdFun \u03ba \u03b7 a s\n\ncase hc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 IsSFiniteKernel \u03ba \u2227 IsSFiniteKernel \u03b7", "state_after": "case hc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 IsSFiniteKernel \u03ba \u2227 IsSFiniteKernel \u03b7\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191{\n                val := fun a =>\n                  Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)\n                    (_ :\n                      \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),\n                        (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n                          Pairwise (Disjoint on f) \u2192\n                            compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i)),\n                property :=\n                  (_ :\n                    Measurable fun a =>\n                      Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)\n                        (_ :\n                          \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),\n                            (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n                              Pairwise (Disjoint on f) \u2192\n                                compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i))) }\n            a)\n      s =\n    compProdFun \u03ba \u03b7 a s"}, {"tactic": "change\n  Measure.ofMeasurable (fun s _ => compProdFun \u03ba \u03b7 a s) (compProdFun_empty \u03ba \u03b7 a)\n      (compProdFun_iUnion \u03ba \u03b7 a) s =\n    \u222b\u207b b, \u03b7 (a, b) {c | (b, c) \u2208 s} \u2202\u03ba a", "annotated_tactic": ["change\n    <a>Measure.ofMeasurable</a> (fun s _ => <a>compProdFun</a> \u03ba \u03b7 a s) (<a>compProdFun_empty</a> \u03ba \u03b7 a)\n        (<a>compProdFun_iUnion</a> \u03ba \u03b7 a) s =\n      \u222b\u207b b, \u03b7 (a, b) {c | (b, c) \u2208 s} \u2202\u03ba a", [{"full_name": "MeasureTheory.Measure.ofMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [103, 5], "def_end_pos": [103, 17]}, {"full_name": "ProbabilityTheory.kernel.compProdFun", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [88, 19], "def_end_pos": [88, 30]}, {"full_name": "ProbabilityTheory.kernel.compProdFun_empty", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [93, 9], "def_end_pos": [93, 26]}, {"full_name": "ProbabilityTheory.kernel.compProdFun_iUnion", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [99, 9], "def_end_pos": [99, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(\u2191{\n                val := fun a =>\n                  Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)\n                    (_ :\n                      \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),\n                        (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n                          Pairwise (Disjoint on f) \u2192\n                            compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i)),\n                property :=\n                  (_ :\n                    Measurable fun a =>\n                      Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)\n                        (_ :\n                          \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),\n                            (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n                              Pairwise (Disjoint on f) \u2192\n                                compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i))) }\n            a)\n      s =\n    compProdFun \u03ba \u03b7 a s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)\n            (_ :\n              \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),\n                (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n                  Pairwise (Disjoint on f) \u2192 compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i)))\n      s =\n    \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 s} \u2202\u2191\u03ba a"}, {"tactic": "rw [Measure.ofMeasurable_apply _ hs]", "annotated_tactic": ["rw [<a>Measure.ofMeasurable_apply</a> _ hs]", [{"full_name": "MeasureTheory.Measure.ofMeasurable_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [120, 9], "def_end_pos": [120, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.ofMeasurable (fun s x => compProdFun \u03ba \u03b7 a s) (_ : compProdFun \u03ba \u03b7 a \u2205 = 0)\n            (_ :\n              \u2200 (f : \u2115 \u2192 Set (\u03b2 \u00d7 \u03b3)),\n                (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192\n                  Pairwise (Disjoint on f) \u2192 compProdFun \u03ba \u03b7 a (\u22c3 i, f i) = \u2211' (i : \u2115), compProdFun \u03ba \u03b7 a (f i)))\n      s =\n    \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 s} \u2202\u2191\u03ba a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 compProdFun \u03ba \u03b7 a s = \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 s} \u2202\u2191\u03ba a"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 compProdFun \u03ba \u03b7 a s = \u222b\u207b (b : \u03b2), \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 s} \u2202\u2191\u03ba a", "state_after": "no goals"}, {"tactic": "constructor <;> infer_instance", "annotated_tactic": ["constructor <;> infer_instance", []], "state_before": "case hc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nhs : MeasurableSet s\n\u22a2 IsSFiniteKernel \u03ba \u2227 IsSFiniteKernel \u03b7", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "full_name": "IsUnit.mul_left_eq_zero", "start": [62, 1], "end": [64, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Multiset/Bind.lean", "full_name": "Multiset.coe_bind", "start": [98, 1], "end": [100, 6], "traced_tactics": [{"tactic": "rw [List.bind, \u2190 coe_join, List.map_map]", "annotated_tactic": ["rw [<a>List.bind</a>, \u2190 <a>coe_join</a>, <a>List.map_map</a>]", [{"full_name": "List.bind", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [643, 25], "def_end_pos": [643, 29]}, {"full_name": "Multiset.coe_join", "def_path": "Mathlib/Data/Multiset/Bind.lean", "def_pos": [39, 9], "def_end_pos": [39, 17]}, {"full_name": "List.map_map", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [96, 17], "def_end_pos": [96, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\na : \u03b1\ns t : Multiset \u03b1\nf\u271d g : \u03b1 \u2192 Multiset \u03b2\nl : List \u03b1\nf : \u03b1 \u2192 List \u03b2\n\u22a2 (bind \u2191l fun a => \u2191(f a)) = \u2191(List.bind l f)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\na : \u03b1\ns t : Multiset \u03b1\nf\u271d g : \u03b1 \u2192 Multiset \u03b2\nl : List \u03b1\nf : \u03b1 \u2192 List \u03b2\n\u22a2 (bind \u2191l fun a => \u2191(f a)) = join \u2191(List.map (ofList \u2218 f) l)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\na : \u03b1\ns t : Multiset \u03b1\nf\u271d g : \u03b1 \u2192 Multiset \u03b2\nl : List \u03b1\nf : \u03b1 \u2192 List \u03b2\n\u22a2 (bind \u2191l fun a => \u2191(f a)) = join \u2191(List.map (ofList \u2218 f) l)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "full_name": "smul_vsub_rev_vadd_mem_affineSpan_pair", "start": [1344, 1], "end": [1346, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "full_name": "Filter.EventuallyEq.differentiableAt_iff", "start": [929, 1], "end": [931, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Sheaves/Stalks.lean", "full_name": "TopCat.Presheaf.stalkPushforward.id", "start": [177, 1], "end": [185, 40], "traced_tactics": [{"tactic": "ext1 j", "annotated_tactic": ["ext1 j", []], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\n\u2131 : Presheaf C X\nx : \u2191X\n\u22a2 stalkPushforward C (\ud835\udfd9 X) \u2131 x = (stalkFunctor C x).map (Pushforward.id \u2131).hom", "state_after": "case w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\n\u2131 : Presheaf C X\nx : \u2191X\nj : (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56\n\u22a2 colimit.\u03b9\n        (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj\n          (\ud835\udfd9 X _* \u2131))\n        j \u226b\n      stalkPushforward C (\ud835\udfd9 X) \u2131 x =\n    colimit.\u03b9\n        (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj\n          (\ud835\udfd9 X _* \u2131))\n        j \u226b\n      (stalkFunctor C x).map (Pushforward.id \u2131).hom"}, {"tactic": "induction' j with j", "annotated_tactic": ["induction' j with j", []], "state_before": "case w\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\n\u2131 : Presheaf C X\nx : \u2191X\nj : (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56\n\u22a2 colimit.\u03b9\n        (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj\n          (\ud835\udfd9 X _* \u2131))\n        j \u226b\n      stalkPushforward C (\ud835\udfd9 X) \u2131 x =\n    colimit.\u03b9\n        (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj\n          (\ud835\udfd9 X _* \u2131))\n        j \u226b\n      (stalkFunctor C x).map (Pushforward.id \u2131).hom", "state_after": "case w.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\n\u2131 : Presheaf C X\nx : \u2191X\nj : OpenNhds (\u2191(\ud835\udfd9 X) x)\n\u22a2 colimit.\u03b9\n        (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj\n          (\ud835\udfd9 X _* \u2131))\n        (op j) \u226b\n      stalkPushforward C (\ud835\udfd9 X) \u2131 x =\n    colimit.\u03b9\n        (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj\n          (\ud835\udfd9 X _* \u2131))\n        (op j) \u226b\n      (stalkFunctor C x).map (Pushforward.id \u2131).hom"}, {"tactic": "rcases j with \u27e8\u27e8_, _\u27e9, _\u27e9", "annotated_tactic": ["rcases j with \u27e8\u27e8_, _\u27e9, _\u27e9", []], "state_before": "case w.h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\n\u2131 : Presheaf C X\nx : \u2191X\nj : OpenNhds (\u2191(\ud835\udfd9 X) x)\n\u22a2 colimit.\u03b9\n        (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj\n          (\ud835\udfd9 X _* \u2131))\n        (op j) \u226b\n      stalkPushforward C (\ud835\udfd9 X) \u2131 x =\n    colimit.\u03b9\n        (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj\n          (\ud835\udfd9 X _* \u2131))\n        (op j) \u226b\n      (stalkFunctor C x).map (Pushforward.id \u2131).hom", "state_after": "case w.h.mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\n\u2131 : Presheaf C X\nx : \u2191X\ncarrier\u271d : Set \u2191X\nis_open'\u271d : IsOpen carrier\u271d\nproperty\u271d : \u2191(\ud835\udfd9 X) x \u2208 { carrier := carrier\u271d, is_open' := is_open'\u271d }\n\u22a2 colimit.\u03b9\n        (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj\n          (\ud835\udfd9 X _* \u2131))\n        (op { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d }) \u226b\n      stalkPushforward C (\ud835\udfd9 X) \u2131 x =\n    colimit.\u03b9\n        (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj\n          (\ud835\udfd9 X _* \u2131))\n        (op { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d }) \u226b\n      (stalkFunctor C x).map (Pushforward.id \u2131).hom"}, {"tactic": "erw [colimit.\u03b9_map_assoc]", "annotated_tactic": ["erw [<a>colimit.\u03b9_map_assoc</a>]", [{"full_name": "CategoryTheory.Limits.colimit.\u03b9_map_assoc", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [1110, 3], "def_end_pos": [1110, 10]}]], "state_before": "case w.h.mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\n\u2131 : Presheaf C X\nx : \u2191X\ncarrier\u271d : Set \u2191X\nis_open'\u271d : IsOpen carrier\u271d\nproperty\u271d : \u2191(\ud835\udfd9 X) x \u2208 { carrier := carrier\u271d, is_open' := is_open'\u271d }\n\u22a2 colimit.\u03b9\n        (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj\n          (\ud835\udfd9 X _* \u2131))\n        (op { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d }) \u226b\n      stalkPushforward C (\ud835\udfd9 X) \u2131 x =\n    colimit.\u03b9\n        (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj\n          (\ud835\udfd9 X _* \u2131))\n        (op { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d }) \u226b\n      (stalkFunctor C x).map (Pushforward.id \u2131).hom", "state_after": "case w.h.mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\n\u2131 : Presheaf C X\nx : \u2191X\ncarrier\u271d : Set \u2191X\nis_open'\u271d : IsOpen carrier\u271d\nproperty\u271d : \u2191(\ud835\udfd9 X) x \u2208 { carrier := carrier\u271d, is_open' := is_open'\u271d }\n\u22a2 (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso (\ud835\udfd9 X) x).inv) \u2131).app\n        (op { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d }) \u226b\n      colimit.\u03b9\n          ((OpenNhds.map (\ud835\udfd9 X) x).op \u22d9\n            ((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj \u2131)\n          (op { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d }) \u226b\n        colimit.pre (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj \u2131)\n          (OpenNhds.map (\ud835\udfd9 X) x).op =\n    colimit.\u03b9\n        (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj\n          (\ud835\udfd9 X _* \u2131))\n        (op { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d }) \u226b\n      (stalkFunctor C x).map (Pushforward.id \u2131).hom"}, {"tactic": "simp [stalkFunctor, stalkPushforward]", "annotated_tactic": ["simp [<a>stalkFunctor</a>, <a>stalkPushforward</a>]", [{"full_name": "TopCat.Presheaf.stalkFunctor", "def_path": "Mathlib/Topology/Sheaves/Stalks.lean", "def_pos": [77, 5], "def_end_pos": [77, 17]}, {"full_name": "TopCat.Presheaf.stalkPushforward", "def_path": "Mathlib/Topology/Sheaves/Stalks.lean", "def_pos": [142, 5], "def_end_pos": [142, 21]}]], "state_before": "case w.h.mk.mk\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasColimits C\nX Y Z : TopCat\n\u2131 : Presheaf C X\nx : \u2191X\ncarrier\u271d : Set \u2191X\nis_open'\u271d : IsOpen carrier\u271d\nproperty\u271d : \u2191(\ud835\udfd9 X) x \u2208 { carrier := carrier\u271d, is_open' := is_open'\u271d }\n\u22a2 (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso (\ud835\udfd9 X) x).inv) \u2131).app\n        (op { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d }) \u226b\n      colimit.\u03b9\n          ((OpenNhds.map (\ud835\udfd9 X) x).op \u22d9\n            ((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj \u2131)\n          (op { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d }) \u226b\n        colimit.pre (((whiskeringLeft (OpenNhds x)\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion x).op).obj \u2131)\n          (OpenNhds.map (\ud835\udfd9 X) x).op =\n    colimit.\u03b9\n        (((whiskeringLeft (OpenNhds (\u2191(\ud835\udfd9 X) x))\u1d52\u1d56 (Opens \u2191X)\u1d52\u1d56 C).obj (OpenNhds.inclusion (\u2191(\ud835\udfd9 X) x)).op).obj\n          (\ud835\udfd9 X _* \u2131))\n        (op { obj := { carrier := carrier\u271d, is_open' := is_open'\u271d }, property := property\u271d }) \u226b\n      (stalkFunctor C x).map (Pushforward.id \u2131).hom", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "full_name": "nonempty_measurable_superset", "start": [246, 1], "end": [247, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.iInter_plift_down", "start": [338, 1], "end": [339, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean", "full_name": "MeasureTheory.exists_lt_lowerSemicontinuous_integral_gt_nnreal", "start": [270, 1], "end": [313, 78], "traced_tactics": [{"tactic": "have fmeas : AEMeasurable f \u03bc := by\n  convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable\n  simp only [Real.toNNReal_coe]", "annotated_tactic": ["have fmeas : <a>AEMeasurable</a> f \u03bc := by\n    convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable\n    simp only [<a>Real.toNNReal_coe</a>]", [{"full_name": "AEMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [708, 5], "def_end_pos": [708, 17]}, {"full_name": "Real.toNNReal_coe", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [391, 9], "def_end_pos": [391, 33]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nfmeas : AEMeasurable f\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5"}, {"tactic": "lift \u03b5 to \u211d\u22650 using \u03b5pos.le", "annotated_tactic": ["lift \u03b5 to \u211d\u22650 using \u03b5pos.le", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nfmeas : AEMeasurable f\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u03b5", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5"}, {"tactic": "obtain \u27e8\u03b4, \u03b4pos, h\u03b4\u03b5\u27e9 : \u2203 \u03b4 : \u211d\u22650, 0 < \u03b4 \u2227 \u03b4 < \u03b5", "annotated_tactic": ["obtain \u27e8\u03b4, \u03b4pos, h\u03b4\u03b5\u27e9 : \u2203 \u03b4 : \u211d\u22650, 0 < \u03b4 \u2227 \u03b4 < \u03b5", []], "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 \u03b4 < \u03b5\n\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5"}, {"tactic": "exact exists_between \u03b5pos", "annotated_tactic": ["exact <a>exists_between</a> \u03b5pos", [{"full_name": "exists_between", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [1294, 9], "def_end_pos": [1294, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 \u03b4 < \u03b5\n\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5"}, {"tactic": "have int_f_ne_top : (\u222b\u207b a : \u03b1, f a \u2202\u03bc) \u2260 \u221e :=\n  (hasFiniteIntegral_iff_ofNNReal.1 fint.hasFiniteIntegral).ne", "annotated_tactic": ["have int_f_ne_top : (\u222b\u207b a : \u03b1, f a \u2202\u03bc) \u2260 \u221e :=\n    (<a>hasFiniteIntegral_iff_ofNNReal</a>.1 fint.hasFiniteIntegral).<a>ne</a>", [{"full_name": "MeasureTheory.hasFiniteIntegral_iff_ofNNReal", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [132, 9], "def_end_pos": [132, 39]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5"}, {"tactic": "rcases exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable \u03bc f fmeas\n    (ENNReal.coe_ne_zero.2 \u03b4pos.ne') with\n  \u27e8g, f_lt_g, gcont, gint\u27e9", "annotated_tactic": ["rcases <a>exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable</a> \u03bc f fmeas\n      (<a>ENNReal.coe_ne_zero</a>.2 \u03b4pos.ne') with\n    \u27e8g, f_lt_g, gcont, gint\u27e9", [{"full_name": "MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable", "def_path": "Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean", "def_pos": [231, 9], "def_end_pos": [231, 67]}, {"full_name": "ENNReal.coe_ne_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [383, 9], "def_end_pos": [383, 20]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5"}, {"tactic": "have gint_ne : (\u222b\u207b x : \u03b1, g x \u2202\u03bc) \u2260 \u221e := ne_top_of_le_ne_top (by simpa) gint", "annotated_tactic": ["have gint_ne : (\u222b\u207b x : \u03b1, g x \u2202\u03bc) \u2260 \u221e := <a>ne_top_of_le_ne_top</a> (by simpa) gint", [{"full_name": "ne_top_of_le_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [194, 9], "def_end_pos": [194, 28]}]], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5"}, {"tactic": "have g_lt_top : \u2200\u1d50 x : \u03b1 \u2202\u03bc, g x < \u221e := ae_lt_top gcont.measurable gint_ne", "annotated_tactic": ["have g_lt_top : \u2200\u1d50 x : \u03b1 \u2202\u03bc, g x < \u221e := <a>ae_lt_top</a> gcont.measurable gint_ne", [{"full_name": "MeasureTheory.ae_lt_top", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1522, 9], "def_end_pos": [1522, 18]}]], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5"}, {"tactic": "have Ig : (\u222b\u207b a : \u03b1, ENNReal.ofReal (g a).toReal \u2202\u03bc) = \u222b\u207b a : \u03b1, g a \u2202\u03bc := by\n  apply lintegral_congr_ae\n  filter_upwards [g_lt_top] with _ hx\n  simp only [hx.ne, ENNReal.ofReal_toReal, Ne.def, not_false_iff]", "annotated_tactic": ["have Ig : (\u222b\u207b a : \u03b1, <a>ENNReal.ofReal</a> (g a).<a>toReal</a> \u2202\u03bc) = \u222b\u207b a : \u03b1, g a \u2202\u03bc := by\n    apply <a>lintegral_congr_ae</a>\n    filter_upwards [g_lt_top] with _ hx\n    simp only [hx.ne, <a>ENNReal.ofReal_toReal</a>, <a>Ne.def</a>, <a>not_false_iff</a>]", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}, {"full_name": "ENNReal.ofReal_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5", "state_after": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5"}, {"tactic": "refine' \u27e8g, f_lt_g, gcont, g_lt_top, _, _\u27e9", "annotated_tactic": ["refine' \u27e8g, f_lt_g, gcont, g_lt_top, _, _\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u2203 g,\n    (\u2200 (x : \u03b1), \u2191(f x) < g x) \u2227\n      LowerSemicontinuous g \u2227\n        (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4) \u2227\n          (Integrable fun x => ENNReal.toReal (g x)) \u2227 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5", "state_after": "case intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 Integrable fun x => ENNReal.toReal (g x)\n\ncase intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5"}, {"tactic": "convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable", "annotated_tactic": ["convert fint.aestronglyMeasurable.real_toNNReal.aemeasurable", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 AEMeasurable f", "state_after": "case h.e'_5.h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nx\u271d : \u03b1\n\u22a2 f x\u271d = Real.toNNReal \u2191(f x\u271d)"}, {"tactic": "simp only [Real.toNNReal_coe]", "annotated_tactic": ["simp only [<a>Real.toNNReal_coe</a>]", [{"full_name": "Real.toNNReal_coe", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [391, 9], "def_end_pos": [391, 33]}]], "state_before": "case h.e'_5.h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nx\u271d : \u03b1\n\u22a2 f x\u271d = Real.toNNReal \u2191(f x\u271d)", "state_after": "no goals"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "apply lintegral_congr_ae", "annotated_tactic": ["apply <a>lintegral_congr_ae</a>", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\n\u22a2 (fun a => ENNReal.ofReal (ENNReal.toReal (g a))) =\u1da0[ae \u03bc] fun a => g a"}, {"tactic": "filter_upwards [g_lt_top] with _ hx", "annotated_tactic": ["filter_upwards [g_lt_top] with _ hx", []], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\n\u22a2 (fun a => ENNReal.ofReal (ENNReal.toReal (g a))) =\u1da0[ae \u03bc] fun a => g a", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\na\u271d : \u03b1\nhx : g a\u271d < \u22a4\n\u22a2 ENNReal.ofReal (ENNReal.toReal (g a\u271d)) = g a\u271d"}, {"tactic": "simp only [hx.ne, ENNReal.ofReal_toReal, Ne.def, not_false_iff]", "annotated_tactic": ["simp only [hx.ne, <a>ENNReal.ofReal_toReal</a>, <a>Ne.def</a>, <a>not_false_iff</a>]", [{"full_name": "ENNReal.ofReal_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [186, 9], "def_end_pos": [186, 22]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\na\u271d : \u03b1\nhx : g a\u271d < \u22a4\n\u22a2 ENNReal.ofReal (ENNReal.toReal (g a\u271d)) = g a\u271d", "state_after": "no goals"}, {"tactic": "refine' \u27e8gcont.measurable.ennreal_toReal.aemeasurable.aestronglyMeasurable, _\u27e9", "annotated_tactic": ["refine' \u27e8gcont.measurable.ennreal_toReal.aemeasurable.aestronglyMeasurable, _\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 Integrable fun x => ENNReal.toReal (g x)", "state_after": "case intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 HasFiniteIntegral fun x => ENNReal.toReal (g x)"}, {"tactic": "simp only [hasFiniteIntegral_iff_norm, Real.norm_eq_abs, abs_of_nonneg ENNReal.toReal_nonneg]", "annotated_tactic": ["simp only [<a>hasFiniteIntegral_iff_norm</a>, <a>Real.norm_eq_abs</a>, <a>abs_of_nonneg</a> <a>ENNReal.toReal_nonneg</a>]", [{"full_name": "MeasureTheory.hasFiniteIntegral_iff_norm", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [117, 9], "def_end_pos": [117, 35]}, {"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 20]}, {"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [107, 9], "def_end_pos": [107, 22]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [221, 17], "def_end_pos": [221, 30]}]], "state_before": "case intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 HasFiniteIntegral fun x => ENNReal.toReal (g x)", "state_after": "case intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc < \u22a4"}, {"tactic": "convert gint_ne.lt_top using 1", "annotated_tactic": ["convert gint_ne.lt_top using 1", []], "state_before": "case intro.intro.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc < \u22a4", "state_after": "no goals"}, {"tactic": "rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae]", "annotated_tactic": ["rw [<a>integral_eq_lintegral_of_nonneg_ae</a>, <a>integral_eq_lintegral_of_nonneg_ae</a>]", [{"full_name": "MeasureTheory.integral_eq_lintegral_of_nonneg_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1158, 9], "def_end_pos": [1158, 43]}, {"full_name": "MeasureTheory.integral_eq_lintegral_of_nonneg_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1158, 9], "def_end_pos": [1158, 43]}]], "state_before": "case intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u222b (x : \u03b1), ENNReal.toReal (g x) \u2202\u03bc < \u222b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b5", "state_after": "case intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc) <\n    ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc) + \u2191\u03b5\n\ncase intro.intro.intro.intro.intro.intro.refine'_2.hf\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 0 \u2264\u1da0[ae \u03bc] fun x => \u2191(f x)\n\ncase intro.intro.intro.intro.intro.intro.refine'_2.hfm\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 AEStronglyMeasurable (fun x => \u2191(f x)) \u03bc\n\ncase intro.intro.intro.intro.intro.intro.refine'_2.hf\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 0 \u2264\u1da0[ae \u03bc] fun x => ENNReal.toReal (g x)\n\ncase intro.intro.intro.intro.intro.intro.refine'_2.hfm\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 AEStronglyMeasurable (fun x => ENNReal.toReal (g x)) \u03bc"}, {"tactic": "calc\n  ENNReal.toReal (\u222b\u207b a : \u03b1, ENNReal.ofReal (g a).toReal \u2202\u03bc) =\n      ENNReal.toReal (\u222b\u207b a : \u03b1, g a \u2202\u03bc) :=\n    by congr 1\n  _ \u2264 ENNReal.toReal ((\u222b\u207b a : \u03b1, f a \u2202\u03bc) + \u03b4) := by\n    apply ENNReal.toReal_mono _ gint\n    simpa using int_f_ne_top\n  _ = ENNReal.toReal (\u222b\u207b a : \u03b1, f a \u2202\u03bc) + \u03b4 := by\n    rw [ENNReal.toReal_add int_f_ne_top ENNReal.coe_ne_top, ENNReal.coe_toReal]\n  _ < ENNReal.toReal (\u222b\u207b a : \u03b1, f a \u2202\u03bc) + \u03b5 := (add_lt_add_left h\u03b4\u03b5 _)\n  _ = (\u222b\u207b a : \u03b1, ENNReal.ofReal \u2191(f a) \u2202\u03bc).toReal + \u03b5 := by simp", "annotated_tactic": ["calc\n        <a>ENNReal.toReal</a> (\u222b\u207b a : \u03b1, <a>ENNReal.ofReal</a> (g a).<a>toReal</a> \u2202\u03bc) =\n            <a>ENNReal.toReal</a> (\u222b\u207b a : \u03b1, g a \u2202\u03bc) :=\n          by congr 1\n        _ \u2264 <a>ENNReal.toReal</a> ((\u222b\u207b a : \u03b1, f a \u2202\u03bc) + \u03b4) := by\n          apply <a>ENNReal.toReal_mono</a> _ gint\n          simpa using int_f_ne_top\n        _ = <a>ENNReal.toReal</a> (\u222b\u207b a : \u03b1, f a \u2202\u03bc) + \u03b4 := by\n          rw [<a>ENNReal.toReal_add</a> int_f_ne_top <a>ENNReal.coe_ne_top</a>, <a>ENNReal.coe_toReal</a>]\n        _ < <a>ENNReal.toReal</a> (\u222b\u207b a : \u03b1, f a \u2202\u03bc) + \u03b5 := (<a>add_lt_add_left</a> h\u03b4\u03b5 _)\n        _ = (\u222b\u207b a : \u03b1, <a>ENNReal.ofReal</a> \u2191(f a) \u2202\u03bc).<a>toReal</a> + \u03b5 := by simp", [{"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "ENNReal.toReal_mono", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2042, 9], "def_end_pos": [2042, 20]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "ENNReal.toReal_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1997, 9], "def_end_pos": [1997, 19]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "ENNReal.coe_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [236, 17], "def_end_pos": [236, 27]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "add_lt_add_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [120, 15], "def_end_pos": [120, 30]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}]], "state_before": "case intro.intro.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc) <\n    ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc) + \u2191\u03b5", "state_after": "no goals"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc) = ENNReal.toReal (\u222b\u207b (a : \u03b1), g a \u2202\u03bc)", "state_after": "no goals"}, {"tactic": "apply ENNReal.toReal_mono _ gint", "annotated_tactic": ["apply <a>ENNReal.toReal_mono</a> _ gint", [{"full_name": "ENNReal.toReal_mono", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2042, 9], "def_end_pos": [2042, 20]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), g a \u2202\u03bc) \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc + \u2191\u03b4)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 \u2260 \u22a4"}, {"tactic": "simpa using int_f_ne_top", "annotated_tactic": ["simpa using int_f_ne_top", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4 \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "rw [ENNReal.toReal_add int_f_ne_top ENNReal.coe_ne_top, ENNReal.coe_toReal]", "annotated_tactic": ["rw [<a>ENNReal.toReal_add</a> int_f_ne_top <a>ENNReal.coe_ne_top</a>, <a>ENNReal.coe_toReal</a>]", [{"full_name": "ENNReal.toReal_add", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1997, 9], "def_end_pos": [1997, 19]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "ENNReal.coe_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [236, 17], "def_end_pos": [236, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc + \u2191\u03b4) = ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc) + \u2191\u03b4", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc) + \u2191\u03b5 = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc) + \u2191\u03b5", "state_after": "no goals"}, {"tactic": "apply Filter.eventually_of_forall fun x => _", "annotated_tactic": ["apply <a>Filter.eventually_of_forall</a> fun x => _", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case intro.intro.intro.intro.intro.intro.refine'_2.hf\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 0 \u2264\u1da0[ae \u03bc] fun x => \u2191(f x)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 (fun x => \u2191(f x)) x"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 (fun x => \u2191(f x)) x", "state_after": "no goals"}, {"tactic": "exact fmeas.coe_nnreal_real.aestronglyMeasurable", "annotated_tactic": ["exact fmeas.coe_nnreal_real.aestronglyMeasurable", []], "state_before": "case intro.intro.intro.intro.intro.intro.refine'_2.hfm\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 AEStronglyMeasurable (fun x => \u2191(f x)) \u03bc", "state_after": "no goals"}, {"tactic": "apply Filter.eventually_of_forall fun x => _", "annotated_tactic": ["apply <a>Filter.eventually_of_forall</a> fun x => _", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}]], "state_before": "case intro.intro.intro.intro.intro.intro.refine'_2.hf\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 0 \u2264\u1da0[ae \u03bc] fun x => ENNReal.toReal (g x)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 (fun x => ENNReal.toReal (g x)) x"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 (fun x => ENNReal.toReal (g x)) x", "state_after": "no goals"}, {"tactic": "apply gcont.measurable.ennreal_toReal.aemeasurable.aestronglyMeasurable", "annotated_tactic": ["apply gcont.measurable.ennreal_toReal.aemeasurable.aestronglyMeasurable", []], "state_before": "case intro.intro.intro.intro.intro.intro.refine'_2.hfm\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : WeaklyRegular \u03bc\ninst\u271d : SigmaFinite \u03bc\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable fun x => \u2191(f x)\nfmeas : AEMeasurable f\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u2191\u03b5\n\u03b4 : \u211d\u22650\n\u03b4pos : 0 < \u03b4\nh\u03b4\u03b5 : \u03b4 < \u03b5\nint_f_ne_top : \u222b\u207b (a : \u03b1), \u2191(f a) \u2202\u03bc \u2260 \u22a4\ng : \u03b1 \u2192 \u211d\u22650\u221e\nf_lt_g : \u2200 (x : \u03b1), \u2191(f x) < g x\ngcont : LowerSemicontinuous g\ngint : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), \u2191(f x) \u2202\u03bc + \u2191\u03b4\ngint_ne : \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2260 \u22a4\ng_lt_top : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, g x < \u22a4\nIg : \u222b\u207b (a : \u03b1), ENNReal.ofReal (ENNReal.toReal (g a)) \u2202\u03bc = \u222b\u207b (a : \u03b1), g a \u2202\u03bc\n\u22a2 AEStronglyMeasurable (fun x => ENNReal.toReal (g x)) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/SimpleGraph/Density.lean", "full_name": "SimpleGraph.edgeDensity_le_one", "start": [404, 1], "end": [405, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean", "full_name": "List.Nat.antidiagonalTuple_pairwise_pi_lex", "start": [152, 1], "end": [173, 17], "traced_tactics": [{"tactic": "simp_rw [antidiagonalTuple, List.pairwise_bind, List.pairwise_map, List.mem_map,\n  forall_exists_index, and_imp, forall_apply_eq_imp_iff\u2082]", "annotated_tactic": ["simp_rw [<a>antidiagonalTuple</a>, <a>List.pairwise_bind</a>, <a>List.pairwise_map</a>, <a>List.mem_map</a>,\n      <a>forall_exists_index</a>, <a>and_imp</a>, <a>forall_apply_eq_imp_iff\u2082</a>]", [{"full_name": "List.Nat.antidiagonalTuple", "def_path": "Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean", "def_pos": [63, 5], "def_end_pos": [63, 22]}, {"full_name": "List.pairwise_bind", "def_path": "lake-packages/std/Std/Data/List/Pairwise.lean", "def_pos": [167, 9], "def_end_pos": [167, 22]}, {"full_name": "List.pairwise_map", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1451, 9], "def_end_pos": [1451, 21]}, {"full_name": "List.mem_map", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [159, 17], "def_end_pos": [159, 24]}, {"full_name": "forall_exists_index", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [356, 17], "def_end_pos": [356, 36]}, {"full_name": "and_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [313, 17], "def_end_pos": [313, 24]}, {"full_name": "forall_apply_eq_imp_iff\u2082", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [505, 17], "def_end_pos": [505, 41]}]], "state_before": "k n : \u2115\n\u22a2 Pairwise (Pi.Lex (fun x x_1 => x < x_1) fun x x x_1 => x < x_1) (antidiagonalTuple (k + 1) n)", "state_after": "k n : \u2115\n\u22a2 (\u2200 (a : \u2115 \u00d7 \u2115),\n      a \u2208 antidiagonal n \u2192\n        Pairwise\n          (fun a_2 b => Pi.Lex (fun x x_1 => x < x_1) (fun x x x_1 => x < x_1) (Fin.cons a.1 a_2) (Fin.cons a.1 b))\n          (antidiagonalTuple (Nat.add k 0) a.2)) \u2227\n    Pairwise\n      (fun a\u2081 a\u2082 =>\n        \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n          a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n            \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n              a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n                Pi.Lex (fun x x_1 => x < x_1) (fun x x x_1 => x < x_1) (Fin.cons a\u2081.1 a) (Fin.cons a\u2082.1 a_2))\n      (antidiagonal n)"}, {"tactic": "simp only [mem_antidiagonal, Prod.forall, and_imp, forall_apply_eq_imp_iff\u2082]", "annotated_tactic": ["simp only [<a>mem_antidiagonal</a>, <a>Prod.forall</a>, <a>and_imp</a>, <a>forall_apply_eq_imp_iff\u2082</a>]", [{"full_name": "List.Nat.mem_antidiagonal", "def_path": "Mathlib/Data/List/NatAntidiagonal.lean", "def_pos": [38, 9], "def_end_pos": [38, 25]}, {"full_name": "Prod.forall", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [36, 9], "def_end_pos": [36, 17]}, {"full_name": "and_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [313, 17], "def_end_pos": [313, 24]}, {"full_name": "forall_apply_eq_imp_iff\u2082", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [505, 17], "def_end_pos": [505, 41]}]], "state_before": "k n : \u2115\n\u22a2 (\u2200 (a : \u2115 \u00d7 \u2115),\n      a \u2208 antidiagonal n \u2192\n        Pairwise\n          (fun a_2 b => Pi.Lex (fun x x_1 => x < x_1) (fun x x x_1 => x < x_1) (Fin.cons a.1 a_2) (Fin.cons a.1 b))\n          (antidiagonalTuple (Nat.add k 0) a.2)) \u2227\n    Pairwise\n      (fun a\u2081 a\u2082 =>\n        \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n          a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n            \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n              a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n                Pi.Lex (fun x x_1 => x < x_1) (fun x x x_1 => x < x_1) (Fin.cons a\u2081.1 a) (Fin.cons a\u2082.1 a_2))\n      (antidiagonal n)", "state_after": "k n : \u2115\n\u22a2 (\u2200 (a b : \u2115),\n      a + b = n \u2192\n        Pairwise (fun a_2 b => Pi.Lex (fun x x_1 => x < x_1) (fun x x x_1 => x < x_1) (Fin.cons a a_2) (Fin.cons a b))\n          (antidiagonalTuple (Nat.add k 0) b)) \u2227\n    Pairwise\n      (fun a\u2081 a\u2082 =>\n        \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n          a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n            \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n              a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n                Pi.Lex (fun x x_1 => x < x_1) (fun x x x_1 => x < x_1) (Fin.cons a\u2081.1 a) (Fin.cons a\u2082.1 a_2))\n      (antidiagonal n)"}, {"tactic": "simp only [Fin.pi_lex_lt_cons_cons, eq_self_iff_true, true_and_iff, lt_self_iff_false,\n  false_or_iff]", "annotated_tactic": ["simp only [<a>Fin.pi_lex_lt_cons_cons</a>, <a>eq_self_iff_true</a>, <a>true_and_iff</a>, <a>lt_self_iff_false</a>,\n      <a>false_or_iff</a>]", [{"full_name": "Fin.pi_lex_lt_cons_cons", "def_path": "Mathlib/Data/Fin/Tuple/Basic.lean", "def_pos": [269, 9], "def_end_pos": [269, 28]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "lt_self_iff_false", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [175, 9], "def_end_pos": [175, 26]}, {"full_name": "false_or_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [186, 9], "def_end_pos": [186, 21]}]], "state_before": "k n : \u2115\n\u22a2 (\u2200 (a b : \u2115),\n      a + b = n \u2192\n        Pairwise (fun a_2 b => Pi.Lex (fun x x_1 => x < x_1) (fun x x x_1 => x < x_1) (Fin.cons a a_2) (Fin.cons a b))\n          (antidiagonalTuple (Nat.add k 0) b)) \u2227\n    Pairwise\n      (fun a\u2081 a\u2082 =>\n        \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n          a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n            \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n              a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n                Pi.Lex (fun x x_1 => x < x_1) (fun x x x_1 => x < x_1) (Fin.cons a\u2081.1 a) (Fin.cons a\u2082.1 a_2))\n      (antidiagonal n)", "state_after": "k n : \u2115\n\u22a2 (\u2200 (a b : \u2115),\n      a + b = n \u2192\n        Pairwise (fun a b => Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a b)\n          (antidiagonalTuple (Nat.add k 0) b)) \u2227\n    Pairwise\n      (fun a\u2081 a\u2082 =>\n        \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n          a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n            \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n              a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n                a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n      (antidiagonal n)"}, {"tactic": "refine' \u27e8fun _ _ _ => antidiagonalTuple_pairwise_pi_lex k _, _\u27e9", "annotated_tactic": ["refine' \u27e8fun _ _ _ => antidiagonalTuple_pairwise_pi_lex k _, _\u27e9", []], "state_before": "k n : \u2115\n\u22a2 (\u2200 (a b : \u2115),\n      a + b = n \u2192\n        Pairwise (fun a b => Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a b)\n          (antidiagonalTuple (Nat.add k 0) b)) \u2227\n    Pairwise\n      (fun a\u2081 a\u2082 =>\n        \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n          a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n            \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n              a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n                a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n      (antidiagonal n)", "state_after": "k n : \u2115\n\u22a2 Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)"}, {"tactic": "induction' n with n n_ih", "annotated_tactic": ["induction' n with n n_ih", []], "state_before": "k n : \u2115\n\u22a2 Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)", "state_after": "case zero\nk : \u2115\n\u22a2 Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal Nat.zero)\n\ncase succ\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\n\u22a2 Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal (Nat.succ n))"}, {"tactic": "rw [antidiagonal_zero]", "annotated_tactic": ["rw [<a>antidiagonal_zero</a>]", [{"full_name": "List.Nat.antidiagonal_zero", "def_path": "Mathlib/Data/List/NatAntidiagonal.lean", "def_pos": [58, 9], "def_end_pos": [58, 26]}]], "state_before": "case zero\nk : \u2115\n\u22a2 Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal Nat.zero)", "state_after": "case zero\nk : \u2115\n\u22a2 Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    [(0, 0)]"}, {"tactic": "exact List.pairwise_singleton _ _", "annotated_tactic": ["exact <a>List.pairwise_singleton</a> _ _", [{"full_name": "List.pairwise_singleton", "def_path": "lake-packages/std/Std/Data/List/Pairwise.lean", "def_pos": [104, 9], "def_end_pos": [104, 27]}]], "state_before": "case zero\nk : \u2115\n\u22a2 Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    [(0, 0)]", "state_after": "no goals"}, {"tactic": "rw [antidiagonal_succ, List.pairwise_cons, List.pairwise_map]", "annotated_tactic": ["rw [<a>antidiagonal_succ</a>, <a>List.pairwise_cons</a>, <a>List.pairwise_map</a>]", [{"full_name": "List.Nat.antidiagonal_succ", "def_path": "Mathlib/Data/List/NatAntidiagonal.lean", "def_pos": [68, 9], "def_end_pos": [68, 26]}, {"full_name": "List.pairwise_cons", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1129, 17], "def_end_pos": [1129, 30]}, {"full_name": "List.pairwise_map", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1451, 9], "def_end_pos": [1451, 21]}]], "state_before": "case succ\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\n\u22a2 Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal (Nat.succ n))", "state_after": "case succ\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\n\u22a2 (\u2200 (a' : \u2115 \u00d7 \u2115),\n      a' \u2208 map (Prod.map Nat.succ id) (antidiagonal n) \u2192\n        \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n          a \u2208 antidiagonalTuple (Nat.add k 0) (0, n + 1).2 \u2192\n            \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n              a_2 \u2208 antidiagonalTuple (Nat.add k 0) a'.2 \u2192\n                (0, n + 1).1 < a'.1 \u2228\n                  (0, n + 1).1 = a'.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2) \u2227\n    Pairwise\n      (fun a b =>\n        \u2200 (a_1 : Fin (Nat.add k 0) \u2192 \u2115),\n          a_1 \u2208 antidiagonalTuple (Nat.add k 0) (Prod.map Nat.succ id a).2 \u2192\n            \u2200 (a_3 : Fin (Nat.add k 0) \u2192 \u2115),\n              a_3 \u2208 antidiagonalTuple (Nat.add k 0) (Prod.map Nat.succ id b).2 \u2192\n                (Prod.map Nat.succ id a).1 < (Prod.map Nat.succ id b).1 \u2228\n                  (Prod.map Nat.succ id a).1 = (Prod.map Nat.succ id b).1 \u2227\n                    Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a_1 a_3)\n      (antidiagonal n)"}, {"tactic": "refine' \u27e8fun p hp x hx y hy => _, _\u27e9", "annotated_tactic": ["refine' \u27e8fun p hp x hx y hy => _, _\u27e9", []], "state_before": "case succ\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\n\u22a2 (\u2200 (a' : \u2115 \u00d7 \u2115),\n      a' \u2208 map (Prod.map Nat.succ id) (antidiagonal n) \u2192\n        \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n          a \u2208 antidiagonalTuple (Nat.add k 0) (0, n + 1).2 \u2192\n            \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n              a_2 \u2208 antidiagonalTuple (Nat.add k 0) a'.2 \u2192\n                (0, n + 1).1 < a'.1 \u2228\n                  (0, n + 1).1 = a'.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2) \u2227\n    Pairwise\n      (fun a b =>\n        \u2200 (a_1 : Fin (Nat.add k 0) \u2192 \u2115),\n          a_1 \u2208 antidiagonalTuple (Nat.add k 0) (Prod.map Nat.succ id a).2 \u2192\n            \u2200 (a_3 : Fin (Nat.add k 0) \u2192 \u2115),\n              a_3 \u2208 antidiagonalTuple (Nat.add k 0) (Prod.map Nat.succ id b).2 \u2192\n                (Prod.map Nat.succ id a).1 < (Prod.map Nat.succ id b).1 \u2228\n                  (Prod.map Nat.succ id a).1 = (Prod.map Nat.succ id b).1 \u2227\n                    Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a_1 a_3)\n      (antidiagonal n)", "state_after": "case succ.refine'_1\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\np : \u2115 \u00d7 \u2115\nhp : p \u2208 map (Prod.map Nat.succ id) (antidiagonal n)\nx : Fin (Nat.add k 0) \u2192 \u2115\nhx : x \u2208 antidiagonalTuple (Nat.add k 0) (0, n + 1).2\ny : Fin (Nat.add k 0) \u2192 \u2115\nhy : y \u2208 antidiagonalTuple (Nat.add k 0) p.2\n\u22a2 (0, n + 1).1 < p.1 \u2228 (0, n + 1).1 = p.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) x y\n\ncase succ.refine'_2\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\n\u22a2 Pairwise\n    (fun a b =>\n      \u2200 (a_1 : Fin (Nat.add k 0) \u2192 \u2115),\n        a_1 \u2208 antidiagonalTuple (Nat.add k 0) (Prod.map Nat.succ id a).2 \u2192\n          \u2200 (a_3 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_3 \u2208 antidiagonalTuple (Nat.add k 0) (Prod.map Nat.succ id b).2 \u2192\n              (Prod.map Nat.succ id a).1 < (Prod.map Nat.succ id b).1 \u2228\n                (Prod.map Nat.succ id a).1 = (Prod.map Nat.succ id b).1 \u2227\n                  Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a_1 a_3)\n    (antidiagonal n)"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "case succ.refine'_2\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\n\u22a2 Pairwise\n    (fun a b =>\n      \u2200 (a_1 : Fin (Nat.add k 0) \u2192 \u2115),\n        a_1 \u2208 antidiagonalTuple (Nat.add k 0) (Prod.map Nat.succ id a).2 \u2192\n          \u2200 (a_3 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_3 \u2208 antidiagonalTuple (Nat.add k 0) (Prod.map Nat.succ id b).2 \u2192\n              (Prod.map Nat.succ id a).1 < (Prod.map Nat.succ id b).1 \u2228\n                (Prod.map Nat.succ id a).1 = (Prod.map Nat.succ id b).1 \u2227\n                  Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a_1 a_3)\n    (antidiagonal n)", "state_after": "case succ.refine'_2\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\n\u22a2 Pairwise\n    (fun a b =>\n      \u2200 (a_1 : Fin k \u2192 \u2115),\n        a_1 \u2208 antidiagonalTuple k a.2 \u2192\n          \u2200 (a_3 : Fin k \u2192 \u2115),\n            a_3 \u2208 antidiagonalTuple k b.2 \u2192\n              Nat.succ a.1 < Nat.succ b.1 \u2228\n                Nat.succ a.1 = Nat.succ b.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a_1 a_3)\n    (antidiagonal n)"}, {"tactic": "simp_rw [Nat.succ_inj', Nat.succ_lt_succ_iff]", "annotated_tactic": ["simp_rw [<a>Nat.succ_inj'</a>, <a>Nat.succ_lt_succ_iff</a>]", [{"full_name": "Nat.succ_inj'", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [244, 9], "def_end_pos": [244, 18]}, {"full_name": "Nat.succ_lt_succ_iff", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [260, 9], "def_end_pos": [260, 25]}]], "state_before": "case succ.refine'_2\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\n\u22a2 Pairwise\n    (fun a b =>\n      \u2200 (a_1 : Fin k \u2192 \u2115),\n        a_1 \u2208 antidiagonalTuple k a.2 \u2192\n          \u2200 (a_3 : Fin k \u2192 \u2115),\n            a_3 \u2208 antidiagonalTuple k b.2 \u2192\n              Nat.succ a.1 < Nat.succ b.1 \u2228\n                Nat.succ a.1 = Nat.succ b.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a_1 a_3)\n    (antidiagonal n)", "state_after": "case succ.refine'_2\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\n\u22a2 Pairwise\n    (fun a b =>\n      \u2200 (a_1 : Fin k \u2192 \u2115),\n        a_1 \u2208 antidiagonalTuple k a.2 \u2192\n          \u2200 (a_3 : Fin k \u2192 \u2115),\n            a_3 \u2208 antidiagonalTuple k b.2 \u2192\n              a.1 < b.1 \u2228 a.1 = b.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a_1 a_3)\n    (antidiagonal n)"}, {"tactic": "exact n_ih", "annotated_tactic": ["exact n_ih", []], "state_before": "case succ.refine'_2\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\n\u22a2 Pairwise\n    (fun a b =>\n      \u2200 (a_1 : Fin k \u2192 \u2115),\n        a_1 \u2208 antidiagonalTuple k a.2 \u2192\n          \u2200 (a_3 : Fin k \u2192 \u2115),\n            a_3 \u2208 antidiagonalTuple k b.2 \u2192\n              a.1 < b.1 \u2228 a.1 = b.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a_1 a_3)\n    (antidiagonal n)", "state_after": "no goals"}, {"tactic": "rw [List.mem_map, Prod.exists] at hp", "annotated_tactic": ["rw [<a>List.mem_map</a>, <a>Prod.exists</a>] at hp", [{"full_name": "List.mem_map", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [159, 17], "def_end_pos": [159, 24]}, {"full_name": "Prod.exists", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [41, 9], "def_end_pos": [41, 17]}]], "state_before": "case succ.refine'_1\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\np : \u2115 \u00d7 \u2115\nhp : p \u2208 map (Prod.map Nat.succ id) (antidiagonal n)\nx : Fin (Nat.add k 0) \u2192 \u2115\nhx : x \u2208 antidiagonalTuple (Nat.add k 0) (0, n + 1).2\ny : Fin (Nat.add k 0) \u2192 \u2115\nhy : y \u2208 antidiagonalTuple (Nat.add k 0) p.2\n\u22a2 (0, n + 1).1 < p.1 \u2228 (0, n + 1).1 = p.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) x y", "state_after": "case succ.refine'_1\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\np : \u2115 \u00d7 \u2115\nhp : \u2203 a b, (a, b) \u2208 antidiagonal n \u2227 Prod.map Nat.succ id (a, b) = p\nx : Fin (Nat.add k 0) \u2192 \u2115\nhx : x \u2208 antidiagonalTuple (Nat.add k 0) (0, n + 1).2\ny : Fin (Nat.add k 0) \u2192 \u2115\nhy : y \u2208 antidiagonalTuple (Nat.add k 0) p.2\n\u22a2 (0, n + 1).1 < p.1 \u2228 (0, n + 1).1 = p.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) x y"}, {"tactic": "obtain \u27e8a, b, _, rfl : (Nat.succ a, b) = p\u27e9 := hp", "annotated_tactic": ["obtain \u27e8a, b, _, rfl : (<a>Nat.succ</a> a, b) = p\u27e9 := hp", [{"full_name": "Nat.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}]], "state_before": "case succ.refine'_1\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\np : \u2115 \u00d7 \u2115\nhp : \u2203 a b, (a, b) \u2208 antidiagonal n \u2227 Prod.map Nat.succ id (a, b) = p\nx : Fin (Nat.add k 0) \u2192 \u2115\nhx : x \u2208 antidiagonalTuple (Nat.add k 0) (0, n + 1).2\ny : Fin (Nat.add k 0) \u2192 \u2115\nhy : y \u2208 antidiagonalTuple (Nat.add k 0) p.2\n\u22a2 (0, n + 1).1 < p.1 \u2228 (0, n + 1).1 = p.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) x y", "state_after": "case succ.refine'_1.intro.intro.intro\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\nx : Fin (Nat.add k 0) \u2192 \u2115\nhx : x \u2208 antidiagonalTuple (Nat.add k 0) (0, n + 1).2\ny : Fin (Nat.add k 0) \u2192 \u2115\na b : \u2115\nleft\u271d : (a, b) \u2208 antidiagonal n\nhy : y \u2208 antidiagonalTuple (Nat.add k 0) (Nat.succ a, b).2\n\u22a2 (0, n + 1).1 < (Nat.succ a, b).1 \u2228\n    (0, n + 1).1 = (Nat.succ a, b).1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) x y"}, {"tactic": "exact Or.inl (Nat.zero_lt_succ _)", "annotated_tactic": ["exact <a>Or.inl</a> (<a>Nat.zero_lt_succ</a> _)", [{"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "Nat.zero_lt_succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1586, 9], "def_end_pos": [1586, 25]}]], "state_before": "case succ.refine'_1.intro.intro.intro\nk n : \u2115\nn_ih :\n  Pairwise\n    (fun a\u2081 a\u2082 =>\n      \u2200 (a : Fin (Nat.add k 0) \u2192 \u2115),\n        a \u2208 antidiagonalTuple (Nat.add k 0) a\u2081.2 \u2192\n          \u2200 (a_2 : Fin (Nat.add k 0) \u2192 \u2115),\n            a_2 \u2208 antidiagonalTuple (Nat.add k 0) a\u2082.2 \u2192\n              a\u2081.1 < a\u2082.1 \u2228 a\u2081.1 = a\u2082.1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) a a_2)\n    (antidiagonal n)\nx : Fin (Nat.add k 0) \u2192 \u2115\nhx : x \u2208 antidiagonalTuple (Nat.add k 0) (0, n + 1).2\ny : Fin (Nat.add k 0) \u2192 \u2115\na b : \u2115\nleft\u271d : (a, b) \u2208 antidiagonal n\nhy : y \u2208 antidiagonalTuple (Nat.add k 0) (Nat.succ a, b).2\n\u22a2 (0, n + 1).1 < (Nat.succ a, b).1 \u2228\n    (0, n + 1).1 = (Nat.succ a, b).1 \u2227 Pi.Lex (fun x x_1 => x < x_1) (fun i x x_1 => x < x_1) x y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finsupp/Basic.lean", "full_name": "Finsupp.not_mem_graph_snd_zero", "start": [94, 1], "end": [95, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.measure_diff'", "start": [247, 1], "end": [249, 97], "traced_tactics": [{"tactic": "rw [add_comm, measure_add_diff hm, union_comm]", "annotated_tactic": ["rw [<a>add_comm</a>, <a>measure_add_diff</a> hm, <a>union_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "MeasureTheory.measure_add_diff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [243, 9], "def_end_pos": [243, 25]}, {"full_name": "Set.union_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [786, 9], "def_end_pos": [786, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t s : Set \u03b1\nhm : MeasurableSet t\nh_fin : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (s \\ t) + \u2191\u2191\u03bc t = \u2191\u2191\u03bc (s \u222a t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Contracting.lean", "full_name": "ContractingWith.isFixedPt_fixedPoint_iterate", "start": [351, 1], "end": [362, 100], "traced_tactics": [{"tactic": "set x := hf.fixedPoint f^[n]", "annotated_tactic": ["set x := hf.fixedPoint f^[n]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b1\nhf\u271d : ContractingWith K f\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : CompleteSpace \u03b1\nn : \u2115\nhf : ContractingWith K f^[n]\n\u22a2 IsFixedPt f (fixedPoint f^[n] hf)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b1\nhf\u271d : ContractingWith K f\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : CompleteSpace \u03b1\nn : \u2115\nhf : ContractingWith K f^[n]\nx : \u03b1 := fixedPoint f^[n] hf\n\u22a2 IsFixedPt f x"}, {"tactic": "have hx : f^[n] x = x := hf.fixedPoint_isFixedPt", "annotated_tactic": ["have hx : f^[n] x = x := hf.fixedPoint_isFixedPt", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b1\nhf\u271d : ContractingWith K f\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : CompleteSpace \u03b1\nn : \u2115\nhf : ContractingWith K f^[n]\nx : \u03b1 := fixedPoint f^[n] hf\n\u22a2 IsFixedPt f x", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b1\nhf\u271d : ContractingWith K f\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : CompleteSpace \u03b1\nn : \u2115\nhf : ContractingWith K f^[n]\nx : \u03b1 := fixedPoint f^[n] hf\nhx : f^[n] x = x\n\u22a2 IsFixedPt f x"}, {"tactic": "have := hf.toLipschitzWith.dist_le_mul x (f x)", "annotated_tactic": ["have := hf.toLipschitzWith.dist_le_mul x (f x)", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b1\nhf\u271d : ContractingWith K f\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : CompleteSpace \u03b1\nn : \u2115\nhf : ContractingWith K f^[n]\nx : \u03b1 := fixedPoint f^[n] hf\nhx : f^[n] x = x\n\u22a2 IsFixedPt f x", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b1\nhf\u271d : ContractingWith K f\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : CompleteSpace \u03b1\nn : \u2115\nhf : ContractingWith K f^[n]\nx : \u03b1 := fixedPoint f^[n] hf\nhx : f^[n] x = x\nthis : dist (f^[n] x) (f^[n] (f x)) \u2264 \u2191K * dist x (f x)\n\u22a2 IsFixedPt f x"}, {"tactic": "rw [\u2190 iterate_succ_apply, iterate_succ_apply', hx] at this", "annotated_tactic": ["rw [\u2190 <a>iterate_succ_apply</a>, <a>iterate_succ_apply'</a>, hx] at this", [{"full_name": "Function.iterate_succ_apply", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [66, 9], "def_end_pos": [66, 27]}, {"full_name": "Function.iterate_succ_apply'", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [190, 9], "def_end_pos": [190, 28]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b1\nhf\u271d : ContractingWith K f\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : CompleteSpace \u03b1\nn : \u2115\nhf : ContractingWith K f^[n]\nx : \u03b1 := fixedPoint f^[n] hf\nhx : f^[n] x = x\nthis : dist (f^[n] x) (f^[n] (f x)) \u2264 \u2191K * dist x (f x)\n\u22a2 IsFixedPt f x", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b1\nhf\u271d : ContractingWith K f\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : CompleteSpace \u03b1\nn : \u2115\nhf : ContractingWith K f^[n]\nx : \u03b1 := fixedPoint f^[n] hf\nhx : f^[n] x = x\nthis : dist x (f x) \u2264 \u2191K * dist x (f x)\n\u22a2 IsFixedPt f x"}, {"tactic": "revert this", "annotated_tactic": ["revert this", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b1\nhf\u271d : ContractingWith K f\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : CompleteSpace \u03b1\nn : \u2115\nhf : ContractingWith K f^[n]\nx : \u03b1 := fixedPoint f^[n] hf\nhx : f^[n] x = x\nthis : dist x (f x) \u2264 \u2191K * dist x (f x)\n\u22a2 IsFixedPt f x", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b1\nhf\u271d : ContractingWith K f\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : CompleteSpace \u03b1\nn : \u2115\nhf : ContractingWith K f^[n]\nx : \u03b1 := fixedPoint f^[n] hf\nhx : f^[n] x = x\n\u22a2 dist x (f x) \u2264 \u2191K * dist x (f x) \u2192 IsFixedPt f x"}, {"tactic": "contrapose!", "annotated_tactic": ["contrapose!", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b1\nhf\u271d : ContractingWith K f\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : CompleteSpace \u03b1\nn : \u2115\nhf : ContractingWith K f^[n]\nx : \u03b1 := fixedPoint f^[n] hf\nhx : f^[n] x = x\n\u22a2 dist x (f x) \u2264 \u2191K * dist x (f x) \u2192 IsFixedPt f x", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b1\nhf\u271d : ContractingWith K f\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : CompleteSpace \u03b1\nn : \u2115\nhf : ContractingWith K f^[n]\nx : \u03b1 := fixedPoint f^[n] hf\nhx : f^[n] x = x\n\u22a2 \u00acIsFixedPt f x \u2192 \u2191K * dist x (f x) < dist x (f x)"}, {"tactic": "intro this", "annotated_tactic": ["intro this", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b1\nhf\u271d : ContractingWith K f\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : CompleteSpace \u03b1\nn : \u2115\nhf : ContractingWith K f^[n]\nx : \u03b1 := fixedPoint f^[n] hf\nhx : f^[n] x = x\n\u22a2 \u00acIsFixedPt f x \u2192 \u2191K * dist x (f x) < dist x (f x)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b1\nhf\u271d : ContractingWith K f\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : CompleteSpace \u03b1\nn : \u2115\nhf : ContractingWith K f^[n]\nx : \u03b1 := fixedPoint f^[n] hf\nhx : f^[n] x = x\nthis : \u00acIsFixedPt f x\n\u22a2 \u2191K * dist x (f x) < dist x (f x)"}, {"tactic": "have := dist_pos.2 (Ne.symm this)", "annotated_tactic": ["have := <a>dist_pos</a>.2 (<a>Ne.symm</a> this)", [{"full_name": "dist_pos", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2940, 9], "def_end_pos": [2940, 17]}, {"full_name": "Ne.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [575, 9], "def_end_pos": [575, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b1\nhf\u271d : ContractingWith K f\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : CompleteSpace \u03b1\nn : \u2115\nhf : ContractingWith K f^[n]\nx : \u03b1 := fixedPoint f^[n] hf\nhx : f^[n] x = x\nthis : \u00acIsFixedPt f x\n\u22a2 \u2191K * dist x (f x) < dist x (f x)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b1\nhf\u271d : ContractingWith K f\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : CompleteSpace \u03b1\nn : \u2115\nhf : ContractingWith K f^[n]\nx : \u03b1 := fixedPoint f^[n] hf\nhx : f^[n] x = x\nthis\u271d : \u00acIsFixedPt f x\nthis : 0 < dist x (f x)\n\u22a2 \u2191K * dist x (f x) < dist x (f x)"}, {"tactic": "simpa only [NNReal.coe_one, one_mul, NNReal.val_eq_coe] using (mul_lt_mul_right this).mpr hf.left", "annotated_tactic": ["simpa only [<a>NNReal.coe_one</a>, <a>one_mul</a>, <a>NNReal.val_eq_coe</a>] using (<a>mul_lt_mul_right</a> this).<a>mpr</a> hf.left", [{"full_name": "NNReal.coe_one", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [176, 19], "def_end_pos": [176, 26]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "NNReal.val_eq_coe", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [87, 9], "def_end_pos": [87, 19]}, {"full_name": "mul_lt_mul_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [203, 9], "def_end_pos": [203, 25]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nK : \u211d\u22650\nf : \u03b1 \u2192 \u03b1\nhf\u271d : ContractingWith K f\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : CompleteSpace \u03b1\nn : \u2115\nhf : ContractingWith K f^[n]\nx : \u03b1 := fixedPoint f^[n] hf\nhx : f^[n] x = x\nthis\u271d : \u00acIsFixedPt f x\nthis : 0 < dist x (f x)\n\u22a2 \u2191K * dist x (f x) < dist x (f x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "full_name": "normUnit_eq_one", "start": [970, 1], "end": [971, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Multiplicity.lean", "full_name": "multiplicity.finite_pow", "start": [518, 1], "end": [520, 84], "traced_tactics": [{"tactic": "simp [mt isUnit_iff_dvd_one.2 hp.2.1]", "annotated_tactic": ["simp [<a>mt</a> <a>isUnit_iff_dvd_one</a>.2 hp.2.1]", [{"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}, {"full_name": "isUnit_iff_dvd_one", "def_path": "Mathlib/Algebra/Divisibility/Units.lean", "def_pos": [123, 9], "def_end_pos": [123, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\np a : \u03b1\nhp : Prime p\nx\u271d : Finite p a\n\u22a2 \u00acp ^ (0 + 1) \u2223 a ^ 0", "state_after": "no goals"}, {"tactic": "rw [_root_.pow_succ]", "annotated_tactic": ["rw [<a>_root_.pow_succ</a>]", [{"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [645, 9], "def_end_pos": [645, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\np a : \u03b1\nhp : Prime p\nk : \u2115\nha : Finite p a\n\u22a2 Finite p (a ^ (k + 1))", "state_after": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\np a : \u03b1\nhp : Prime p\nk : \u2115\nha : Finite p a\n\u22a2 Finite p (a * a ^ k)"}, {"tactic": "exact finite_mul hp ha (finite_pow hp ha)", "annotated_tactic": ["exact <a>finite_mul</a> hp ha (finite_pow hp ha)", [{"full_name": "multiplicity.finite_mul", "def_path": "Mathlib/RingTheory/Multiplicity.lean", "def_pos": [509, 9], "def_end_pos": [509, 19]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\np a : \u03b1\nhp : Prime p\nk : \u2115\nha : Finite p a\n\u22a2 Finite p (a * a ^ k)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Init/CCLemmas.lean", "full_name": "not_imp_eq_of_eq_false_right", "start": [65, 1], "end": [67, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Algebra/Hom.lean", "full_name": "AlgHom.one_apply", "start": [439, 1], "end": [440, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "iInf_iInf_eq_right", "start": [1251, 1], "end": [1252, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean", "full_name": "NonUnitalSubring.toSubring_toNonUnitalSubring", "start": [252, 1], "end": [253, 80], "traced_tactics": [{"tactic": "cases S", "annotated_tactic": ["cases S", []], "state_before": "R : Type u_1\ninst\u271d : Ring R\nS : NonUnitalSubring R\nh1 : 1 \u2208 S\n\u22a2 Subring.toNonUnitalSubring (toSubring S h1) = S", "state_after": "case mk\nR : Type u_1\ninst\u271d : Ring R\ntoNonUnitalSubsemiring\u271d : NonUnitalSubsemiring R\nneg_mem'\u271d : \u2200 {x : R}, x \u2208 toNonUnitalSubsemiring\u271d.carrier \u2192 -x \u2208 toNonUnitalSubsemiring\u271d.carrier\nh1 : 1 \u2208 { toNonUnitalSubsemiring := toNonUnitalSubsemiring\u271d, neg_mem' := neg_mem'\u271d }\n\u22a2 Subring.toNonUnitalSubring\n      (toSubring { toNonUnitalSubsemiring := toNonUnitalSubsemiring\u271d, neg_mem' := neg_mem'\u271d } h1) =\n    { toNonUnitalSubsemiring := toNonUnitalSubsemiring\u271d, neg_mem' := neg_mem'\u271d }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case mk\nR : Type u_1\ninst\u271d : Ring R\ntoNonUnitalSubsemiring\u271d : NonUnitalSubsemiring R\nneg_mem'\u271d : \u2200 {x : R}, x \u2208 toNonUnitalSubsemiring\u271d.carrier \u2192 -x \u2208 toNonUnitalSubsemiring\u271d.carrier\nh1 : 1 \u2208 { toNonUnitalSubsemiring := toNonUnitalSubsemiring\u271d, neg_mem' := neg_mem'\u271d }\n\u22a2 Subring.toNonUnitalSubring\n      (toSubring { toNonUnitalSubsemiring := toNonUnitalSubsemiring\u271d, neg_mem' := neg_mem'\u271d } h1) =\n    { toNonUnitalSubsemiring := toNonUnitalSubsemiring\u271d, neg_mem' := neg_mem'\u271d }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sym.lean", "full_name": "Finset.Nonempty.sym", "start": [171, 11], "end": [173, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Category/GroupCat/Basic.lean", "full_name": "AddCommGroupCat.asHom_apply", "start": [353, 1], "end": [354, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/BigOperators/Basic.lean", "full_name": "nat_abs_sum_le", "start": [2281, 1], "end": [2287, 65], "traced_tactics": [{"tactic": "induction' s using Finset.induction_on with i s his IH", "annotated_tactic": ["induction' s using <a>Finset.induction_on</a> with i s his IH", [{"full_name": "Finset.induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1251, 19], "def_end_pos": [1251, 31]}]], "state_before": "\u03b9\u271d : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\n\u03b9 : Type u_2\ns : Finset \u03b9\nf : \u03b9 \u2192 \u2124\n\u22a2 Int.natAbs (\u2211 i in s, f i) \u2264 \u2211 i in s, Int.natAbs (f i)", "state_after": "case empty\n\u03b9\u271d : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\n\u03b9 : Type u_2\nf : \u03b9 \u2192 \u2124\n\u22a2 Int.natAbs (\u2211 i in \u2205, f i) \u2264 \u2211 i in \u2205, Int.natAbs (f i)\n\ncase insert\n\u03b9\u271d : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\n\u03b9 : Type u_2\nf : \u03b9 \u2192 \u2124\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nIH : Int.natAbs (\u2211 i in s, f i) \u2264 \u2211 i in s, Int.natAbs (f i)\n\u22a2 Int.natAbs (\u2211 i in insert i s, f i) \u2264 \u2211 i in insert i s, Int.natAbs (f i)"}, {"tactic": "simp only [Finset.sum_empty, Int.natAbs_zero]", "annotated_tactic": ["simp only [<a>Finset.sum_empty</a>, <a>Int.natAbs_zero</a>]", [{"full_name": "Finset.sum_empty", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [298, 3], "def_end_pos": [298, 14]}, {"full_name": "Int.natAbs_zero", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [140, 17], "def_end_pos": [140, 28]}]], "state_before": "case empty\n\u03b9\u271d : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\n\u03b9 : Type u_2\nf : \u03b9 \u2192 \u2124\n\u22a2 Int.natAbs (\u2211 i in \u2205, f i) \u2264 \u2211 i in \u2205, Int.natAbs (f i)", "state_after": "no goals"}, {"tactic": "simp only [his, Finset.sum_insert, not_false_iff]", "annotated_tactic": ["simp only [his, <a>Finset.sum_insert</a>, <a>not_false_iff</a>]", [{"full_name": "Finset.sum_insert", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [316, 3], "def_end_pos": [316, 14]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "case insert\n\u03b9\u271d : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\n\u03b9 : Type u_2\nf : \u03b9 \u2192 \u2124\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nIH : Int.natAbs (\u2211 i in s, f i) \u2264 \u2211 i in s, Int.natAbs (f i)\n\u22a2 Int.natAbs (\u2211 i in insert i s, f i) \u2264 \u2211 i in insert i s, Int.natAbs (f i)", "state_after": "case insert\n\u03b9\u271d : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\n\u03b9 : Type u_2\nf : \u03b9 \u2192 \u2124\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nIH : Int.natAbs (\u2211 i in s, f i) \u2264 \u2211 i in s, Int.natAbs (f i)\n\u22a2 Int.natAbs (f i + \u2211 i in s, f i) \u2264 Int.natAbs (f i) + \u2211 i in s, Int.natAbs (f i)"}, {"tactic": "exact (Int.natAbs_add_le _ _).trans (add_le_add le_rfl IH)", "annotated_tactic": ["exact (<a>Int.natAbs_add_le</a> _ _).<a>trans</a> (<a>add_le_add</a> <a>le_rfl</a> IH)", [{"full_name": "Int.natAbs_add_le", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [1337, 9], "def_end_pos": [1337, 22]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case insert\n\u03b9\u271d : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\n\u03b9 : Type u_2\nf : \u03b9 \u2192 \u2124\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nIH : Int.natAbs (\u2211 i in s, f i) \u2264 \u2211 i in s, Int.natAbs (f i)\n\u22a2 Int.natAbs (f i + \u2211 i in s, f i) \u2264 Int.natAbs (f i) + \u2211 i in s, Int.natAbs (f i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/RingHomProperties.lean", "full_name": "RingHom.StableUnderBaseChange.pushout_inl", "start": [170, 1], "end": [182, 10], "traced_tactics": [{"tactic": "rw [\u2190\n  show _ = pushout.inl from\n    colimit.isoColimitCocone_\u03b9_inv \u27e8_, CommRingCat.pushoutCoconeIsColimit f g\u27e9\n      WalkingSpan.left,\n  hP'.cancel_right_isIso]", "annotated_tactic": ["rw [\u2190\n    show _ = <a>pushout.inl</a> from\n      <a>colimit.isoColimitCocone_\u03b9_inv</a> \u27e8_, <a>CommRingCat.pushoutCoconeIsColimit</a> f g\u27e9\n        <a>WalkingSpan.left</a>,\n    hP'.cancel_right_isIso]", [{"full_name": "CategoryTheory.Limits.pushout.inl", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [1123, 8], "def_end_pos": [1123, 19]}, {"full_name": "CategoryTheory.Limits.colimit.isoColimitCocone_\u03b9_inv", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [836, 9], "def_end_pos": [836, 39]}, {"full_name": "CommRingCat.pushoutCoconeIsColimit", "def_path": "Mathlib/Algebra/Category/Ring/Constructions.lean", "def_pos": [85, 5], "def_end_pos": [85, 27]}, {"full_name": "CategoryTheory.Limits.WalkingSpan.left", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [69, 8], "def_end_pos": [69, 24]}]], "state_before": "P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : R \u27f6 T\nH : P g\n\u22a2 P pushout.inl", "state_after": "P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : R \u27f6 T\nH : P g\n\u22a2 P\n    ({ cocone := CommRingCat.pushoutCocone f g, isColimit := CommRingCat.pushoutCoconeIsColimit f g }.cocone.\u03b9.app\n      WalkingSpan.left)"}, {"tactic": "letI := f.toAlgebra", "annotated_tactic": ["letI := f.toAlgebra", []], "state_before": "P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : R \u27f6 T\nH : P g\n\u22a2 P\n    ({ cocone := CommRingCat.pushoutCocone f g, isColimit := CommRingCat.pushoutCoconeIsColimit f g }.cocone.\u03b9.app\n      WalkingSpan.left)", "state_after": "P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : R \u27f6 T\nH : P g\nthis : Algebra \u2191R \u2191S := toAlgebra f\n\u22a2 P\n    ({ cocone := CommRingCat.pushoutCocone f g, isColimit := CommRingCat.pushoutCoconeIsColimit f g }.cocone.\u03b9.app\n      WalkingSpan.left)"}, {"tactic": "letI := g.toAlgebra", "annotated_tactic": ["letI := g.toAlgebra", []], "state_before": "P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : R \u27f6 T\nH : P g\nthis : Algebra \u2191R \u2191S := toAlgebra f\n\u22a2 P\n    ({ cocone := CommRingCat.pushoutCocone f g, isColimit := CommRingCat.pushoutCoconeIsColimit f g }.cocone.\u03b9.app\n      WalkingSpan.left)", "state_after": "P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : R \u27f6 T\nH : P g\nthis\u271d : Algebra \u2191R \u2191S := toAlgebra f\nthis : Algebra \u2191R \u2191T := toAlgebra g\n\u22a2 P\n    ({ cocone := CommRingCat.pushoutCocone f g, isColimit := CommRingCat.pushoutCoconeIsColimit f g }.cocone.\u03b9.app\n      WalkingSpan.left)"}, {"tactic": "dsimp only [CommRingCat.pushoutCocone_inl, PushoutCocone.\u03b9_app_left]", "annotated_tactic": ["dsimp only [<a>CommRingCat.pushoutCocone_inl</a>, <a>PushoutCocone.\u03b9_app_left</a>]", [{"full_name": "CommRingCat.pushoutCocone_inl", "def_path": "Mathlib/Algebra/Category/Ring/Constructions.lean", "def_pos": [55, 9], "def_end_pos": [55, 26]}, {"full_name": "CategoryTheory.Limits.PushoutCocone.\u03b9_app_left", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [784, 9], "def_end_pos": [784, 19]}]], "state_before": "P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : R \u27f6 T\nH : P g\nthis\u271d : Algebra \u2191R \u2191S := toAlgebra f\nthis : Algebra \u2191R \u2191T := toAlgebra g\n\u22a2 P\n    ({ cocone := CommRingCat.pushoutCocone f g, isColimit := CommRingCat.pushoutCoconeIsColimit f g }.cocone.\u03b9.app\n      WalkingSpan.left)", "state_after": "P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : R \u27f6 T\nH : P g\nthis\u271d : Algebra \u2191R \u2191S := toAlgebra f\nthis : Algebra \u2191R \u2191T := toAlgebra g\n\u22a2 P Algebra.TensorProduct.includeLeftRingHom"}, {"tactic": "apply hP R T S (TensorProduct R S T)", "annotated_tactic": ["apply hP R T S (<a>TensorProduct</a> R S T)", [{"full_name": "TensorProduct", "def_path": "Mathlib/LinearAlgebra/TensorProduct.lean", "def_pos": [76, 5], "def_end_pos": [76, 18]}]], "state_before": "P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : R \u27f6 T\nH : P g\nthis\u271d : Algebra \u2191R \u2191S := toAlgebra f\nthis : Algebra \u2191R \u2191T := toAlgebra g\n\u22a2 P Algebra.TensorProduct.includeLeftRingHom", "state_after": "P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : R \u27f6 T\nH : P g\nthis\u271d : Algebra \u2191R \u2191S := toAlgebra f\nthis : Algebra \u2191R \u2191T := toAlgebra g\n\u22a2 P (algebraMap \u2191R \u2191T)"}, {"tactic": "exact H", "annotated_tactic": ["exact H", []], "state_before": "P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : R \u27f6 T\nH : P g\nthis\u271d : Algebra \u2191R \u2191S := toAlgebra f\nthis : Algebra \u2191R \u2191T := toAlgebra g\n\u22a2 P (algebraMap \u2191R \u2191T)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Bounded.lean", "full_name": "Set.unbounded_lt_inter_le", "start": [372, 1], "end": [375, 20], "traced_tactics": [{"tactic": "convert @unbounded_lt_inter_not_lt _ s _ a", "annotated_tactic": ["convert @<a>unbounded_lt_inter_not_lt</a> _ s _ a", [{"full_name": "Set.unbounded_lt_inter_not_lt", "def_path": "Mathlib/Order/Bounded.lean", "def_pos": [360, 9], "def_end_pos": [360, 34]}]], "state_before": "\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\n\u22a2 Unbounded (fun x x_1 => x < x_1) (s \u2229 {b | a \u2264 b}) \u2194 Unbounded (fun x x_1 => x < x_1) s", "state_after": "case h.e'_1.h.e'_3.h.e'_4.h.e'_2.h.a\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\na x\u271d : \u03b1\n\u22a2 a \u2264 x\u271d \u2194 \u00acx\u271d < a"}, {"tactic": "exact not_lt.symm", "annotated_tactic": ["exact not_lt.symm", []], "state_before": "case h.e'_1.h.e'_3.h.e'_4.h.e'_2.h.a\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\na x\u271d : \u03b1\n\u22a2 a \u2264 x\u271d \u2194 \u00acx\u271d < a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Orientation.lean", "full_name": "Orientation.map_eq_neg_iff_det_neg", "start": [423, 1], "end": [435, 93], "traced_tactics": [{"tactic": "cases isEmpty_or_nonempty \u03b9", "annotated_tactic": ["cases <a>isEmpty_or_nonempty</a> \u03b9", [{"full_name": "isEmpty_or_nonempty", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [207, 9], "def_end_pos": [207, 28]}]], "state_before": "R : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx : Orientation R M \u03b9\nf : M \u2243\u2097[R] M\nh : Fintype.card \u03b9 = finrank R M\n\u22a2 \u2191(map \u03b9 f) x = -x \u2194 \u2191LinearMap.det \u2191f < 0", "state_after": "case inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx : Orientation R M \u03b9\nf : M \u2243\u2097[R] M\nh : Fintype.card \u03b9 = finrank R M\nh\u271d : IsEmpty \u03b9\n\u22a2 \u2191(map \u03b9 f) x = -x \u2194 \u2191LinearMap.det \u2191f < 0\n\ncase inr\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx : Orientation R M \u03b9\nf : M \u2243\u2097[R] M\nh : Fintype.card \u03b9 = finrank R M\nh\u271d : Nonempty \u03b9\n\u22a2 \u2191(map \u03b9 f) x = -x \u2194 \u2191LinearMap.det \u2191f < 0"}, {"tactic": "have H : 0 < finrank R M := by\n  rw [\u2190 h]\n  exact Fintype.card_pos", "annotated_tactic": ["have H : 0 < <a>finrank</a> R M := by\n    rw [\u2190 h]\n    exact <a>Fintype.card_pos</a>", [{"full_name": "FiniteDimensional.finrank", "def_path": "Mathlib/LinearAlgebra/Finrank.lean", "def_pos": [58, 19], "def_end_pos": [58, 26]}, {"full_name": "Fintype.card_pos", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [558, 9], "def_end_pos": [558, 17]}]], "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx : Orientation R M \u03b9\nf : M \u2243\u2097[R] M\nh : Fintype.card \u03b9 = finrank R M\nh\u271d : Nonempty \u03b9\n\u22a2 \u2191(map \u03b9 f) x = -x \u2194 \u2191LinearMap.det \u2191f < 0", "state_after": "case inr\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx : Orientation R M \u03b9\nf : M \u2243\u2097[R] M\nh : Fintype.card \u03b9 = finrank R M\nh\u271d : Nonempty \u03b9\nH : 0 < finrank R M\n\u22a2 \u2191(map \u03b9 f) x = -x \u2194 \u2191LinearMap.det \u2191f < 0"}, {"tactic": "haveI : FiniteDimensional R M := finiteDimensional_of_finrank H", "annotated_tactic": ["haveI : <a>FiniteDimensional</a> R M := <a>finiteDimensional_of_finrank</a> H", [{"full_name": "FiniteDimensional", "def_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "def_pos": [87, 5], "def_end_pos": [87, 22]}, {"full_name": "FiniteDimensional.finiteDimensional_of_finrank", "def_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "def_pos": [201, 9], "def_end_pos": [201, 37]}]], "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx : Orientation R M \u03b9\nf : M \u2243\u2097[R] M\nh : Fintype.card \u03b9 = finrank R M\nh\u271d : Nonempty \u03b9\nH : 0 < finrank R M\n\u22a2 \u2191(map \u03b9 f) x = -x \u2194 \u2191LinearMap.det \u2191f < 0", "state_after": "case inr\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx : Orientation R M \u03b9\nf : M \u2243\u2097[R] M\nh : Fintype.card \u03b9 = finrank R M\nh\u271d : Nonempty \u03b9\nH : 0 < finrank R M\nthis : FiniteDimensional R M\n\u22a2 \u2191(map \u03b9 f) x = -x \u2194 \u2191LinearMap.det \u2191f < 0"}, {"tactic": "rw [map_eq_det_inv_smul _ _ h, units_inv_smul, units_smul_eq_neg_iff, LinearEquiv.coe_det]", "annotated_tactic": ["rw [<a>map_eq_det_inv_smul</a> _ _ h, <a>units_inv_smul</a>, <a>units_smul_eq_neg_iff</a>, <a>LinearEquiv.coe_det</a>]", [{"full_name": "Orientation.map_eq_det_inv_smul", "def_path": "Mathlib/LinearAlgebra/Orientation.lean", "def_pos": [400, 9], "def_end_pos": [400, 28]}, {"full_name": "units_inv_smul", "def_path": "Mathlib/LinearAlgebra/Ray.lean", "def_pos": [518, 9], "def_end_pos": [518, 23]}, {"full_name": "units_smul_eq_neg_iff", "def_path": "Mathlib/LinearAlgebra/Ray.lean", "def_pos": [582, 9], "def_end_pos": [582, 30]}, {"full_name": "LinearEquiv.coe_det", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [376, 9], "def_end_pos": [376, 16]}]], "state_before": "case inr\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx : Orientation R M \u03b9\nf : M \u2243\u2097[R] M\nh : Fintype.card \u03b9 = finrank R M\nh\u271d : Nonempty \u03b9\nH : 0 < finrank R M\nthis : FiniteDimensional R M\n\u22a2 \u2191(map \u03b9 f) x = -x \u2194 \u2191LinearMap.det \u2191f < 0", "state_after": "no goals"}, {"tactic": "have H : finrank R M = 0 := by\n  refine' h.symm.trans _\n  convert @Fintype.card_of_isEmpty \u03b9 _", "annotated_tactic": ["have H : <a>finrank</a> R M = 0 := by\n      refine' h.symm.trans _\n      convert @<a>Fintype.card_of_isEmpty</a> \u03b9 _", [{"full_name": "FiniteDimensional.finrank", "def_path": "Mathlib/LinearAlgebra/Finrank.lean", "def_pos": [58, 19], "def_end_pos": [58, 26]}, {"full_name": "Fintype.card_of_isEmpty", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [228, 9], "def_end_pos": [228, 24]}]], "state_before": "case inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx : Orientation R M \u03b9\nf : M \u2243\u2097[R] M\nh : Fintype.card \u03b9 = finrank R M\nh\u271d : IsEmpty \u03b9\n\u22a2 \u2191(map \u03b9 f) x = -x \u2194 \u2191LinearMap.det \u2191f < 0", "state_after": "case inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx : Orientation R M \u03b9\nf : M \u2243\u2097[R] M\nh : Fintype.card \u03b9 = finrank R M\nh\u271d : IsEmpty \u03b9\nH : finrank R M = 0\n\u22a2 \u2191(map \u03b9 f) x = -x \u2194 \u2191LinearMap.det \u2191f < 0"}, {"tactic": "simp [LinearMap.det_eq_one_of_finrank_eq_zero H, Module.Ray.ne_neg_self x]", "annotated_tactic": ["simp [<a>LinearMap.det_eq_one_of_finrank_eq_zero</a> H, <a>Module.Ray.ne_neg_self</a> x]", [{"full_name": "LinearMap.det_eq_one_of_finrank_eq_zero", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [296, 9], "def_end_pos": [296, 38]}, {"full_name": "Module.Ray.ne_neg_self", "def_path": "Mathlib/LinearAlgebra/Ray.lean", "def_pos": [474, 9], "def_end_pos": [474, 20]}]], "state_before": "case inl\nR : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx : Orientation R M \u03b9\nf : M \u2243\u2097[R] M\nh : Fintype.card \u03b9 = finrank R M\nh\u271d : IsEmpty \u03b9\nH : finrank R M = 0\n\u22a2 \u2191(map \u03b9 f) x = -x \u2194 \u2191LinearMap.det \u2191f < 0", "state_after": "no goals"}, {"tactic": "refine' h.symm.trans _", "annotated_tactic": ["refine' h.symm.trans _", []], "state_before": "R : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx : Orientation R M \u03b9\nf : M \u2243\u2097[R] M\nh : Fintype.card \u03b9 = finrank R M\nh\u271d : IsEmpty \u03b9\n\u22a2 finrank R M = 0", "state_after": "R : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx : Orientation R M \u03b9\nf : M \u2243\u2097[R] M\nh : Fintype.card \u03b9 = finrank R M\nh\u271d : IsEmpty \u03b9\n\u22a2 Fintype.card \u03b9 = 0"}, {"tactic": "convert @Fintype.card_of_isEmpty \u03b9 _", "annotated_tactic": ["convert @<a>Fintype.card_of_isEmpty</a> \u03b9 _", [{"full_name": "Fintype.card_of_isEmpty", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [228, 9], "def_end_pos": [228, 24]}]], "state_before": "R : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx : Orientation R M \u03b9\nf : M \u2243\u2097[R] M\nh : Fintype.card \u03b9 = finrank R M\nh\u271d : IsEmpty \u03b9\n\u22a2 Fintype.card \u03b9 = 0", "state_after": "no goals"}, {"tactic": "rw [\u2190 h]", "annotated_tactic": ["rw [\u2190 h]", []], "state_before": "R : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx : Orientation R M \u03b9\nf : M \u2243\u2097[R] M\nh : Fintype.card \u03b9 = finrank R M\nh\u271d : Nonempty \u03b9\n\u22a2 0 < finrank R M", "state_after": "R : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx : Orientation R M \u03b9\nf : M \u2243\u2097[R] M\nh : Fintype.card \u03b9 = finrank R M\nh\u271d : Nonempty \u03b9\n\u22a2 0 < Fintype.card \u03b9"}, {"tactic": "exact Fintype.card_pos", "annotated_tactic": ["exact <a>Fintype.card_pos</a>", [{"full_name": "Fintype.card_pos", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [558, 9], "def_end_pos": [558, 17]}]], "state_before": "R : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx : Orientation R M \u03b9\nf : M \u2243\u2097[R] M\nh : Fintype.card \u03b9 = finrank R M\nh\u271d : Nonempty \u03b9\n\u22a2 0 < Fintype.card \u03b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/WittVector/MulCoeff.lean", "full_name": "WittVector.mul_polyOfInterest_aux3", "start": [150, 1], "end": [181, 7], "traced_tactics": [{"tactic": "have mvpz : (p : \ud835\udd44) ^ (n + 1) = MvPolynomial.C ((p : \u2124) ^ (n + 1)) := by simp only; norm_cast", "annotated_tactic": ["have mvpz : (p : \ud835\udd44) ^ (n + 1) = <a>MvPolynomial.C</a> ((p : \u2124) ^ (n + 1)) := by simp only; norm_cast", [{"full_name": "MvPolynomial.C", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [180, 5], "def_end_pos": [180, 6]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d : CommRing k\nn : \u2115\n\u22a2 wittPolyProd p (n + 1) =\n    -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n          \u2191p ^ (n + 1) * X (0, n + 1) * \u2191(rename (Prod.mk 1)) (wittPolynomial p \u2124 (n + 1)) +\n        \u2191p ^ (n + 1) * X (1, n + 1) * \u2191(rename (Prod.mk 0)) (wittPolynomial p \u2124 (n + 1)) +\n      remainder p n", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d : CommRing k\nn : \u2115\nmvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1))\n\u22a2 wittPolyProd p (n + 1) =\n    -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n          \u2191p ^ (n + 1) * X (0, n + 1) * \u2191(rename (Prod.mk 1)) (wittPolynomial p \u2124 (n + 1)) +\n        \u2191p ^ (n + 1) * X (1, n + 1) * \u2191(rename (Prod.mk 0)) (wittPolynomial p \u2124 (n + 1)) +\n      remainder p n"}, {"tactic": "rw [wittPolyProd, wittPolynomial, AlgHom.map_sum, AlgHom.map_sum]", "annotated_tactic": ["rw [<a>wittPolyProd</a>, <a>wittPolynomial</a>, <a>AlgHom.map_sum</a>, <a>AlgHom.map_sum</a>]", [{"full_name": "WittVector.wittPolyProd", "def_path": "Mathlib/RingTheory/WittVector/MulCoeff.lean", "def_pos": [56, 5], "def_end_pos": [56, 17]}, {"full_name": "wittPolynomial", "def_path": "Mathlib/RingTheory/WittVector/WittPolynomial.lean", "def_pos": [80, 19], "def_end_pos": [80, 33]}, {"full_name": "AlgHom.map_sum", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [269, 19], "def_end_pos": [269, 26]}, {"full_name": "AlgHom.map_sum", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [269, 19], "def_end_pos": [269, 26]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d : CommRing k\nn : \u2115\nmvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1))\n\u22a2 wittPolyProd p (n + 1) =\n    -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n          \u2191p ^ (n + 1) * X (0, n + 1) * \u2191(rename (Prod.mk 1)) (wittPolynomial p \u2124 (n + 1)) +\n        \u2191p ^ (n + 1) * X (1, n + 1) * \u2191(rename (Prod.mk 0)) (wittPolynomial p \u2124 (n + 1)) +\n      remainder p n", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d : CommRing k\nn : \u2115\nmvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1))\n\u22a2 (\u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) *\n      \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) =\n    -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n          \u2191p ^ (n + 1) * X (0, n + 1) *\n            \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (1, n + 1) *\n          \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n      remainder p n"}, {"tactic": "conv_lhs =>\n  arg 1\n  rw [sum_range_succ, \u2190 C_mul_X_pow_eq_monomial, tsub_self, pow_zero, pow_one, map_mul,\n    rename_C, rename_X, \u2190 mvpz]", "annotated_tactic": ["conv_lhs =>\n    arg 1\n    rw [<a>sum_range_succ</a>, \u2190 <a>C_mul_X_pow_eq_monomial</a>, <a>tsub_self</a>, <a>pow_zero</a>, <a>pow_one</a>, <a>map_mul</a>,\n      <a>rename_C</a>, <a>rename_X</a>, \u2190 mvpz]", [{"full_name": "Finset.sum_range_succ", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1219, 3], "def_end_pos": [1219, 14]}, {"full_name": "MvPolynomial.C_mul_X_pow_eq_monomial", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [338, 9], "def_end_pos": [338, 32]}, {"full_name": "tsub_self", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [333, 9], "def_end_pos": [333, 18]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [639, 9], "def_end_pos": [639, 17]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}, {"full_name": "map_mul", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [299, 9], "def_end_pos": [299, 16]}, {"full_name": "MvPolynomial.rename_C", "def_path": "Mathlib/Data/MvPolynomial/Rename.lean", "def_pos": [61, 9], "def_end_pos": [61, 17]}, {"full_name": "MvPolynomial.rename_X", "def_path": "Mathlib/Data/MvPolynomial/Rename.lean", "def_pos": [67, 9], "def_end_pos": [67, 17]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d : CommRing k\nn : \u2115\nmvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1))\n\u22a2 (\u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) *\n      \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) =\n    -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n          \u2191p ^ (n + 1) * X (0, n + 1) *\n            \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (1, n + 1) *\n          \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n      remainder p n", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d : CommRing k\nn : \u2115\nmvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1))\n\u22a2 (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (0, n + 1)) *\n      \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) =\n    -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n          \u2191p ^ (n + 1) * X (0, n + 1) *\n            \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (1, n + 1) *\n          \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n      remainder p n"}, {"tactic": "conv_lhs =>\n  arg 2\n  rw [sum_range_succ, \u2190 C_mul_X_pow_eq_monomial, tsub_self, pow_zero, pow_one, map_mul,\n    rename_C, rename_X, \u2190 mvpz]", "annotated_tactic": ["conv_lhs =>\n    arg 2\n    rw [<a>sum_range_succ</a>, \u2190 <a>C_mul_X_pow_eq_monomial</a>, <a>tsub_self</a>, <a>pow_zero</a>, <a>pow_one</a>, <a>map_mul</a>,\n      <a>rename_C</a>, <a>rename_X</a>, \u2190 mvpz]", [{"full_name": "Finset.sum_range_succ", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1219, 3], "def_end_pos": [1219, 14]}, {"full_name": "MvPolynomial.C_mul_X_pow_eq_monomial", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [338, 9], "def_end_pos": [338, 32]}, {"full_name": "tsub_self", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [333, 9], "def_end_pos": [333, 18]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [639, 9], "def_end_pos": [639, 17]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}, {"full_name": "map_mul", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [299, 9], "def_end_pos": [299, 16]}, {"full_name": "MvPolynomial.rename_C", "def_path": "Mathlib/Data/MvPolynomial/Rename.lean", "def_pos": [61, 9], "def_end_pos": [61, 17]}, {"full_name": "MvPolynomial.rename_X", "def_path": "Mathlib/Data/MvPolynomial/Rename.lean", "def_pos": [67, 9], "def_end_pos": [67, 17]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d : CommRing k\nn : \u2115\nmvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1))\n\u22a2 (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (0, n + 1)) *\n      \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) =\n    -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n          \u2191p ^ (n + 1) * X (0, n + 1) *\n            \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (1, n + 1) *\n          \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n      remainder p n", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d : CommRing k\nn : \u2115\nmvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1))\n\u22a2 (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (0, n + 1)) *\n      (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (1, n + 1)) =\n    -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n          \u2191p ^ (n + 1) * X (0, n + 1) *\n            \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (1, n + 1) *\n          \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n      remainder p n"}, {"tactic": "conv_rhs =>\n  enter [1, 1, 2, 2]\n  rw [sum_range_succ, \u2190 C_mul_X_pow_eq_monomial, tsub_self, pow_zero, pow_one, map_mul,\n    rename_C, rename_X, \u2190 mvpz]", "annotated_tactic": ["conv_rhs =>\n    enter [1, 1, 2, 2]\n    rw [<a>sum_range_succ</a>, \u2190 <a>C_mul_X_pow_eq_monomial</a>, <a>tsub_self</a>, <a>pow_zero</a>, <a>pow_one</a>, <a>map_mul</a>,\n      <a>rename_C</a>, <a>rename_X</a>, \u2190 mvpz]", [{"full_name": "Finset.sum_range_succ", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1219, 3], "def_end_pos": [1219, 14]}, {"full_name": "MvPolynomial.C_mul_X_pow_eq_monomial", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [338, 9], "def_end_pos": [338, 32]}, {"full_name": "tsub_self", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [333, 9], "def_end_pos": [333, 18]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [639, 9], "def_end_pos": [639, 17]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}, {"full_name": "map_mul", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [299, 9], "def_end_pos": [299, 16]}, {"full_name": "MvPolynomial.rename_C", "def_path": "Mathlib/Data/MvPolynomial/Rename.lean", "def_pos": [61, 9], "def_end_pos": [61, 17]}, {"full_name": "MvPolynomial.rename_X", "def_path": "Mathlib/Data/MvPolynomial/Rename.lean", "def_pos": [67, 9], "def_end_pos": [67, 17]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d : CommRing k\nn : \u2115\nmvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1))\n\u22a2 (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (0, n + 1)) *\n      (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (1, n + 1)) =\n    -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n          \u2191p ^ (n + 1) * X (0, n + 1) *\n            \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (1, n + 1) *\n          \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n      remainder p n", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d : CommRing k\nn : \u2115\nmvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1))\n\u22a2 (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (0, n + 1)) *\n      (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (1, n + 1)) =\n    -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n          \u2191p ^ (n + 1) * X (0, n + 1) *\n            (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n              \u2191p ^ (n + 1) * X (1, n + 1)) +\n        \u2191p ^ (n + 1) * X (1, n + 1) *\n          \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n      remainder p n"}, {"tactic": "conv_rhs =>\n  enter [1, 2, 2]\n  rw [sum_range_succ, \u2190 C_mul_X_pow_eq_monomial, tsub_self, pow_zero, pow_one, map_mul,\n    rename_C, rename_X, \u2190 mvpz]", "annotated_tactic": ["conv_rhs =>\n    enter [1, 2, 2]\n    rw [<a>sum_range_succ</a>, \u2190 <a>C_mul_X_pow_eq_monomial</a>, <a>tsub_self</a>, <a>pow_zero</a>, <a>pow_one</a>, <a>map_mul</a>,\n      <a>rename_C</a>, <a>rename_X</a>, \u2190 mvpz]", [{"full_name": "Finset.sum_range_succ", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1219, 3], "def_end_pos": [1219, 14]}, {"full_name": "MvPolynomial.C_mul_X_pow_eq_monomial", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [338, 9], "def_end_pos": [338, 32]}, {"full_name": "tsub_self", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [333, 9], "def_end_pos": [333, 18]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [639, 9], "def_end_pos": [639, 17]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}, {"full_name": "map_mul", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [299, 9], "def_end_pos": [299, 16]}, {"full_name": "MvPolynomial.rename_C", "def_path": "Mathlib/Data/MvPolynomial/Rename.lean", "def_pos": [61, 9], "def_end_pos": [61, 17]}, {"full_name": "MvPolynomial.rename_X", "def_path": "Mathlib/Data/MvPolynomial/Rename.lean", "def_pos": [67, 9], "def_end_pos": [67, 17]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d : CommRing k\nn : \u2115\nmvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1))\n\u22a2 (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (0, n + 1)) *\n      (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (1, n + 1)) =\n    -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n          \u2191p ^ (n + 1) * X (0, n + 1) *\n            (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n              \u2191p ^ (n + 1) * X (1, n + 1)) +\n        \u2191p ^ (n + 1) * X (1, n + 1) *\n          \u2211 x in range (n + 1 + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n      remainder p n", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d : CommRing k\nn : \u2115\nmvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1))\n\u22a2 (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (0, n + 1)) *\n      (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (1, n + 1)) =\n    -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n          \u2191p ^ (n + 1) * X (0, n + 1) *\n            (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n              \u2191p ^ (n + 1) * X (1, n + 1)) +\n        \u2191p ^ (n + 1) * X (1, n + 1) *\n          (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n            \u2191p ^ (n + 1) * X (0, n + 1)) +\n      remainder p n"}, {"tactic": "simp only [add_mul, mul_add]", "annotated_tactic": ["simp only [<a>add_mul</a>, <a>mul_add</a>]", [{"full_name": "add_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [91, 7], "def_end_pos": [91, 14]}, {"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d : CommRing k\nn : \u2115\nmvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1))\n\u22a2 (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (0, n + 1)) *\n      (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (1, n + 1)) =\n    -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n          \u2191p ^ (n + 1) * X (0, n + 1) *\n            (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n              \u2191p ^ (n + 1) * X (1, n + 1)) +\n        \u2191p ^ (n + 1) * X (1, n + 1) *\n          (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n            \u2191p ^ (n + 1) * X (0, n + 1)) +\n      remainder p n", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d : CommRing k\nn : \u2115\nmvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1))\n\u22a2 (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) *\n          \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (0, n + 1) *\n          \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n      ((\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) *\n          (\u2191p ^ (n + 1) * X (1, n + 1)) +\n        \u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1))) =\n    -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n          (\u2191p ^ (n + 1) * X (0, n + 1) *\n              \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n            \u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1))) +\n        (\u2191p ^ (n + 1) * X (1, n + 1) *\n            \u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n          \u2191p ^ (n + 1) * X (1, n + 1) * (\u2191p ^ (n + 1) * X (0, n + 1))) +\n      remainder p n"}, {"tactic": "rw [add_comm _ (remainder p n)]", "annotated_tactic": ["rw [<a>add_comm</a> _ (<a>remainder</a> p n)]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "WittVector.remainder", "def_path": "Mathlib/RingTheory/WittVector/MulCoeff.lean", "def_pos": [97, 5], "def_end_pos": [97, 14]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d : CommRing k\nn : \u2115\nmvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1))\n\u22a2 (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) *\n          \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (0, n + 1) *\n          \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n      ((\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) *\n          (\u2191p ^ (n + 1) * X (1, n + 1)) +\n        \u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1))) =\n    -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n          (\u2191p ^ (n + 1) * X (0, n + 1) *\n              \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n            \u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1))) +\n        (\u2191p ^ (n + 1) * X (1, n + 1) *\n            \u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n          \u2191p ^ (n + 1) * X (1, n + 1) * (\u2191p ^ (n + 1) * X (0, n + 1))) +\n      remainder p n", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d : CommRing k\nn : \u2115\nmvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1))\n\u22a2 (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) *\n          \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (0, n + 1) *\n          \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n      ((\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) *\n          (\u2191p ^ (n + 1) * X (1, n + 1)) +\n        \u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1))) =\n    remainder p n +\n      (-(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n          (\u2191p ^ (n + 1) * X (0, n + 1) *\n              \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n            \u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1))) +\n        (\u2191p ^ (n + 1) * X (1, n + 1) *\n            \u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n          \u2191p ^ (n + 1) * X (1, n + 1) * (\u2191p ^ (n + 1) * X (0, n + 1))))"}, {"tactic": "simp only [add_assoc]", "annotated_tactic": ["simp only [<a>add_assoc</a>]", [{"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d : CommRing k\nn : \u2115\nmvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1))\n\u22a2 (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) *\n          \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        \u2191p ^ (n + 1) * X (0, n + 1) *\n          \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n      ((\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) *\n          (\u2191p ^ (n + 1) * X (1, n + 1)) +\n        \u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1))) =\n    remainder p n +\n      (-(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n          (\u2191p ^ (n + 1) * X (0, n + 1) *\n              \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n            \u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1))) +\n        (\u2191p ^ (n + 1) * X (1, n + 1) *\n            \u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n          \u2191p ^ (n + 1) * X (1, n + 1) * (\u2191p ^ (n + 1) * X (0, n + 1))))", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d : CommRing k\nn : \u2115\nmvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1))\n\u22a2 (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) *\n        \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n      (\u2191p ^ (n + 1) * X (0, n + 1) *\n          \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        ((\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) *\n            (\u2191p ^ (n + 1) * X (1, n + 1)) +\n          \u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1)))) =\n    remainder p n +\n      (-(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n        (\u2191p ^ (n + 1) * X (0, n + 1) *\n            \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n          (\u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n            (\u2191p ^ (n + 1) * X (1, n + 1) *\n                \u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n              \u2191p ^ (n + 1) * X (1, n + 1) * (\u2191p ^ (n + 1) * X (0, n + 1))))))"}, {"tactic": "apply congrArg (Add.add _)", "annotated_tactic": ["apply <a>congrArg</a> (<a>Add.add</a> _)", [{"full_name": "congrArg", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [349, 9], "def_end_pos": [349, 17]}, {"full_name": "Add.add", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1265, 3], "def_end_pos": [1265, 6]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d : CommRing k\nn : \u2115\nmvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1))\n\u22a2 (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) *\n        \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n      (\u2191p ^ (n + 1) * X (0, n + 1) *\n          \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        ((\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) *\n            (\u2191p ^ (n + 1) * X (1, n + 1)) +\n          \u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1)))) =\n    remainder p n +\n      (-(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n        (\u2191p ^ (n + 1) * X (0, n + 1) *\n            \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n          (\u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n            (\u2191p ^ (n + 1) * X (1, n + 1) *\n                \u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n              \u2191p ^ (n + 1) * X (1, n + 1) * (\u2191p ^ (n + 1) * X (0, n + 1))))))", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d : CommRing k\nn : \u2115\nmvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1))\n\u22a2 \u2191p ^ (n + 1) * X (0, n + 1) *\n        \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n      ((\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) *\n          (\u2191p ^ (n + 1) * X (1, n + 1)) +\n        \u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1))) =\n    -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n      (\u2191p ^ (n + 1) * X (0, n + 1) *\n          \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        (\u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n          (\u2191p ^ (n + 1) * X (1, n + 1) *\n              \u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n            \u2191p ^ (n + 1) * X (1, n + 1) * (\u2191p ^ (n + 1) * X (0, n + 1)))))"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d : CommRing k\nn : \u2115\nmvpz : \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1))\n\u22a2 \u2191p ^ (n + 1) * X (0, n + 1) *\n        \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n      ((\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) *\n          (\u2191p ^ (n + 1) * X (1, n + 1)) +\n        \u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1))) =\n    -(\u2191p ^ (n + 1) * X (0, n + 1)) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n      (\u2191p ^ (n + 1) * X (0, n + 1) *\n          \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n        (\u2191p ^ (n + 1) * X (0, n + 1) * (\u2191p ^ (n + 1) * X (1, n + 1)) +\n          (\u2191p ^ (n + 1) * X (1, n + 1) *\n              \u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x)) +\n            \u2191p ^ (n + 1) * X (1, n + 1) * (\u2191p ^ (n + 1) * X (0, n + 1)))))", "state_after": "no goals"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d : CommRing k\nn : \u2115\n\u22a2 \u2191p ^ (n + 1) = \u2191C (\u2191p ^ (n + 1))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/ContinuedFractions/ContinuantsRecurrence.lean", "full_name": "GeneralizedContinuedFraction.continuantsAux_recurrence", "start": [26, 1], "end": [30, 79], "traced_tactics": [{"tactic": "simp [*, continuantsAux, nextContinuants, nextDenominator, nextNumerator]", "annotated_tactic": ["simp [*, <a>continuantsAux</a>, <a>nextContinuants</a>, <a>nextDenominator</a>, <a>nextNumerator</a>]", [{"full_name": "GeneralizedContinuedFraction.continuantsAux", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [354, 5], "def_end_pos": [354, 19]}, {"full_name": "GeneralizedContinuedFraction.nextContinuants", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [349, 5], "def_end_pos": [349, 20]}, {"full_name": "GeneralizedContinuedFraction.nextDenominator", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [341, 5], "def_end_pos": [341, 20]}, {"full_name": "GeneralizedContinuedFraction.nextNumerator", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [334, 5], "def_end_pos": [334, 18]}]], "state_before": "K : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp ppred pred : Pair K\nnth_s_eq : Stream'.Seq.get? g.s n = some gp\nnth_conts_aux_eq : continuantsAux g n = ppred\nsucc_nth_conts_aux_eq : continuantsAux g (n + 1) = pred\n\u22a2 continuantsAux g (n + 2) = { a := gp.b * pred.a + gp.a * ppred.a, b := gp.b * pred.b + gp.a * ppred.b }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/ZFC/Basic.lean", "full_name": "PSet.Equiv.symm", "start": [146, 11], "end": [147, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Separation.lean", "full_name": "SeparatedNhds.preimage", "start": [128, 1], "end": [132, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Bool/Basic.lean", "full_name": "Bool.xor_iff_ne", "start": [302, 1], "end": [302, 73], "traced_tactics": [{"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "\u22a2 \u2200 {x y : Bool}, xor x y = true \u2194 x \u2260 y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/Factorial/DoubleFactorial.lean", "full_name": "Nat.doubleFactorial_add_one", "start": [43, 1], "end": [43, 94], "traced_tactics": [{"tactic": "cases n <;> rfl", "annotated_tactic": ["cases n <;> rfl", []], "state_before": "n : \u2115\n\u22a2 (n + 1)\u203c = (n + 1) * (n - 1)\u203c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Category/Ring/Constructions.lean", "full_name": "CommRingCat.pushoutCocone_inl", "start": [55, 1], "end": [60, 6], "traced_tactics": [{"tactic": "letI := f.toAlgebra", "annotated_tactic": ["letI := f.toAlgebra", []], "state_before": "R A B : CommRingCat\nf : R \u27f6 A\ng : R \u27f6 B\n\u22a2 A \u27f6 (pushoutCocone f g).pt", "state_after": "R A B : CommRingCat\nf : R \u27f6 A\ng : R \u27f6 B\nthis : Algebra \u2191R \u2191A := RingHom.toAlgebra f\n\u22a2 A \u27f6 (pushoutCocone f g).pt"}, {"tactic": "letI := g.toAlgebra", "annotated_tactic": ["letI := g.toAlgebra", []], "state_before": "R A B : CommRingCat\nf : R \u27f6 A\ng : R \u27f6 B\nthis : Algebra \u2191R \u2191A := RingHom.toAlgebra f\n\u22a2 A \u27f6 (pushoutCocone f g).pt", "state_after": "R A B : CommRingCat\nf : R \u27f6 A\ng : R \u27f6 B\nthis\u271d : Algebra \u2191R \u2191A := RingHom.toAlgebra f\nthis : Algebra \u2191R \u2191B := RingHom.toAlgebra g\n\u22a2 A \u27f6 (pushoutCocone f g).pt"}, {"tactic": "exact Algebra.TensorProduct.includeLeftRingHom", "annotated_tactic": ["exact <a>Algebra.TensorProduct.includeLeftRingHom</a>", [{"full_name": "Algebra.TensorProduct.includeLeftRingHom", "def_path": "Mathlib/RingTheory/TensorProduct.lean", "def_pos": [355, 5], "def_end_pos": [355, 23]}]], "state_before": "R A B : CommRingCat\nf : R \u27f6 A\ng : R \u27f6 B\nthis\u271d : Algebra \u2191R \u2191A := RingHom.toAlgebra f\nthis : Algebra \u2191R \u2191B := RingHom.toAlgebra g\n\u22a2 A \u27f6 (pushoutCocone f g).pt", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean", "full_name": "CliffordAlgebraQuaternion.toQuaternion_ofQuaternion", "start": [365, 1], "end": [366, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.get?_set_ne", "start": [919, 1], "end": [920, 56], "traced_tactics": [{"tactic": "simp only [set_eq_modifyNth, get?_modifyNth_ne _ _ h]", "annotated_tactic": ["simp only [<a>set_eq_modifyNth</a>, <a>get?_modifyNth_ne</a> _ _ h]", [{"full_name": "List.set_eq_modifyNth", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [889, 9], "def_end_pos": [889, 25]}, {"full_name": "List.get?_modifyNth_ne", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [877, 17], "def_end_pos": [877, 34]}]], "state_before": "\u03b1 : Type u_1\na : \u03b1\nm n : Nat\nl : List \u03b1\nh : m \u2260 n\n\u22a2 get? (set l m a) n = get? l n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Matrix/BilinearForm.lean", "full_name": "Matrix.toBilin_basisFun", "start": [332, 1], "end": [334, 76], "traced_tactics": [{"tactic": "ext M", "annotated_tactic": ["ext M", []], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : AddCommMonoid M\ninst\u271d\u00b9\u2075 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2074 : Ring R\u2081\ninst\u271d\u00b9\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b2 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b9 : CommSemiring R\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2078 : CommRing R\u2083\ninst\u271d\u2077 : AddCommGroup M\u2083\ninst\u271d\u2076 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq n\nb : Basis n R\u2082 M\u2082\n\u22a2 toBilin (Pi.basisFun R\u2082 n) = toBilin'", "state_after": "case h.H\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2075 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2074 : Ring R\u2081\ninst\u271d\u00b9\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b2 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b9 : CommSemiring R\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2078 : CommRing R\u2083\ninst\u271d\u2077 : AddCommGroup M\u2083\ninst\u271d\u2076 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM : Matrix n n R\u2082\nx\u271d y\u271d : n \u2192 R\u2082\n\u22a2 bilin (\u2191(toBilin (Pi.basisFun R\u2082 n)) M) x\u271d y\u271d = bilin (\u2191toBilin' M) x\u271d y\u271d"}, {"tactic": "simp only [Matrix.toBilin_apply, Matrix.toBilin'_apply, Pi.basisFun_repr]", "annotated_tactic": ["simp only [<a>Matrix.toBilin_apply</a>, <a>Matrix.toBilin'_apply</a>, <a>Pi.basisFun_repr</a>]", [{"full_name": "Matrix.toBilin_apply", "def_path": "Mathlib/LinearAlgebra/Matrix/BilinearForm.lean", "def_pos": [309, 9], "def_end_pos": [309, 29]}, {"full_name": "Matrix.toBilin'_apply", "def_path": "Mathlib/LinearAlgebra/Matrix/BilinearForm.lean", "def_pos": [167, 9], "def_end_pos": [167, 30]}, {"full_name": "Pi.basisFun_repr", "def_path": "Mathlib/LinearAlgebra/StdBasis.lean", "def_pos": [286, 9], "def_end_pos": [286, 22]}]], "state_before": "case h.H\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2075 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2074 : Ring R\u2081\ninst\u271d\u00b9\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b2 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b9 : CommSemiring R\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2078 : CommRing R\u2083\ninst\u271d\u2077 : AddCommGroup M\u2083\ninst\u271d\u2076 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM : Matrix n n R\u2082\nx\u271d y\u271d : n \u2192 R\u2082\n\u22a2 bilin (\u2191(toBilin (Pi.basisFun R\u2082 n)) M) x\u271d y\u271d = bilin (\u2191toBilin' M) x\u271d y\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "full_name": "MeasureTheory.map_withDensity_abs_det_fderiv_eq_addHaar", "start": [1118, 1], "end": [1127, 26], "traced_tactics": [{"tactic": "apply Measure.ext fun t ht => ?_", "annotated_tactic": ["apply <a>Measure.ext</a> fun t ht => ?_", [{"full_name": "MeasureTheory.Measure.ext", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [135, 9], "def_end_pos": [135, 12]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nh'f : Measurable f\n\u22a2 Measure.map f (withDensity (Measure.restrict \u03bc s) fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|) =\n    Measure.restrict \u03bc (f '' s)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nh'f : Measurable f\nt : Set E\nht : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.map f (withDensity (Measure.restrict \u03bc s) fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) t =\n    \u2191\u2191(Measure.restrict \u03bc (f '' s)) t"}, {"tactic": "rw [map_apply h'f ht, withDensity_apply _ (h'f ht), Measure.restrict_apply ht,\n  restrict_restrict (h'f ht),\n  lintegral_abs_det_fderiv_eq_addHaar_image \u03bc ((h'f ht).inter hs)\n    (fun x hx => (hf' x hx.2).mono (inter_subset_right _ _)) (hf.mono (inter_subset_right _ _)),\n  image_preimage_inter]", "annotated_tactic": ["rw [<a>map_apply</a> h'f ht, <a>withDensity_apply</a> _ (h'f ht), <a>Measure.restrict_apply</a> ht,\n    <a>restrict_restrict</a> (h'f ht),\n    <a>lintegral_abs_det_fderiv_eq_addHaar_image</a> \u03bc ((h'f ht).<a>inter</a> hs)\n      (fun x hx => (hf' x hx.2).<a>mono</a> (<a>inter_subset_right</a> _ _)) (hf.mono (<a>inter_subset_right</a> _ _)),\n    <a>image_preimage_inter</a>]", [{"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 18]}, {"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}, {"full_name": "MeasureTheory.Measure.restrict_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1533, 9], "def_end_pos": [1533, 23]}, {"full_name": "MeasureTheory.Measure.restrict_restrict", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1645, 9], "def_end_pos": [1645, 26]}, {"full_name": "MeasureTheory.lintegral_abs_det_fderiv_eq_addHaar_image", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [1104, 9], "def_end_pos": [1104, 50]}, {"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "HasFDerivWithinAt.mono", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [380, 16], "def_end_pos": [380, 38]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 27]}, {"full_name": "Set.image_preimage_inter", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [538, 9], "def_end_pos": [538, 29]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : IsAddHaarMeasure \u03bc\nhs : MeasurableSet s\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhf : InjOn f s\nh'f : Measurable f\nt : Set E\nht : MeasurableSet t\n\u22a2 \u2191\u2191(Measure.map f (withDensity (Measure.restrict \u03bc s) fun x => ENNReal.ofReal |ContinuousLinearMap.det (f' x)|)) t =\n    \u2191\u2191(Measure.restrict \u03bc (f '' s)) t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finsupp/Antidiagonal.lean", "full_name": "Finsupp.prod_antidiagonal_swap", "start": [93, 1], "end": [98, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Types.lean", "full_name": "CategoryTheory.FunctorToTypes.map_id_apply", "start": [148, 1], "end": [148, 79], "traced_tactics": [{"tactic": "simp [types_id]", "annotated_tactic": ["simp [<a>types_id</a>]", [{"full_name": "CategoryTheory.types_id", "def_path": "Mathlib/CategoryTheory/Types.lean", "def_pos": [63, 9], "def_end_pos": [63, 17]}]], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nF G H : C \u2964 Type w\nX Y Z : C\n\u03c3 : F \u27f6 G\n\u03c4 : G \u27f6 H\na : F.obj X\n\u22a2 F.map (\ud835\udfd9 X) a = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Abelian/Exact.lean", "full_name": "CategoryTheory.Abelian.tfae_epi", "start": [272, 1], "end": [283, 14], "traced_tactics": [{"tactic": "tfae_have 3 \u2192 2", "annotated_tactic": ["tfae_have 3 \u2192 2", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 TFAE [Epi f, cokernel.\u03c0 f = 0, Exact f 0]", "state_after": "case tfae_3_to_2\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 Exact f 0 \u2192 cokernel.\u03c0 f = 0\n\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0\n\u22a2 TFAE [Epi f, cokernel.\u03c0 f = 0, Exact f 0]"}, {"tactic": "tfae_have 1 \u2192 3", "annotated_tactic": ["tfae_have 1 \u2192 3", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0\n\u22a2 TFAE [Epi f, cokernel.\u03c0 f = 0, Exact f 0]", "state_after": "case tfae_1_to_3\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0\n\u22a2 Epi f \u2192 Exact f 0\n\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0\ntfae_1_to_3 : Epi f \u2192 Exact f 0\n\u22a2 TFAE [Epi f, cokernel.\u03c0 f = 0, Exact f 0]"}, {"tactic": "tfae_have 2 \u2192 1", "annotated_tactic": ["tfae_have 2 \u2192 1", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0\ntfae_1_to_3 : Epi f \u2192 Exact f 0\n\u22a2 TFAE [Epi f, cokernel.\u03c0 f = 0, Exact f 0]", "state_after": "case tfae_2_to_1\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0\ntfae_1_to_3 : Epi f \u2192 Exact f 0\n\u22a2 cokernel.\u03c0 f = 0 \u2192 Epi f\n\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0\ntfae_1_to_3 : Epi f \u2192 Exact f 0\ntfae_2_to_1 : cokernel.\u03c0 f = 0 \u2192 Epi f\n\u22a2 TFAE [Epi f, cokernel.\u03c0 f = 0, Exact f 0]"}, {"tactic": "tfae_finish", "annotated_tactic": ["tfae_finish", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0\ntfae_1_to_3 : Epi f \u2192 Exact f 0\ntfae_2_to_1 : cokernel.\u03c0 f = 0 \u2192 Epi f\n\u22a2 TFAE [Epi f, cokernel.\u03c0 f = 0, Exact f 0]", "state_after": "no goals"}, {"tactic": "rw [exact_iff]", "annotated_tactic": ["rw [<a>exact_iff</a>]", [{"full_name": "CategoryTheory.Abelian.exact_iff", "def_path": "Mathlib/CategoryTheory/Abelian/Exact.lean", "def_pos": [66, 9], "def_end_pos": [66, 18]}]], "state_before": "case tfae_3_to_2\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 Exact f 0 \u2192 cokernel.\u03c0 f = 0", "state_after": "case tfae_3_to_2\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 f \u226b 0 = 0 \u2227 kernel.\u03b9 0 \u226b cokernel.\u03c0 f = 0 \u2192 cokernel.\u03c0 f = 0"}, {"tactic": "rintro \u27e8-, h\u27e9", "annotated_tactic": ["rintro \u27e8-, h\u27e9", []], "state_before": "case tfae_3_to_2\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 f \u226b 0 = 0 \u2227 kernel.\u03b9 0 \u226b cokernel.\u03c0 f = 0 \u2192 cokernel.\u03c0 f = 0", "state_after": "case tfae_3_to_2.intro\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nh : kernel.\u03b9 0 \u226b cokernel.\u03c0 f = 0\n\u22a2 cokernel.\u03c0 f = 0"}, {"tactic": "exact zero_of_epi_comp _ h", "annotated_tactic": ["exact <a>zero_of_epi_comp</a> _ h", [{"full_name": "CategoryTheory.Limits.zero_of_epi_comp", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean", "def_pos": [146, 9], "def_end_pos": [146, 25]}]], "state_before": "case tfae_3_to_2.intro\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nh : kernel.\u03b9 0 \u226b cokernel.\u03c0 f = 0\n\u22a2 cokernel.\u03c0 f = 0", "state_after": "no goals"}, {"tactic": "rw [exact_iff]", "annotated_tactic": ["rw [<a>exact_iff</a>]", [{"full_name": "CategoryTheory.Abelian.exact_iff", "def_path": "Mathlib/CategoryTheory/Abelian/Exact.lean", "def_pos": [66, 9], "def_end_pos": [66, 18]}]], "state_before": "case tfae_1_to_3\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0\n\u22a2 Epi f \u2192 Exact f 0", "state_after": "case tfae_1_to_3\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0\n\u22a2 Epi f \u2192 f \u226b 0 = 0 \u2227 kernel.\u03b9 0 \u226b cokernel.\u03c0 f = 0"}, {"tactic": "intro", "annotated_tactic": ["intro", []], "state_before": "case tfae_1_to_3\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0\n\u22a2 Epi f \u2192 f \u226b 0 = 0 \u2227 kernel.\u03b9 0 \u226b cokernel.\u03c0 f = 0", "state_after": "case tfae_1_to_3\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0\n\u271d : Epi f\n\u22a2 f \u226b 0 = 0 \u2227 kernel.\u03b9 0 \u226b cokernel.\u03c0 f = 0"}, {"tactic": "exact \u27e8by simp, by simp [cokernel.\u03c0_of_epi]\u27e9", "annotated_tactic": ["exact \u27e8by simp, by simp [<a>cokernel.\u03c0_of_epi</a>]\u27e9", [{"full_name": "CategoryTheory.Limits.cokernel.\u03c0_of_epi", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean", "def_pos": [936, 9], "def_end_pos": [936, 26]}]], "state_before": "case tfae_1_to_3\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0\n\u271d : Epi f\n\u22a2 f \u226b 0 = 0 \u2227 kernel.\u03b9 0 \u226b cokernel.\u03c0 f = 0", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\ntfae_3_to_2 : Exact f 0 \u2192 cokernel.\u03c0 f = 0\n\u271d : Epi f\n\u22a2 f \u226b 0 = 0", "state_after": "no goals"}, {"tactic": "simp [cokernel.\u03c0_of_epi]", "annotated_tactic": ["simp [<a>cokernel.\u03c0_of_epi</a>]", [{"full_name": "CategoryTheory.Limits.cokernel.\u03c0_of_epi", "def_path": 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"def_pos": [1152, 9], "def_end_pos": [1152, 83]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 inner x y = (\u2191\u2016x + y\u2016 ^ 2 - \u2191\u2016x - y\u2016 ^ 2 + (\u2191\u2016x - I \u2022 y\u2016 ^ 2 - \u2191\u2016x + I \u2022 y\u2016 ^ 2) * I) / 4", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191((\u2016x + y\u2016 * \u2016x + y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016) / 4) + \u2191((\u2016x - I \u2022 y\u2016 * \u2016x - I \u2022 y\u2016 - \u2016x + I \u2022 y\u2016 * \u2016x + I \u2022 y\u2016) / 4) * I =\n    (\u2191\u2016x + y\u2016 ^ 2 - \u2191\u2016x - y\u2016 ^ 2 + (\u2191\u2016x - I \u2022 y\u2016 ^ 2 - \u2191\u2016x + I \u2022 y\u2016 ^ 2) * I) / 4"}, {"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191((\u2016x + y\u2016 * \u2016x + y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016) / 4) + \u2191((\u2016x - I \u2022 y\u2016 * \u2016x - I \u2022 y\u2016 - \u2016x + I \u2022 y\u2016 * \u2016x + I \u2022 y\u2016) / 4) * I =\n    (\u2191\u2016x + y\u2016 ^ 2 - \u2191\u2016x - y\u2016 ^ 2 + (\u2191\u2016x - I \u2022 y\u2016 ^ 2 - \u2191\u2016x + I \u2022 y\u2016 ^ 2) * I) / 4", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 (\u2191\u2016x + y\u2016 * \u2191\u2016x + y\u2016 - \u2191\u2016x - y\u2016 * \u2191\u2016x - y\u2016) / 4 +\n      (\u2191\u2016x - I \u2022 y\u2016 * \u2191\u2016x - I \u2022 y\u2016 - \u2191\u2016x + I \u2022 y\u2016 * \u2191\u2016x + I \u2022 y\u2016) / 4 * I =\n    (\u2191\u2016x + y\u2016 ^ 2 - \u2191\u2016x - y\u2016 ^ 2 + (\u2191\u2016x - I \u2022 y\u2016 ^ 2 - \u2191\u2016x + I \u2022 y\u2016 ^ 2) * I) / 4"}, {"tactic": "simp only [sq, \u2190 mul_div_right_comm, \u2190 add_div]", "annotated_tactic": ["simp only [<a>sq</a>, \u2190 <a>mul_div_right_comm</a>, \u2190 <a>add_div</a>]", [{"full_name": "sq", "def_path": 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(\u2191\u2016x - I \u2022 y\u2016 ^ 2 - \u2191\u2016x + I \u2022 y\u2016 ^ 2) * I) / 4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.IndepSets.union_iff", "start": [283, 1], "end": [285, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/ModelTheory/Substructures.lean", "full_name": "FirstOrder.Language.Substructure.map_inf_comap_of_surjective", "start": [627, 1], "end": [629, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean", "full_name": "CategoryTheory.Limits.hasZeroObject_of_hasInitial_object", "start": [547, 1], "end": 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"state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasInitial C\nX : C\n\u22a2 \u2200 (a : \u22a5_ C \u27f6 X), a = default", "state_after": "no goals"}, {"tactic": "congr!", "annotated_tactic": ["congr!", []], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\ninst\u271d\u00b9 : HasZeroMorphisms C\ninst\u271d : HasInitial C\nX : C\nf : X \u27f6 \u22a5_ C\n\u22a2 f \u226b \ud835\udfd9 (\u22a5_ C) = f \u226b 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/FieldTheory/Adjoin.lean", "full_name": "IntermediateField.adjoin_simple_le_iff", "start": [584, 1], "end": [584, 87], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\nS : Set E\n\u03b1 : E\nK : IntermediateField F E\n\u22a2 F\u27ee\u03b1\u27ef \u2264 K \u2194 \u03b1 \u2208 K", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/Coeff.lean", "full_name": "Polynomial.coeff_C_mul_X", "start": [156, 1], "end": [157, 38], "traced_tactics": [{"tactic": "rw [\u2190 pow_one X, coeff_C_mul_X_pow]", "annotated_tactic": ["rw [\u2190 <a>pow_one</a> <a>X</a>, <a>coeff_C_mul_X_pow</a>]", [{"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}, {"full_name": "Polynomial.X", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [566, 5], "def_end_pos": [566, 6]}, {"full_name": "Polynomial.coeff_C_mul_X_pow", "def_path": "Mathlib/Data/Polynomial/Coeff.lean", "def_pos": [149, 9], "def_end_pos": [149, 26]}]], "state_before": "R : Type u\nS : Type v\na b : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nx : R\nn : \u2115\n\u22a2 coeff (\u2191C x * X) n = if n = 1 then x else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Determinant.lean", "full_name": "LinearEquiv.det_symm_mul_det", "start": [421, 1], "end": [423, 33], "traced_tactics": [{"tactic": "simp [\u2190 LinearMap.det_comp]", "annotated_tactic": ["simp [\u2190 <a>LinearMap.det_comp</a>]", [{"full_name": "LinearMap.det_comp", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [247, 9], "def_end_pos": [247, 17]}]], "state_before": "R : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\nf : M \u2243\u2097[A] M\n\u22a2 \u2191LinearMap.det \u2191(symm f) * \u2191LinearMap.det \u2191f = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Distribution/SchwartzSpace.lean", "full_name": "SchwartzMap.isBigO_cocompact_zpow_neg_nat", "start": [145, 1], "end": [153, 58], "traced_tactics": [{"tactic": "obtain \u27e8d, _, hd'\u27e9 := f.decay k 0", "annotated_tactic": ["obtain \u27e8d, _, hd'\u27e9 := f.decay k 0", []], "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\n\u22a2 \u2191f =O[cocompact E] fun x => \u2016x\u2016 ^ (-\u2191k)", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\nd : \u211d\nleft\u271d : 0 < d\nhd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d 0 (\u2191f) x\u2016 \u2264 d\n\u22a2 \u2191f =O[cocompact E] fun x => \u2016x\u2016 ^ (-\u2191k)"}, {"tactic": "simp only [norm_iteratedFDeriv_zero] at hd'", "annotated_tactic": ["simp only [<a>norm_iteratedFDeriv_zero</a>] at hd'", [{"full_name": "norm_iteratedFDeriv_zero", "def_path": "Mathlib/Analysis/Calculus/ContDiffDef.lean", "def_pos": [1535, 9], "def_end_pos": [1535, 33]}]], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\nd : \u211d\nleft\u271d : 0 < d\nhd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d 0 (\u2191f) x\u2016 \u2264 d\n\u22a2 \u2191f =O[cocompact E] fun x => \u2016x\u2016 ^ (-\u2191k)", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\nd : \u211d\nleft\u271d : 0 < d\nhd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d\n\u22a2 \u2191f =O[cocompact E] fun x => \u2016x\u2016 ^ (-\u2191k)"}, {"tactic": "simp_rw [Asymptotics.IsBigO, Asymptotics.IsBigOWith]", "annotated_tactic": ["simp_rw [<a>Asymptotics.IsBigO</a>, <a>Asymptotics.IsBigOWith</a>]", [{"full_name": "Asymptotics.IsBigO", "def_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "def_pos": [104, 17], "def_end_pos": [104, 23]}, {"full_name": "Asymptotics.IsBigOWith", "def_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "def_pos": [88, 17], "def_end_pos": [88, 27]}]], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\nd : \u211d\nleft\u271d : 0 < d\nhd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d\n\u22a2 \u2191f =O[cocompact E] fun x => \u2016x\u2016 ^ (-\u2191k)", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\nd : \u211d\nleft\u271d : 0 < d\nhd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d\n\u22a2 \u2203 c, \u2200\u1da0 (x : E) in cocompact E, \u2016\u2191f x\u2016 \u2264 c * \u2016\u2016x\u2016 ^ (-\u2191k)\u2016"}, {"tactic": "refine' \u27e8d, Filter.Eventually.filter_mono Filter.cocompact_le_cofinite _\u27e9", "annotated_tactic": ["refine' \u27e8d, <a>Filter.Eventually.filter_mono</a> <a>Filter.cocompact_le_cofinite</a> _\u27e9", [{"full_name": "Filter.Eventually.filter_mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1093, 9], "def_end_pos": [1093, 31]}, {"full_name": "Filter.cocompact_le_cofinite", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [535, 9], "def_end_pos": [535, 30]}]], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\nd : \u211d\nleft\u271d : 0 < d\nhd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d\n\u22a2 \u2203 c, \u2200\u1da0 (x : E) in cocompact E, \u2016\u2191f x\u2016 \u2264 c * \u2016\u2016x\u2016 ^ (-\u2191k)\u2016", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\nd : \u211d\nleft\u271d : 0 < d\nhd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d\n\u22a2 \u2200\u1da0 (x : E) in cofinite, \u2016\u2191f x\u2016 \u2264 d * \u2016\u2016x\u2016 ^ (-\u2191k)\u2016"}, {"tactic": "refine' (Filter.eventually_cofinite_ne 0).mono fun x hx => _", "annotated_tactic": ["refine' (<a>Filter.eventually_cofinite_ne</a> 0).<a>mono</a> fun x hx => _", [{"full_name": "Filter.eventually_cofinite_ne", "def_path": "Mathlib/Order/Filter/Cofinite.lean", "def_pos": [90, 9], "def_end_pos": [90, 31]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}]], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\nd : \u211d\nleft\u271d : 0 < d\nhd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d\n\u22a2 \u2200\u1da0 (x : E) in cofinite, \u2016\u2191f x\u2016 \u2264 d * \u2016\u2016x\u2016 ^ (-\u2191k)\u2016", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\nd : \u211d\nleft\u271d : 0 < d\nhd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d\nx : E\nhx : x \u2260 0\n\u22a2 \u2016\u2191f x\u2016 \u2264 d * \u2016\u2016x\u2016 ^ (-\u2191k)\u2016"}, {"tactic": "rw [Real.norm_of_nonneg (zpow_nonneg (norm_nonneg _) _), zpow_neg, \u2190 div_eq_mul_inv, le_div_iff']", "annotated_tactic": ["rw [<a>Real.norm_of_nonneg</a> (<a>zpow_nonneg</a> (<a>norm_nonneg</a> _) _), <a>zpow_neg</a>, \u2190 <a>div_eq_mul_inv</a>, <a>le_div_iff'</a>]", [{"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1768, 9], "def_end_pos": [1768, 23]}, {"full_name": "zpow_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [109, 9], "def_end_pos": [109, 20]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"full_name": "zpow_neg", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}, {"full_name": "le_div_iff'", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [141, 9], "def_end_pos": [141, 20]}]], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\nd : \u211d\nleft\u271d : 0 < d\nhd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d\nx : E\nhx : x \u2260 0\n\u22a2 \u2016\u2191f x\u2016 \u2264 d * \u2016\u2016x\u2016 ^ (-\u2191k)\u2016", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\nd : \u211d\nleft\u271d : 0 < d\nhd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d\nx : E\nhx : x \u2260 0\n\u22a2 \u2016x\u2016 ^ \u2191k * \u2016\u2191f x\u2016 \u2264 d\n\ncase intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\nd : \u211d\nleft\u271d : 0 < d\nhd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d\nx : E\nhx : x \u2260 0\n\u22a2 0 < \u2016x\u2016 ^ \u2191k"}, {"tactic": "exacts [hd' x, zpow_pos_of_pos (norm_pos_iff.mpr hx) _]", "annotated_tactic": ["exacts [hd' x, <a>zpow_pos_of_pos</a> (norm_pos_iff.mpr hx) _]", [{"full_name": "zpow_pos_of_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 24]}]], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\nd : \u211d\nleft\u271d : 0 < d\nhd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d\nx : E\nhx : x \u2260 0\n\u22a2 \u2016x\u2016 ^ \u2191k * \u2016\u2191f x\u2016 \u2264 d\n\ncase intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\nd : \u211d\nleft\u271d : 0 < d\nhd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d\nx : E\nhx : x \u2260 0\n\u22a2 0 < \u2016x\u2016 ^ \u2191k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Multiplicity.lean", "full_name": "multiplicity.is_greatest", "start": [113, 1], "end": [114, 94], "traced_tactics": [{"tactic": "rw [PartENat.lt_coe_iff] at hm", "annotated_tactic": ["rw [<a>PartENat.lt_coe_iff</a>] at hm", [{"full_name": "PartENat.lt_coe_iff", "def_path": "Mathlib/Data/Nat/PartENat.lean", "def_pos": [287, 9], "def_end_pos": [287, 19]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\na b : \u03b1\nm : \u2115\nhm : multiplicity a b < \u2191m\nh : a ^ m \u2223 b\n\u22a2 False", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\na b : \u03b1\nm : \u2115\nhm : \u2203 h, Part.get (multiplicity a b) h < m\nh : a ^ m \u2223 b\n\u22a2 False"}, {"tactic": "exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h)", "annotated_tactic": ["exact <a>Nat.find_spec</a> hm.fst ((<a>pow_dvd_pow</a> _ hm.snd).<a>trans</a> h)", [{"full_name": "Nat.find_spec", "def_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "def_pos": [717, 19], "def_end_pos": [717, 28]}, {"full_name": "pow_dvd_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [442, 9], "def_end_pos": [442, 20]}, {"full_name": "Dvd.dvd.trans", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [75, 7], "def_end_pos": [75, 20]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\na b : \u03b1\nm : \u2115\nhm : \u2203 h, Part.get (multiplicity a b) h < m\nh : a ^ m \u2223 b\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/FractionalIdeal.lean", "full_name": "FractionalIdeal.one_mem_one", "start": [355, 1], "end": [356, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Disjoint.lean", "full_name": "isCompl_toDual_iff", "start": [610, 1], "end": [611, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/GroupRingAction/Basic.lean", "full_name": "smul_inv''", "start": [99, 1], "end": [100, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.toNat_apply_of_lt_aleph0", "start": [1705, 1], "end": [1707, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Regular/SMul.lean", "full_name": "IsSMulRegular.of_smul_eq_one", "start": [173, 1], "end": [177, 19], "traced_tactics": [{"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "R : Type u_1\nS : Type u_2\nM : Type u_3\na b : R\ns : S\ninst\u271d\u2074 : Monoid S\ninst\u271d\u00b3 : SMul R M\ninst\u271d\u00b2 : SMul R S\ninst\u271d\u00b9 : MulAction S M\ninst\u271d : IsScalarTower R S M\nh : a \u2022 s = 1\n\u22a2 IsSMulRegular M (a \u2022 s)", "state_after": "R : Type u_1\nS : Type u_2\nM : Type u_3\na b : R\ns : S\ninst\u271d\u2074 : Monoid S\ninst\u271d\u00b3 : SMul R M\ninst\u271d\u00b2 : SMul R S\ninst\u271d\u00b9 : MulAction S M\ninst\u271d : IsScalarTower R S M\nh : a \u2022 s = 1\n\u22a2 IsSMulRegular M 1"}, {"tactic": "exact one M", "annotated_tactic": ["exact <a>one</a> M", [{"full_name": "IsSMulRegular.one", "def_path": "Mathlib/Algebra/Regular/SMul.lean", "def_pos": [135, 9], "def_end_pos": [135, 12]}]], "state_before": "R : Type u_1\nS : Type u_2\nM : Type u_3\na b : R\ns : S\ninst\u271d\u2074 : Monoid S\ninst\u271d\u00b3 : SMul R M\ninst\u271d\u00b2 : SMul R S\ninst\u271d\u00b9 : MulAction S M\ninst\u271d : IsScalarTower R S M\nh : a \u2022 s = 1\n\u22a2 IsSMulRegular M 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Equicontinuity.lean", "full_name": "Metric.equicontinuous_of_continuity_modulus", "start": [119, 1], "end": [122, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "full_name": "MeasureTheory.AEEqFun.coeFn_comp\u2082Measurable", "start": [428, 1], "end": [431, 17], "traced_tactics": [{"tactic": "rw [comp\u2082Measurable_eq_mk]", "annotated_tactic": ["rw [<a>comp\u2082Measurable_eq_mk</a>]", [{"full_name": "MeasureTheory.AEEqFun.comp\u2082Measurable_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [420, 9], "def_end_pos": [420, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9\u2074 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b4\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b3\ninst\u271d\u2076 : PseudoMetrizableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : SecondCountableTopologyEither \u03b2 \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b4\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b4\ninst\u271d : SecondCountableTopology \u03b4\ng : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nhg : Measurable (uncurry g)\nf\u2081 : \u03b1 \u2192\u2098[\u03bc] \u03b2\nf\u2082 : \u03b1 \u2192\u2098[\u03bc] \u03b3\n\u22a2 \u2191(comp\u2082Measurable g hg f\u2081 f\u2082) =\u1d50[\u03bc] fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9\u2074 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b4\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b3\ninst\u271d\u2076 : PseudoMetrizableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : SecondCountableTopologyEither \u03b2 \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b4\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b4\ninst\u271d : SecondCountableTopology \u03b4\ng : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nhg : Measurable (uncurry g)\nf\u2081 : \u03b1 \u2192\u2098[\u03bc] \u03b2\nf\u2082 : \u03b1 \u2192\u2098[\u03bc] \u03b3\n\u22a2 \u2191(mk (fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a)) (_ : AEStronglyMeasurable (uncurry g \u2218 fun x => (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc)) =\u1d50[\u03bc] fun a =>\n    g (\u2191f\u2081 a) (\u2191f\u2082 a)"}, {"tactic": "apply coeFn_mk", "annotated_tactic": ["apply <a>coeFn_mk</a>", [{"full_name": "MeasureTheory.AEEqFun.coeFn_mk", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [182, 9], "def_end_pos": [182, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9\u2074 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9\u00b9 : TopologicalSpace \u03b4\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03b2\ninst\u271d\u2079 : PseudoMetrizableSpace \u03b2\ninst\u271d\u2078 : BorelSpace \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b3\ninst\u271d\u2076 : PseudoMetrizableSpace \u03b3\ninst\u271d\u2075 : BorelSpace \u03b3\ninst\u271d\u2074 : SecondCountableTopologyEither \u03b2 \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b4\ninst\u271d\u00b2 : PseudoMetrizableSpace \u03b4\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b4\ninst\u271d : SecondCountableTopology \u03b4\ng : \u03b2 \u2192 \u03b3 \u2192 \u03b4\nhg : Measurable (uncurry g)\nf\u2081 : \u03b1 \u2192\u2098[\u03bc] \u03b2\nf\u2082 : \u03b1 \u2192\u2098[\u03bc] \u03b3\n\u22a2 \u2191(mk (fun a => g (\u2191f\u2081 a) (\u2191f\u2082 a)) (_ : AEStronglyMeasurable (uncurry g \u2218 fun x => (\u2191f\u2081 x, \u2191f\u2082 x)) \u03bc)) =\u1d50[\u03bc] fun a =>\n    g (\u2191f\u2081 a) (\u2191f\u2082 a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Star/Subalgebra.lean", "full_name": "StarSubalgebra.centralizer_le", "start": [318, 1], "end": [319, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Holor.lean", "full_name": "Holor.cprankMax_nil", "start": [324, 1], "end": [326, 34], "traced_tactics": [{"tactic": "have h := CPRankMax.succ 0 x 0 (CPRankMax1.nil x) CPRankMax.zero", "annotated_tactic": ["have h := <a>CPRankMax.succ</a> 0 x 0 (<a>CPRankMax1.nil</a> x) <a>CPRankMax.zero</a>", [{"full_name": "Holor.CPRankMax.succ", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [320, 5], "def_end_pos": [320, 9]}, {"full_name": "Holor.CPRankMax1.nil", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [312, 5], "def_end_pos": [312, 8]}, {"full_name": "Holor.CPRankMax.zero", "def_path": "Mathlib/Data/Holor.lean", "def_pos": [319, 5], "def_end_pos": [319, 9]}]], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : AddMonoid \u03b1\nx : Holor \u03b1 []\n\u22a2 CPRankMax 1 x", "state_after": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : AddMonoid \u03b1\nx : Holor \u03b1 []\nh : CPRankMax (0 + 1) (x + 0)\n\u22a2 CPRankMax 1 x"}, {"tactic": "rwa [add_zero x, zero_add] at h", "annotated_tactic": ["rwa [<a>add_zero</a> x, <a>zero_add</a>] at h", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : AddMonoid \u03b1\nx : Holor \u03b1 []\nh : CPRankMax (0 + 1) (x + 0)\n\u22a2 CPRankMax 1 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/NodupEquivFin.lean", "full_name": "List.Sorted.get_strictMono", "start": [85, 1], "end": [85, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "full_name": "eq_of_norm_le_re_inner_eq_norm_sq", "start": [1727, 1], "end": [1733, 78], "traced_tactics": [{"tactic": "suffices H : re \u27eax - y, x - y\u27eb \u2264 0", "annotated_tactic": ["suffices H : <a>re</a> \u27eax - y, x - y\u27eb \u2264 0", [{"full_name": "IsROrC.re", "def_path": "Mathlib/Data/IsROrC/Basic.lean", "def_pos": [58, 3], "def_end_pos": [58, 5]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nhle : \u2016x\u2016 \u2264 \u2016y\u2016\nh : \u2191re (inner x y) = \u2016y\u2016 ^ 2\n\u22a2 x = y", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nhle : \u2016x\u2016 \u2264 \u2016y\u2016\nh : \u2191re (inner x y) = \u2016y\u2016 ^ 2\nH : \u2191re (inner (x - y) (x - y)) \u2264 0\n\u22a2 x = y\n\ncase H\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nhle : \u2016x\u2016 \u2264 \u2016y\u2016\nh : \u2191re (inner x y) = \u2016y\u2016 ^ 2\n\u22a2 \u2191re (inner (x - y) (x - y)) \u2264 0"}, {"tactic": "have H\u2081 : \u2016x\u2016 ^ 2 \u2264 \u2016y\u2016 ^ 2 := by gcongr", "annotated_tactic": ["have H\u2081 : \u2016x\u2016 ^ 2 \u2264 \u2016y\u2016 ^ 2 := by gcongr", []], "state_before": "case H\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nhle : \u2016x\u2016 \u2264 \u2016y\u2016\nh : \u2191re (inner x y) = \u2016y\u2016 ^ 2\n\u22a2 \u2191re (inner (x - y) (x - y)) \u2264 0", "state_after": "case H\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nhle : \u2016x\u2016 \u2264 \u2016y\u2016\nh : \u2191re (inner x y) = \u2016y\u2016 ^ 2\nH\u2081 : \u2016x\u2016 ^ 2 \u2264 \u2016y\u2016 ^ 2\n\u22a2 \u2191re (inner (x - y) (x - y)) \u2264 0"}, {"tactic": "have H\u2082 : re \u27eay, x\u27eb = \u2016y\u2016 ^ 2 := by rwa [\u2190 inner_conj_symm, conj_re]", "annotated_tactic": ["have H\u2082 : <a>re</a> \u27eay, x\u27eb = \u2016y\u2016 ^ 2 := by rwa [\u2190 <a>inner_conj_symm</a>, <a>conj_re</a>]", [{"full_name": "IsROrC.re", "def_path": "Mathlib/Data/IsROrC/Basic.lean", "def_pos": [58, 3], "def_end_pos": [58, 5]}, {"full_name": "inner_conj_symm", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [438, 9], "def_end_pos": [438, 24]}, {"full_name": "IsROrC.conj_re", "def_path": "Mathlib/Data/IsROrC/Basic.lean", "def_pos": [345, 9], "def_end_pos": [345, 16]}]], "state_before": "case H\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nhle : \u2016x\u2016 \u2264 \u2016y\u2016\nh : \u2191re (inner x y) = \u2016y\u2016 ^ 2\nH\u2081 : \u2016x\u2016 ^ 2 \u2264 \u2016y\u2016 ^ 2\n\u22a2 \u2191re (inner (x - y) (x - y)) \u2264 0", "state_after": "case H\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nhle : \u2016x\u2016 \u2264 \u2016y\u2016\nh : \u2191re (inner x y) = \u2016y\u2016 ^ 2\nH\u2081 : \u2016x\u2016 ^ 2 \u2264 \u2016y\u2016 ^ 2\nH\u2082 : \u2191re (inner y x) = \u2016y\u2016 ^ 2\n\u22a2 \u2191re (inner (x - y) (x - y)) \u2264 0"}, {"tactic": "simpa [inner_sub_left, inner_sub_right, \u2190 norm_sq_eq_inner, h, H\u2082] using H\u2081", "annotated_tactic": ["simpa [<a>inner_sub_left</a>, <a>inner_sub_right</a>, \u2190 <a>norm_sq_eq_inner</a>, h, H\u2082] using H\u2081", [{"full_name": "inner_sub_left", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [646, 9], "def_end_pos": [646, 23]}, {"full_name": "inner_sub_right", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [650, 9], "def_end_pos": [650, 24]}, {"full_name": "InnerProductSpace.norm_sq_eq_inner", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [110, 3], "def_end_pos": [110, 19]}]], "state_before": "case H\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nhle : \u2016x\u2016 \u2264 \u2016y\u2016\nh : \u2191re (inner x y) = \u2016y\u2016 ^ 2\nH\u2081 : \u2016x\u2016 ^ 2 \u2264 \u2016y\u2016 ^ 2\nH\u2082 : \u2191re (inner y x) = \u2016y\u2016 ^ 2\n\u22a2 \u2191re (inner (x - y) (x - y)) \u2264 0", "state_after": "no goals"}, {"tactic": "rwa [inner_self_nonpos, sub_eq_zero] at H", "annotated_tactic": ["rwa [<a>inner_self_nonpos</a>, <a>sub_eq_zero</a>] at H", [{"full_name": "inner_self_nonpos", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [617, 9], "def_end_pos": [617, 26]}, {"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [801, 3], "def_end_pos": [801, 14]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nhle : \u2016x\u2016 \u2264 \u2016y\u2016\nh : \u2191re (inner x y) = \u2016y\u2016 ^ 2\nH : \u2191re (inner (x - y) (x - y)) \u2264 0\n\u22a2 x = y", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nhle : \u2016x\u2016 \u2264 \u2016y\u2016\nh : \u2191re (inner x y) = \u2016y\u2016 ^ 2\n\u22a2 \u2016x\u2016 ^ 2 \u2264 \u2016y\u2016 ^ 2", "state_after": "no goals"}, {"tactic": "rwa [\u2190 inner_conj_symm, conj_re]", "annotated_tactic": ["rwa [\u2190 <a>inner_conj_symm</a>, <a>conj_re</a>]", [{"full_name": "inner_conj_symm", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [438, 9], "def_end_pos": [438, 24]}, {"full_name": "IsROrC.conj_re", "def_path": "Mathlib/Data/IsROrC/Basic.lean", "def_pos": [345, 9], "def_end_pos": [345, 16]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nhle : \u2016x\u2016 \u2264 \u2016y\u2016\nh : \u2191re (inner x y) = \u2016y\u2016 ^ 2\nH\u2081 : \u2016x\u2016 ^ 2 \u2264 \u2016y\u2016 ^ 2\n\u22a2 \u2191re (inner y x) = \u2016y\u2016 ^ 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Contracting.lean", "full_name": "ContractingWith.efixedPoint_mem'", "start": [195, 1], "end": [198, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Manifold/PartitionOfUnity.lean", "full_name": "SmoothPartitionOfUnity.exists_isSubordinate_chartAt_source", "start": [542, 1], "end": [545, 51], "traced_tactics": [{"tactic": "apply exists_isSubordinate _ isClosed_univ _ (fun i \u21a6 (chartAt H _).open_source) (fun x _ \u21a6 ?_)", "annotated_tactic": ["apply <a>exists_isSubordinate</a> _ <a>isClosed_univ</a> _ (fun i \u21a6 (<a>chartAt</a> H _).<a>open_source</a>) (fun x _ \u21a6 ?_)", [{"full_name": "SmoothPartitionOfUnity.exists_isSubordinate", "def_path": "Mathlib/Geometry/Manifold/PartitionOfUnity.lean", "def_pos": [521, 9], "def_end_pos": [521, 29]}, {"full_name": "isClosed_univ", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [215, 17], "def_end_pos": [215, 30]}, {"full_name": "chartAt", "def_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "def_pos": [571, 8], "def_end_pos": [571, 15]}, {"full_name": "LocalHomeomorph.open_source", "def_path": "Mathlib/Topology/LocalHomeomorph.lean", "def_pos": [59, 3], "def_end_pos": [59, 14]}]], "state_before": "\u03b9 : Type u\u03b9\nE : Type uE\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : FiniteDimensional \u211d E\nF : Type uF\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\nH : Type uH\ninst\u271d\u2075 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace H M\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : T2Space M\ninst\u271d : SigmaCompactSpace M\n\u22a2 \u2203 f, IsSubordinate f fun x => (chartAt H x).toLocalEquiv.source", "state_after": "\u03b9 : Type u\u03b9\nE : Type uE\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : FiniteDimensional \u211d E\nF : Type uF\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\nH : Type uH\ninst\u271d\u2075 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace H M\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : T2Space M\ninst\u271d : SigmaCompactSpace M\nx : M\nx\u271d : x \u2208 univ\n\u22a2 x \u2208 \u22c3 i, (chartAt H i).toLocalEquiv.source"}, {"tactic": "exact mem_iUnion_of_mem x (mem_chart_source H x)", "annotated_tactic": ["exact <a>mem_iUnion_of_mem</a> x (<a>mem_chart_source</a> H x)", [{"full_name": "Set.mem_iUnion_of_mem", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [221, 9], "def_end_pos": [221, 26]}, {"full_name": "mem_chart_source", "def_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "def_pos": [576, 7], "def_end_pos": [576, 23]}]], "state_before": "\u03b9 : Type u\u03b9\nE : Type uE\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : FiniteDimensional \u211d E\nF : Type uF\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\nH : Type uH\ninst\u271d\u2075 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace H M\ninst\u271d\u00b2 : SmoothManifoldWithCorners I M\ninst\u271d\u00b9 : T2Space M\ninst\u271d : SigmaCompactSpace M\nx : M\nx\u271d : x \u2208 univ\n\u22a2 x \u2208 \u22c3 i, (chartAt H i).toLocalEquiv.source", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.Integrable.norm", "start": [706, 1], "end": [707, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Sum/Order.lean", "full_name": "Sum.Lex.inl_lt_inl_iff", "start": [352, 1], "end": [353, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "full_name": "MeasureTheory.ProbabilityMeasure.tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds", "start": [272, 1], "end": [275, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/LinearPMap.lean", "full_name": "Submodule.toLinearPMap_graph_eq", "start": [1049, 1], "end": [1065, 84], "traced_tactics": [{"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "R : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\n\u22a2 LinearPMap.graph (toLinearPMap g) = g", "state_after": "case h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx : E \u00d7 F\n\u22a2 x \u2208 LinearPMap.graph (toLinearPMap g) \u2194 x \u2208 g"}, {"tactic": "constructor <;> intro hx", "annotated_tactic": ["constructor <;> intro hx", []], "state_before": "case h\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx : E \u00d7 F\n\u22a2 x \u2208 LinearPMap.graph (toLinearPMap g) \u2194 x \u2208 g", "state_after": "case h.mp\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx : E \u00d7 F\nhx : x \u2208 LinearPMap.graph (toLinearPMap g)\n\u22a2 x \u2208 g\n\ncase h.mpr\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx : E \u00d7 F\nhx : x \u2208 g\n\u22a2 x \u2208 LinearPMap.graph (toLinearPMap g)"}, {"tactic": "rw [LinearPMap.mem_graph_iff]", "annotated_tactic": ["rw [<a>LinearPMap.mem_graph_iff</a>]", [{"full_name": "LinearPMap.mem_graph_iff", "def_path": "Mathlib/LinearAlgebra/LinearPMap.lean", "def_pos": [776, 9], "def_end_pos": [776, 22]}]], "state_before": "case h.mpr\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx : E \u00d7 F\nhx : x \u2208 g\n\u22a2 x \u2208 LinearPMap.graph (toLinearPMap g)", "state_after": "case h.mpr\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx : E \u00d7 F\nhx : x \u2208 g\n\u22a2 \u2203 y, \u2191y = x.1 \u2227 \u2191(toLinearPMap g) y = x.2"}, {"tactic": "cases' x with x_fst x_snd", "annotated_tactic": ["cases' x with x_fst x_snd", []], "state_before": "case h.mpr\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx : E \u00d7 F\nhx : x \u2208 g\n\u22a2 \u2203 y, \u2191y = x.1 \u2227 \u2191(toLinearPMap g) y = x.2", "state_after": "case h.mpr.mk\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx_fst : E\nx_snd : F\nhx : (x_fst, x_snd) \u2208 g\n\u22a2 \u2203 y, \u2191y = (x_fst, x_snd).1 \u2227 \u2191(toLinearPMap g) y = (x_fst, x_snd).2"}, {"tactic": "have hx_fst : x_fst \u2208 g.map (LinearMap.fst R E F) := by\n  simp only [mem_map, LinearMap.fst_apply, Prod.exists, exists_and_right, exists_eq_right]\n  exact \u27e8x_snd, hx\u27e9", "annotated_tactic": ["have hx_fst : x_fst \u2208 g.map (<a>LinearMap.fst</a> R E F) := by\n    simp only [<a>mem_map</a>, <a>LinearMap.fst_apply</a>, <a>Prod.exists</a>, <a>exists_and_right</a>, <a>exists_eq_right</a>]\n    exact \u27e8x_snd, hx\u27e9", [{"full_name": "LinearMap.fst", "def_path": "Mathlib/LinearAlgebra/Prod.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Submodule.mem_map", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [598, 9], "def_end_pos": [598, 16]}, {"full_name": "LinearMap.fst_apply", "def_path": "Mathlib/LinearAlgebra/Prod.lean", "def_pos": [75, 9], "def_end_pos": [75, 18]}, {"full_name": "Prod.exists", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [41, 9], "def_end_pos": [41, 17]}, {"full_name": "exists_and_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [468, 17], "def_end_pos": [468, 33]}, {"full_name": "exists_eq_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [462, 17], "def_end_pos": [462, 32]}]], "state_before": "case h.mpr.mk\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx_fst : E\nx_snd : F\nhx : (x_fst, x_snd) \u2208 g\n\u22a2 \u2203 y, \u2191y = (x_fst, x_snd).1 \u2227 \u2191(toLinearPMap g) y = (x_fst, x_snd).2", "state_after": "case h.mpr.mk\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx_fst : E\nx_snd : F\nhx : (x_fst, x_snd) \u2208 g\nhx_fst : x_fst \u2208 map (LinearMap.fst R E F) g\n\u22a2 \u2203 y, \u2191y = (x_fst, x_snd).1 \u2227 \u2191(toLinearPMap g) y = (x_fst, x_snd).2"}, {"tactic": "refine' \u27e8\u27e8x_fst, hx_fst\u27e9, Subtype.coe_mk x_fst hx_fst, _\u27e9", "annotated_tactic": ["refine' \u27e8\u27e8x_fst, hx_fst\u27e9, <a>Subtype.coe_mk</a> x_fst hx_fst, _\u27e9", [{"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}]], "state_before": "case h.mpr.mk\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx_fst : E\nx_snd : F\nhx : (x_fst, x_snd) \u2208 g\nhx_fst : x_fst \u2208 map (LinearMap.fst R E F) g\n\u22a2 \u2203 y, \u2191y = (x_fst, x_snd).1 \u2227 \u2191(toLinearPMap g) y = (x_fst, x_snd).2", "state_after": "case h.mpr.mk\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx_fst : E\nx_snd : F\nhx : (x_fst, x_snd) \u2208 g\nhx_fst : x_fst \u2208 map (LinearMap.fst R E F) g\n\u22a2 \u2191(toLinearPMap g) { val := x_fst, property := hx_fst } = (x_fst, x_snd).2"}, {"tactic": "rw [toLinearPMap_apply_aux hg]", "annotated_tactic": ["rw [<a>toLinearPMap_apply_aux</a> hg]", [{"full_name": "Submodule.toLinearPMap_apply_aux", "def_path": "Mathlib/LinearAlgebra/LinearPMap.lean", "def_pos": [1031, 9], "def_end_pos": [1031, 31]}]], "state_before": "case h.mpr.mk\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx_fst : E\nx_snd : F\nhx : (x_fst, x_snd) \u2208 g\nhx_fst : x_fst \u2208 map (LinearMap.fst R E F) g\n\u22a2 \u2191(toLinearPMap g) { val := x_fst, property := hx_fst } = (x_fst, x_snd).2", "state_after": "case h.mpr.mk\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx_fst : E\nx_snd : F\nhx : (x_fst, x_snd) \u2208 g\nhx_fst : x_fst \u2208 map (LinearMap.fst R E F) g\n\u22a2 valFromGraph hg (_ : \u2191{ val := x_fst, property := hx_fst } \u2208 map (LinearMap.fst R E F) g) = (x_fst, x_snd).2"}, {"tactic": "exact (existsUnique_from_graph @hg hx_fst).unique (valFromGraph_mem hg hx_fst) hx", "annotated_tactic": ["exact (<a>existsUnique_from_graph</a> @hg hx_fst).<a>unique</a> (<a>valFromGraph_mem</a> hg hx_fst) hx", [{"full_name": "Submodule.existsUnique_from_graph", "def_path": "Mathlib/LinearAlgebra/LinearPMap.lean", "def_pos": [970, 9], "def_end_pos": [970, 32]}, {"full_name": "ExistsUnique.unique", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [252, 9], "def_end_pos": [252, 28]}, {"full_name": "Submodule.valFromGraph_mem", "def_path": "Mathlib/LinearAlgebra/LinearPMap.lean", "def_pos": [990, 9], "def_end_pos": [990, 25]}]], "state_before": "case h.mpr.mk\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx_fst : E\nx_snd : F\nhx : (x_fst, x_snd) \u2208 g\nhx_fst : x_fst \u2208 map (LinearMap.fst R E F) g\n\u22a2 valFromGraph hg (_ : \u2191{ val := x_fst, property := hx_fst } \u2208 map (LinearMap.fst R E F) g) = (x_fst, x_snd).2", "state_after": "no goals"}, {"tactic": "rw [LinearPMap.mem_graph_iff] at hx", "annotated_tactic": ["rw [<a>LinearPMap.mem_graph_iff</a>] at hx", [{"full_name": "LinearPMap.mem_graph_iff", "def_path": "Mathlib/LinearAlgebra/LinearPMap.lean", "def_pos": [776, 9], "def_end_pos": [776, 22]}]], "state_before": "case h.mp\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx : E \u00d7 F\nhx : x \u2208 LinearPMap.graph (toLinearPMap g)\n\u22a2 x \u2208 g", "state_after": "case h.mp\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx : E \u00d7 F\nhx : \u2203 y, \u2191y = x.1 \u2227 \u2191(toLinearPMap g) y = x.2\n\u22a2 x \u2208 g"}, {"tactic": "rcases hx with \u27e8y, hx1, hx2\u27e9", "annotated_tactic": ["rcases hx with \u27e8y, hx1, hx2\u27e9", []], "state_before": "case h.mp\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx : E \u00d7 F\nhx : \u2203 y, \u2191y = x.1 \u2227 \u2191(toLinearPMap g) y = x.2\n\u22a2 x \u2208 g", "state_after": "case h.mp.intro.intro\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx : E \u00d7 F\ny : { x // x \u2208 (toLinearPMap g).domain }\nhx1 : \u2191y = x.1\nhx2 : \u2191(toLinearPMap g) y = x.2\n\u22a2 x \u2208 g"}, {"tactic": "convert g.mem_graph_toLinearPMap hg y using 1", "annotated_tactic": ["convert g.mem_graph_toLinearPMap hg y using 1", []], "state_before": "case h.mp.intro.intro\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx : E \u00d7 F\ny : { x // x \u2208 (toLinearPMap g).domain }\nhx1 : \u2191y = x.1\nhx2 : \u2191(toLinearPMap g) y = x.2\n\u22a2 x \u2208 g", "state_after": "case h.e'_4\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx : E \u00d7 F\ny : { x // x \u2208 (toLinearPMap g).domain }\nhx1 : \u2191y = x.1\nhx2 : \u2191(toLinearPMap g) y = x.2\n\u22a2 x = (\u2191y, \u2191(toLinearPMap g) y)"}, {"tactic": "exact Prod.ext hx1.symm hx2.symm", "annotated_tactic": ["exact <a>Prod.ext</a> hx1.symm hx2.symm", [{"full_name": "Prod.ext", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [129, 9], "def_end_pos": [129, 12]}]], "state_before": "case h.e'_4\nR : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx : E \u00d7 F\ny : { x // x \u2208 (toLinearPMap g).domain }\nhx1 : \u2191y = x.1\nhx2 : \u2191(toLinearPMap g) y = x.2\n\u22a2 x = (\u2191y, \u2191(toLinearPMap g) y)", "state_after": "no goals"}, {"tactic": "simp only [mem_map, LinearMap.fst_apply, Prod.exists, exists_and_right, exists_eq_right]", "annotated_tactic": ["simp only [<a>mem_map</a>, <a>LinearMap.fst_apply</a>, <a>Prod.exists</a>, <a>exists_and_right</a>, <a>exists_eq_right</a>]", [{"full_name": "Submodule.mem_map", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [598, 9], "def_end_pos": [598, 16]}, {"full_name": "LinearMap.fst_apply", "def_path": "Mathlib/LinearAlgebra/Prod.lean", "def_pos": [75, 9], "def_end_pos": [75, 18]}, {"full_name": "Prod.exists", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [41, 9], "def_end_pos": [41, 17]}, {"full_name": "exists_and_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [468, 17], "def_end_pos": [468, 33]}, {"full_name": "exists_eq_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [462, 17], "def_end_pos": [462, 32]}]], "state_before": "R : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx_fst : E\nx_snd : F\nhx : (x_fst, x_snd) \u2208 g\n\u22a2 x_fst \u2208 map (LinearMap.fst R E F) g", "state_after": "R : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx_fst : E\nx_snd : F\nhx : (x_fst, x_snd) \u2208 g\n\u22a2 \u2203 x, (x_fst, x) \u2208 g"}, {"tactic": "exact \u27e8x_snd, hx\u27e9", "annotated_tactic": ["exact \u27e8x_snd, hx\u27e9", []], "state_before": "R : Type u_1\ninst\u271d\u2076 : Ring R\nE : Type u_2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\nF : Type u_3\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module R F\nG : Type u_4\ninst\u271d\u00b9 : AddCommGroup G\ninst\u271d : Module R G\ng : Submodule R (E \u00d7 F)\nhg : \u2200 (x : E \u00d7 F), x \u2208 g \u2192 x.1 = 0 \u2192 x.2 = 0\nx_fst : E\nx_snd : F\nhx : (x_fst, x_snd) \u2208 g\n\u22a2 \u2203 x, (x_fst, x) \u2208 g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Count.lean", "full_name": "List.Sublist.count_le", "start": [153, 1], "end": [153, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "full_name": "min_one", "start": [385, 1], "end": [386, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Slice.lean", "full_name": "Finset.eq_of_mem_slice", "start": [155, 1], "end": [156, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "full_name": "derivWithin_Ioi_eq_Ici", "start": [559, 1], "end": [567, 50], "traced_tactics": [{"tactic": "by_cases H : DifferentiableWithinAt \u211d f (Ioi x) x", "annotated_tactic": ["by_cases H : <a>DifferentiableWithinAt</a> \u211d f (<a>Ioi</a> x) x", [{"full_name": "DifferentiableWithinAt", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [172, 5], "def_end_pos": [172, 27]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}]], "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE\u271d : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\nx : \u211d\n\u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x", "state_after": "case pos\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE\u271d : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\nx : \u211d\nH : DifferentiableWithinAt \u211d f (Ioi x) x\n\u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x\n\ncase neg\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE\u271d : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\nx : \u211d\nH : \u00acDifferentiableWithinAt \u211d f (Ioi x) x\n\u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x"}, {"tactic": "have A := H.hasDerivWithinAt.Ici_of_Ioi", "annotated_tactic": ["have A := H.hasDerivWithinAt.Ici_of_Ioi", []], "state_before": "case pos\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE\u271d : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\nx : \u211d\nH : DifferentiableWithinAt \u211d f (Ioi x) x\n\u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x", "state_after": "case pos\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE\u271d : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\nx : \u211d\nH : DifferentiableWithinAt \u211d f (Ioi x) x\nA : HasDerivWithinAt f (derivWithin f (Ioi x) x) (Ici x) x\n\u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x"}, {"tactic": "have B := (differentiableWithinAt_Ioi_iff_Ici.1 H).hasDerivWithinAt", "annotated_tactic": ["have B := (<a>differentiableWithinAt_Ioi_iff_Ici</a>.1 H).<a>hasDerivWithinAt</a>", [{"full_name": "differentiableWithinAt_Ioi_iff_Ici", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [552, 9], "def_end_pos": [552, 43]}, {"full_name": "DifferentiableWithinAt.hasDerivWithinAt", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [430, 9], "def_end_pos": [430, 48]}]], "state_before": "case pos\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE\u271d : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\nx : \u211d\nH : DifferentiableWithinAt \u211d f (Ioi x) x\nA : HasDerivWithinAt f (derivWithin f (Ioi x) x) (Ici x) x\n\u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x", "state_after": "case pos\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE\u271d : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\nx : \u211d\nH : DifferentiableWithinAt \u211d f (Ioi x) x\nA : HasDerivWithinAt f (derivWithin f (Ioi x) x) (Ici x) x\nB : HasDerivWithinAt f (derivWithin f (Ici x) x) (Ici x) x\n\u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x"}, {"tactic": "simpa using (uniqueDiffOn_Ici x).eq left_mem_Ici A B", "annotated_tactic": ["simpa using (<a>uniqueDiffOn_Ici</a> x).<a>eq</a> <a>left_mem_Ici</a> A B", [{"full_name": "uniqueDiffOn_Ici", "def_path": "Mathlib/Analysis/Calculus/TangentCone.lean", "def_pos": [393, 9], "def_end_pos": [393, 25]}, {"full_name": "UniqueDiffOn.eq", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [297, 9], "def_end_pos": [297, 24]}, {"full_name": "Set.left_mem_Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [199, 9], "def_end_pos": [199, 21]}]], "state_before": "case pos\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE\u271d : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\nx : \u211d\nH : DifferentiableWithinAt \u211d f (Ioi x) x\nA : HasDerivWithinAt f (derivWithin f (Ioi x) x) (Ici x) x\nB : HasDerivWithinAt f (derivWithin f (Ici x) x) (Ici x) x\n\u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x", "state_after": "no goals"}, {"tactic": "rw [derivWithin_zero_of_not_differentiableWithinAt H,\n  derivWithin_zero_of_not_differentiableWithinAt]", "annotated_tactic": ["rw [<a>derivWithin_zero_of_not_differentiableWithinAt</a> H,\n      <a>derivWithin_zero_of_not_differentiableWithinAt</a>]", [{"full_name": "derivWithin_zero_of_not_differentiableWithinAt", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [230, 9], "def_end_pos": [230, 55]}, {"full_name": "derivWithin_zero_of_not_differentiableWithinAt", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [230, 9], "def_end_pos": [230, 55]}]], "state_before": "case neg\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE\u271d : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\nx : \u211d\nH : \u00acDifferentiableWithinAt \u211d f (Ioi x) x\n\u22a2 derivWithin f (Ioi x) x = derivWithin f (Ici x) x", "state_after": "case neg\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE\u271d : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\nx : \u211d\nH : \u00acDifferentiableWithinAt \u211d f (Ioi x) x\n\u22a2 \u00acDifferentiableWithinAt \u211d f (Ici x) x"}, {"tactic": "rwa [differentiableWithinAt_Ioi_iff_Ici] at H", "annotated_tactic": ["rwa [<a>differentiableWithinAt_Ioi_iff_Ici</a>] at H", [{"full_name": "differentiableWithinAt_Ioi_iff_Ici", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [552, 9], "def_end_pos": [552, 43]}]], "state_before": "case neg\n\ud835\udd5c : Type u\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nE\u271d : Type w\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\u271d\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf' f\u2080' f\u2081' g' : F\nx\u271d : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nf : \u211d \u2192 E\nx : \u211d\nH : \u00acDifferentiableWithinAt \u211d f (Ioi x) x\n\u22a2 \u00acDifferentiableWithinAt \u211d f (Ici x) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/Bernoulli.lean", "full_name": "bernoulli'_eq_bernoulli", "start": [203, 1], "end": [204, 72], "traced_tactics": [{"tactic": "simp [bernoulli, \u2190 mul_assoc, \u2190 sq, \u2190 pow_mul, mul_comm n 2, pow_mul]", "annotated_tactic": ["simp [<a>bernoulli</a>, \u2190 <a>mul_assoc</a>, \u2190 <a>sq</a>, \u2190 <a>pow_mul</a>, <a>mul_comm</a> n 2, <a>pow_mul</a>]", [{"full_name": "bernoulli", "def_path": "Mathlib/NumberTheory/Bernoulli.lean", "def_pos": [199, 5], "def_end_pos": [199, 14]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "sq", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [106, 7], "def_end_pos": [106, 9]}, {"full_name": "pow_mul", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [125, 9], "def_end_pos": [125, 16]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "pow_mul", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [125, 9], "def_end_pos": [125, 16]}]], "state_before": "A : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\nn : \u2115\n\u22a2 bernoulli' n = (-1) ^ n * bernoulli n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean", "full_name": "MvPolynomial.IsWeightedHomogeneous.add", "start": [264, 1], "end": [266, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ContinuousMapDense.lean", "full_name": "MeasureTheory.Mem\u2112p.exists_boundedContinuous_snorm_sub_le", "start": [239, 1], "end": [286, 92], "traced_tactics": [{"tactic": "suffices H :\n  \u2203 g : \u03b1 \u2192 E, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u03bc \u2227 IsBounded (range g)", "annotated_tactic": ["suffices H :\n    \u2203 g : \u03b1 \u2192 E, <a>snorm</a> (f - g) p \u03bc \u2264 \u03b5 \u2227 <a>Continuous</a> g \u2227 <a>Mem\u2112p</a> g p \u03bc \u2227 <a>IsBounded</a> (<a>range</a> g)", [{"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "Continuous", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1591, 11], "def_end_pos": [1591, 21]}, {"full_name": "MeasureTheory.Mem\u2112p", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [108, 5], "def_end_pos": [108, 10]}, {"full_name": "Bornology.IsBounded", "def_path": "Mathlib/Topology/Bornology/Basic.lean", "def_pos": [140, 5], "def_end_pos": [140, 14]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 g, snorm (f - \u2191g) p \u03bc \u2264 \u03b5 \u2227 Mem\u2112p (\u2191g) p", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nH : \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)\n\u22a2 \u2203 g, snorm (f - \u2191g) p \u03bc \u2264 \u03b5 \u2227 Mem\u2112p (\u2191g) p\n\ncase H\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)"}, {"tactic": "apply hf.induction_dense hp _ _ _ _ h\u03b5", "annotated_tactic": ["apply hf.induction_dense hp _ _ _ _ h\u03b5", []], "state_before": "case H\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s < \u22a4 \u2192\n        \u2200 {\u03b5 : \u211d\u22650\u221e},\n          \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)\n\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2200 (f g : \u03b1 \u2192 E),\n    Continuous f \u2227 Mem\u2112p f p \u2227 IsBounded (range f) \u2192\n      Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g) \u2192 Continuous (f + g) \u2227 Mem\u2112p (f + g) p \u2227 IsBounded (range (f + g))\n\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2200 (f : \u03b1 \u2192 E), Continuous f \u2227 Mem\u2112p f p \u2227 IsBounded (range f) \u2192 AEStronglyMeasurable f \u03bc"}, {"tactic": "rotate_left", "annotated_tactic": ["rotate_left", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s < \u22a4 \u2192\n        \u2200 {\u03b5 : \u211d\u22650\u221e},\n          \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)\n\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2200 (f g : \u03b1 \u2192 E),\n    Continuous f \u2227 Mem\u2112p f p \u2227 IsBounded (range f) \u2192\n      Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g) \u2192 Continuous (f + g) \u2227 Mem\u2112p (f + g) p \u2227 IsBounded (range (f + g))\n\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2200 (f : \u03b1 \u2192 E), Continuous f \u2227 Mem\u2112p f p \u2227 IsBounded (range f) \u2192 AEStronglyMeasurable f \u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2200 (f g : \u03b1 \u2192 E),\n    Continuous f \u2227 Mem\u2112p f p \u2227 IsBounded (range f) \u2192\n      Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g) \u2192 Continuous (f + g) \u2227 Mem\u2112p (f + g) p \u2227 IsBounded (range (f + g))\n\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2200 (f : \u03b1 \u2192 E), Continuous f \u2227 Mem\u2112p f p \u2227 IsBounded (range f) \u2192 AEStronglyMeasurable f \u03bc\n\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s < \u22a4 \u2192\n        \u2200 {\u03b5 : \u211d\u22650\u221e},\n          \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)"}, {"tactic": "intro c t ht ht\u03bc \u03b5 h\u03b5", "annotated_tactic": ["intro c t ht ht\u03bc \u03b5 h\u03b5", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s < \u22a4 \u2192\n        \u2200 {\u03b5 : \u211d\u22650\u221e},\n          \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 g, snorm (g - Set.indicator t fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)"}, {"tactic": "rcases exists_Lp_half E \u03bc p h\u03b5 with \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9", "annotated_tactic": ["rcases <a>exists_Lp_half</a> E \u03bc p h\u03b5 with \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9", [{"full_name": "MeasureTheory.exists_Lp_half", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [862, 9], "def_end_pos": [862, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 g, snorm (g - Set.indicator t fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u22a2 \u2203 g, snorm (g - Set.indicator t fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)"}, {"tactic": "obtain \u27e8\u03b7, \u03b7pos, h\u03b7\u27e9 :\n  \u2203 \u03b7 : \u211d\u22650, 0 < \u03b7 \u2227 \u2200 s : Set \u03b1, \u03bc s \u2264 \u03b7 \u2192 snorm (s.indicator fun _x => c) p \u03bc \u2264 \u03b4", "annotated_tactic": ["obtain \u27e8\u03b7, \u03b7pos, h\u03b7\u27e9 :\n    \u2203 \u03b7 : \u211d\u22650, 0 < \u03b7 \u2227 \u2200 s : <a>Set</a> \u03b1, \u03bc s \u2264 \u03b7 \u2192 <a>snorm</a> (s.indicator fun _x => c) p \u03bc \u2264 \u03b4", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u22a2 \u2203 g, snorm (g - Set.indicator t fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u22a2 \u2203 \u03b7, 0 < \u03b7 \u2227 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\n\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\n\u22a2 \u2203 g, snorm (g - Set.indicator t fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)"}, {"tactic": "exact exists_snorm_indicator_le hp c \u03b4pos.ne'", "annotated_tactic": ["exact <a>exists_snorm_indicator_le</a> hp c \u03b4pos.ne'", [{"full_name": "MeasureTheory.exists_snorm_indicator_le", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [708, 9], "def_end_pos": [708, 34]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u22a2 \u2203 \u03b7, 0 < \u03b7 \u2227 \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\n\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\n\u22a2 \u2203 g, snorm (g - Set.indicator t fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\n\u22a2 \u2203 g, snorm (g - Set.indicator t fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)"}, {"tactic": "have h\u03b7_pos' : (0 : \u211d\u22650\u221e) < \u03b7 := ENNReal.coe_pos.2 \u03b7pos", "annotated_tactic": ["have h\u03b7_pos' : (0 : \u211d\u22650\u221e) < \u03b7 := <a>ENNReal.coe_pos</a>.2 \u03b7pos", [{"full_name": "ENNReal.coe_pos", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [380, 28], "def_end_pos": [380, 35]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\n\u22a2 \u2203 g, snorm (g - Set.indicator t fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\n\u22a2 \u2203 g, snorm (g - Set.indicator t fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)"}, {"tactic": "obtain \u27e8s, st, s_closed, \u03bcs\u27e9 : \u2203 s, s \u2286 t \u2227 IsClosed s \u2227 \u03bc (t \\ s) < \u03b7", "annotated_tactic": ["obtain \u27e8s, st, s_closed, \u03bcs\u27e9 : \u2203 s, s \u2286 t \u2227 <a>IsClosed</a> s \u2227 \u03bc (t \\ s) < \u03b7", [{"full_name": "IsClosed", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [196, 7], "def_end_pos": [196, 15]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\n\u22a2 \u2203 g, snorm (g - Set.indicator t fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\n\u22a2 \u2203 s, s \u2286 t \u2227 IsClosed s \u2227 \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7\n\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\ns : Set \u03b1\nst : s \u2286 t\ns_closed : IsClosed s\n\u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7\n\u22a2 \u2203 g, snorm (g - Set.indicator t fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)"}, {"tactic": "exact ht.exists_isClosed_diff_lt ht\u03bc.ne h\u03b7_pos'.ne'", "annotated_tactic": ["exact ht.exists_isClosed_diff_lt ht\u03bc.ne h\u03b7_pos'.ne'", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\n\u22a2 \u2203 s, s \u2286 t \u2227 IsClosed s \u2227 \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7\n\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\ns : Set \u03b1\nst : s \u2286 t\ns_closed : IsClosed s\n\u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7\n\u22a2 \u2203 g, snorm (g - Set.indicator t fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)", "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\ns : Set \u03b1\nst : s \u2286 t\ns_closed : IsClosed s\n\u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7\n\u22a2 \u2203 g, snorm (g - Set.indicator t fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)"}, {"tactic": "have hs\u03bc : \u03bc s < \u221e := (measure_mono st).trans_lt ht\u03bc", "annotated_tactic": ["have hs\u03bc : \u03bc s < \u221e := (<a>measure_mono</a> st).<a>trans_lt</a> ht\u03bc", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\ns : Set \u03b1\nst : s \u2286 t\ns_closed : IsClosed s\n\u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7\n\u22a2 \u2203 g, snorm (g - Set.indicator t fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)", "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\ns : Set \u03b1\nst : s \u2286 t\ns_closed : IsClosed s\n\u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7\nhs\u03bc : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2203 g, snorm (g - Set.indicator t fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)"}, {"tactic": "have I1 : snorm ((s.indicator fun _y => c) - t.indicator fun _y => c) p \u03bc \u2264 \u03b4 := by\n  rw [\u2190 snorm_neg, neg_sub, \u2190 indicator_diff st]\n  exact h\u03b7 _ \u03bcs.le", "annotated_tactic": ["have I1 : <a>snorm</a> ((s.indicator fun _y => c) - t.indicator fun _y => c) p \u03bc \u2264 \u03b4 := by\n    rw [\u2190 <a>snorm_neg</a>, <a>neg_sub</a>, \u2190 <a>indicator_diff</a> st]\n    exact h\u03b7 _ \u03bcs.le", [{"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "MeasureTheory.snorm_neg", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [261, 9], "def_end_pos": [261, 18]}, {"full_name": "neg_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [400, 3], "def_end_pos": [400, 14]}, {"full_name": "Set.indicator_diff", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [582, 15], "def_end_pos": [582, 29]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\ns : Set \u03b1\nst : s \u2286 t\ns_closed : IsClosed s\n\u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7\nhs\u03bc : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2203 g, snorm (g - Set.indicator t fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)", "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\ns : Set \u03b1\nst : s \u2286 t\ns_closed : IsClosed s\n\u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7\nhs\u03bc : \u2191\u2191\u03bc s < \u22a4\nI1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b4\n\u22a2 \u2203 g, snorm (g - Set.indicator t fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)"}, {"tactic": "rcases exists_continuous_snorm_sub_le_of_closed hp s_closed isOpen_univ (subset_univ _) hs\u03bc.ne c\n    \u03b4pos.ne' with\n  \u27e8f, f_cont, I2, f_bound, -, f_mem\u27e9", "annotated_tactic": ["rcases <a>exists_continuous_snorm_sub_le_of_closed</a> hp s_closed <a>isOpen_univ</a> (<a>subset_univ</a> _) hs\u03bc.ne c\n      \u03b4pos.ne' with\n    \u27e8f, f_cont, I2, f_bound, -, f_mem\u27e9", [{"full_name": "MeasureTheory.exists_continuous_snorm_sub_le_of_closed", "def_path": "Mathlib/MeasureTheory/Function/ContinuousMapDense.lean", "def_pos": [79, 9], "def_end_pos": [79, 49]}, {"full_name": "isOpen_univ", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [123, 17], "def_end_pos": [123, 28]}, {"full_name": "Set.subset_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [691, 9], "def_end_pos": [691, 20]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\ns : Set \u03b1\nst : s \u2286 t\ns_closed : IsClosed s\n\u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7\nhs\u03bc : \u2191\u2191\u03bc s < \u22a4\nI1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b4\n\u22a2 \u2203 g, snorm (g - Set.indicator t fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf\u271d : \u03b1 \u2192 E\nhf : Mem\u2112p f\u271d p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\ns : Set \u03b1\nst : s \u2286 t\ns_closed : IsClosed s\n\u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7\nhs\u03bc : \u2191\u2191\u03bc s < \u22a4\nI1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b4\nf : \u03b1 \u2192 E\nf_cont : Continuous f\nI2 : snorm (fun x => f x - Set.indicator s (fun _y => c) x) p \u03bc \u2264 \u03b4\nf_bound : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 \u2016c\u2016\nf_mem : Mem\u2112p f p\n\u22a2 \u2203 g, snorm (g - Set.indicator t fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)"}, {"tactic": "have I3 : snorm (f - t.indicator fun _y => c) p \u03bc \u2264 \u03b5 := by\n  convert\n    (h\u03b4 _ _\n        (f_mem.aestronglyMeasurable.sub\n          (aestronglyMeasurable_const.indicator s_closed.measurableSet))\n        ((aestronglyMeasurable_const.indicator s_closed.measurableSet).sub\n          (aestronglyMeasurable_const.indicator ht))\n        I2 I1).le using 2\n  simp only [sub_add_sub_cancel]", "annotated_tactic": ["have I3 : <a>snorm</a> (f - t.indicator fun _y => c) p \u03bc \u2264 \u03b5 := by\n    convert\n      (h\u03b4 _ _\n          (f_mem.aestronglyMeasurable.sub\n            (aestronglyMeasurable_const.indicator s_closed.measurableSet))\n          ((aestronglyMeasurable_const.indicator s_closed.measurableSet).<a>sub</a>\n            (aestronglyMeasurable_const.indicator ht))\n          I2 I1).<a>le</a> using 2\n    simp only [<a>sub_add_sub_cancel</a>]", [{"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.sub", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1322, 3], "def_end_pos": [1322, 14]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "sub_add_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [789, 30], "def_end_pos": [789, 48]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf\u271d : \u03b1 \u2192 E\nhf : Mem\u2112p f\u271d p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\ns : Set \u03b1\nst : s \u2286 t\ns_closed : IsClosed s\n\u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7\nhs\u03bc : \u2191\u2191\u03bc s < \u22a4\nI1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b4\nf : \u03b1 \u2192 E\nf_cont : Continuous f\nI2 : snorm (fun x => f x - Set.indicator s (fun _y => c) x) p \u03bc \u2264 \u03b4\nf_bound : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 \u2016c\u2016\nf_mem : Mem\u2112p f p\n\u22a2 \u2203 g, snorm (g - Set.indicator t fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf\u271d : \u03b1 \u2192 E\nhf : Mem\u2112p f\u271d p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\ns : Set \u03b1\nst : s \u2286 t\ns_closed : IsClosed s\n\u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7\nhs\u03bc : \u2191\u2191\u03bc s < \u22a4\nI1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b4\nf : \u03b1 \u2192 E\nf_cont : Continuous f\nI2 : snorm (fun x => f x - Set.indicator s (fun _y => c) x) p \u03bc \u2264 \u03b4\nf_bound : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 \u2016c\u2016\nf_mem : Mem\u2112p f p\nI3 : snorm (f - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b5\n\u22a2 \u2203 g, snorm (g - Set.indicator t fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)"}, {"tactic": "refine' \u27e8f, I3, f_cont, f_mem, _\u27e9", "annotated_tactic": ["refine' \u27e8f, I3, f_cont, f_mem, _\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf\u271d : \u03b1 \u2192 E\nhf : Mem\u2112p f\u271d p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\ns : Set \u03b1\nst : s \u2286 t\ns_closed : IsClosed s\n\u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7\nhs\u03bc : \u2191\u2191\u03bc s < \u22a4\nI1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b4\nf : \u03b1 \u2192 E\nf_cont : Continuous f\nI2 : snorm (fun x => f x - Set.indicator s (fun _y => c) x) p \u03bc \u2264 \u03b4\nf_bound : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 \u2016c\u2016\nf_mem : Mem\u2112p f p\nI3 : snorm (f - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b5\n\u22a2 \u2203 g, snorm (g - Set.indicator t fun x => c) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf\u271d : \u03b1 \u2192 E\nhf : Mem\u2112p f\u271d p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\ns : Set \u03b1\nst : s \u2286 t\ns_closed : IsClosed s\n\u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7\nhs\u03bc : \u2191\u2191\u03bc s < \u22a4\nI1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b4\nf : \u03b1 \u2192 E\nf_cont : Continuous f\nI2 : snorm (fun x => f x - Set.indicator s (fun _y => c) x) p \u03bc \u2264 \u03b4\nf_bound : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 \u2016c\u2016\nf_mem : Mem\u2112p f p\nI3 : snorm (f - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b5\n\u22a2 IsBounded (range f)"}, {"tactic": "exact (BoundedContinuousFunction.ofNormedAddCommGroup f f_cont _ f_bound).isBounded_range", "annotated_tactic": ["exact (<a>BoundedContinuousFunction.ofNormedAddCommGroup</a> f f_cont _ f_bound).<a>isBounded_range</a>", [{"full_name": "BoundedContinuousFunction.ofNormedAddCommGroup", "def_path": "Mathlib/Topology/ContinuousFunction/Bounded.lean", "def_pos": [880, 5], "def_end_pos": [880, 25]}, {"full_name": "BoundedContinuousFunction.isBounded_range", "def_path": "Mathlib/Topology/ContinuousFunction/Bounded.lean", "def_pos": [116, 9], "def_end_pos": [116, 24]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf\u271d : \u03b1 \u2192 E\nhf : Mem\u2112p f\u271d p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\ns : Set \u03b1\nst : s \u2286 t\ns_closed : IsClosed s\n\u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7\nhs\u03bc : \u2191\u2191\u03bc s < \u22a4\nI1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b4\nf : \u03b1 \u2192 E\nf_cont : Continuous f\nI2 : snorm (fun x => f x - Set.indicator s (fun _y => c) x) p \u03bc \u2264 \u03b4\nf_bound : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 \u2016c\u2016\nf_mem : Mem\u2112p f p\nI3 : snorm (f - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b5\n\u22a2 IsBounded (range f)", "state_after": "no goals"}, {"tactic": "rcases H with \u27e8g, hg, g_cont, g_mem, g_bd\u27e9", "annotated_tactic": ["rcases H with \u27e8g, hg, g_cont, g_mem, g_bd\u27e9", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nH : \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g)\n\u22a2 \u2203 g, snorm (f - \u2191g) p \u03bc \u2264 \u03b5 \u2227 Mem\u2112p (\u2191g) p", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\ng : \u03b1 \u2192 E\nhg : snorm (f - g) p \u03bc \u2264 \u03b5\ng_cont : Continuous g\ng_mem : Mem\u2112p g p\ng_bd : IsBounded (range g)\n\u22a2 \u2203 g, snorm (f - \u2191g) p \u03bc \u2264 \u03b5 \u2227 Mem\u2112p (\u2191g) p"}, {"tactic": "exact \u27e8\u27e8\u27e8g, g_cont\u27e9, Metric.isBounded_range_iff.1 g_bd\u27e9, hg, g_mem\u27e9", "annotated_tactic": ["exact \u27e8\u27e8\u27e8g, g_cont\u27e9, <a>Metric.isBounded_range_iff</a>.1 g_bd\u27e9, hg, g_mem\u27e9", [{"full_name": "Metric.isBounded_range_iff", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2486, 9], "def_end_pos": [2486, 28]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\ng : \u03b1 \u2192 E\nhg : snorm (f - g) p \u03bc \u2264 \u03b5\ng_cont : Continuous g\ng_mem : Mem\u2112p g p\ng_bd : IsBounded (range g)\n\u22a2 \u2203 g, snorm (f - \u2191g) p \u03bc \u2264 \u03b5 \u2227 Mem\u2112p (\u2191g) p", "state_after": "no goals"}, {"tactic": "rintro f g \u27e8f_cont, f_mem, f_bd\u27e9 \u27e8g_cont, g_mem, g_bd\u27e9", "annotated_tactic": ["rintro f g \u27e8f_cont, f_mem, f_bd\u27e9 \u27e8g_cont, g_mem, g_bd\u27e9", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2200 (f g : \u03b1 \u2192 E),\n    Continuous f \u2227 Mem\u2112p f p \u2227 IsBounded (range f) \u2192\n      Continuous g \u2227 Mem\u2112p g p \u2227 IsBounded (range g) \u2192 Continuous (f + g) \u2227 Mem\u2112p (f + g) p \u2227 IsBounded (range (f + g))", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf\u271d : \u03b1 \u2192 E\nhf : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf g : \u03b1 \u2192 E\nf_cont : Continuous f\nf_mem : Mem\u2112p f p\nf_bd : IsBounded (range f)\ng_cont : Continuous g\ng_mem : Mem\u2112p g p\ng_bd : IsBounded (range g)\n\u22a2 Continuous (f + g) \u2227 Mem\u2112p (f + g) p \u2227 IsBounded (range (f + g))"}, {"tactic": "refine' \u27e8f_cont.add g_cont, f_mem.add g_mem, _\u27e9", "annotated_tactic": ["refine' \u27e8f_cont.add g_cont, f_mem.add g_mem, _\u27e9", []], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf\u271d : \u03b1 \u2192 E\nhf : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf g : \u03b1 \u2192 E\nf_cont : Continuous f\nf_mem : Mem\u2112p f p\nf_bd : IsBounded (range f)\ng_cont : Continuous g\ng_mem : Mem\u2112p g p\ng_bd : IsBounded (range g)\n\u22a2 Continuous (f + g) \u2227 Mem\u2112p (f + g) p \u2227 IsBounded (range (f + g))", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf\u271d : \u03b1 \u2192 E\nhf : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf g : \u03b1 \u2192 E\nf_cont : Continuous f\nf_mem : Mem\u2112p f p\nf_bd : IsBounded (range f)\ng_cont : Continuous g\ng_mem : Mem\u2112p g p\ng_bd : IsBounded (range g)\n\u22a2 IsBounded (range (f + g))"}, {"tactic": "let f' : \u03b1 \u2192\u1d47 E := \u27e8\u27e8f, f_cont\u27e9, Metric.isBounded_range_iff.1 f_bd\u27e9", "annotated_tactic": ["let f' : \u03b1 \u2192\u1d47 E := \u27e8\u27e8f, f_cont\u27e9, <a>Metric.isBounded_range_iff</a>.1 f_bd\u27e9", [{"full_name": "Metric.isBounded_range_iff", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2486, 9], "def_end_pos": [2486, 28]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf\u271d : \u03b1 \u2192 E\nhf : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf g : \u03b1 \u2192 E\nf_cont : Continuous f\nf_mem : Mem\u2112p f p\nf_bd : IsBounded (range f)\ng_cont : Continuous g\ng_mem : Mem\u2112p g p\ng_bd : IsBounded (range g)\n\u22a2 IsBounded (range (f + g))", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf\u271d : \u03b1 \u2192 E\nhf : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf g : \u03b1 \u2192 E\nf_cont : Continuous f\nf_mem : Mem\u2112p f p\nf_bd : IsBounded (range f)\ng_cont : Continuous g\ng_mem : Mem\u2112p g p\ng_bd : IsBounded (range g)\nf' : \u03b1 \u2192\u1d47 E :=\n  { toContinuousMap := ContinuousMap.mk f,\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : \u03b1),\n            dist (ContinuousMap.toFun (ContinuousMap.mk f) x) (ContinuousMap.toFun (ContinuousMap.mk f) y) \u2264 C) }\n\u22a2 IsBounded (range (f + g))"}, {"tactic": "let g' : \u03b1 \u2192\u1d47 E := \u27e8\u27e8g, g_cont\u27e9, Metric.isBounded_range_iff.1 g_bd\u27e9", "annotated_tactic": ["let g' : \u03b1 \u2192\u1d47 E := \u27e8\u27e8g, g_cont\u27e9, <a>Metric.isBounded_range_iff</a>.1 g_bd\u27e9", [{"full_name": "Metric.isBounded_range_iff", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2486, 9], "def_end_pos": [2486, 28]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf\u271d : \u03b1 \u2192 E\nhf : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf g : \u03b1 \u2192 E\nf_cont : Continuous f\nf_mem : Mem\u2112p f p\nf_bd : IsBounded (range f)\ng_cont : Continuous g\ng_mem : Mem\u2112p g p\ng_bd : IsBounded (range g)\nf' : \u03b1 \u2192\u1d47 E :=\n  { toContinuousMap := ContinuousMap.mk f,\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : \u03b1),\n            dist (ContinuousMap.toFun (ContinuousMap.mk f) x) (ContinuousMap.toFun (ContinuousMap.mk f) y) \u2264 C) }\n\u22a2 IsBounded (range (f + g))", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf\u271d : \u03b1 \u2192 E\nhf : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf g : \u03b1 \u2192 E\nf_cont : Continuous f\nf_mem : Mem\u2112p f p\nf_bd : IsBounded (range f)\ng_cont : Continuous g\ng_mem : Mem\u2112p g p\ng_bd : IsBounded (range g)\nf' : \u03b1 \u2192\u1d47 E :=\n  { toContinuousMap := ContinuousMap.mk f,\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : \u03b1),\n            dist (ContinuousMap.toFun (ContinuousMap.mk f) x) (ContinuousMap.toFun (ContinuousMap.mk f) y) \u2264 C) }\ng' : \u03b1 \u2192\u1d47 E :=\n  { toContinuousMap := ContinuousMap.mk g,\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : \u03b1),\n            dist (ContinuousMap.toFun (ContinuousMap.mk g) x) (ContinuousMap.toFun (ContinuousMap.mk g) y) \u2264 C) }\n\u22a2 IsBounded (range (f + g))"}, {"tactic": "exact (f' + g').isBounded_range", "annotated_tactic": ["exact (f' + g').<a>isBounded_range</a>", [{"full_name": "BoundedContinuousFunction.isBounded_range", "def_path": "Mathlib/Topology/ContinuousFunction/Bounded.lean", "def_pos": [116, 9], "def_end_pos": [116, 24]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf\u271d : \u03b1 \u2192 E\nhf : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nf g : \u03b1 \u2192 E\nf_cont : Continuous f\nf_mem : Mem\u2112p f p\nf_bd : IsBounded (range f)\ng_cont : Continuous g\ng_mem : Mem\u2112p g p\ng_bd : IsBounded (range g)\nf' : \u03b1 \u2192\u1d47 E :=\n  { toContinuousMap := ContinuousMap.mk f,\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : \u03b1),\n            dist (ContinuousMap.toFun (ContinuousMap.mk f) x) (ContinuousMap.toFun (ContinuousMap.mk f) y) \u2264 C) }\ng' : \u03b1 \u2192\u1d47 E :=\n  { toContinuousMap := ContinuousMap.mk g,\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : \u03b1),\n            dist (ContinuousMap.toFun (ContinuousMap.mk g) x) (ContinuousMap.toFun (ContinuousMap.mk g) y) \u2264 C) }\n\u22a2 IsBounded (range (f + g))", "state_after": "no goals"}, {"tactic": "exact fun f \u27e8_, h, _\u27e9 => h.aestronglyMeasurable", "annotated_tactic": ["exact fun f \u27e8_, h, _\u27e9 => h.aestronglyMeasurable", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2200 (f : \u03b1 \u2192 E), Continuous f \u2227 Mem\u2112p f p \u2227 IsBounded (range f) \u2192 AEStronglyMeasurable f \u03bc", "state_after": "no goals"}, {"tactic": "rw [\u2190 snorm_neg, neg_sub, \u2190 indicator_diff st]", "annotated_tactic": ["rw [\u2190 <a>snorm_neg</a>, <a>neg_sub</a>, \u2190 <a>indicator_diff</a> st]", [{"full_name": "MeasureTheory.snorm_neg", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [261, 9], "def_end_pos": [261, 18]}, {"full_name": "neg_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [400, 3], "def_end_pos": [400, 14]}, {"full_name": "Set.indicator_diff", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [582, 15], "def_end_pos": [582, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\ns : Set \u03b1\nst : s \u2286 t\ns_closed : IsClosed s\n\u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7\nhs\u03bc : \u2191\u2191\u03bc s < \u22a4\n\u22a2 snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b4", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\ns : Set \u03b1\nst : s \u2286 t\ns_closed : IsClosed s\n\u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7\nhs\u03bc : \u2191\u2191\u03bc s < \u22a4\n\u22a2 snorm (Set.indicator (t \\ s) fun _y => c) p \u03bc \u2264 \u03b4"}, {"tactic": "exact h\u03b7 _ \u03bcs.le", "annotated_tactic": ["exact h\u03b7 _ \u03bcs.le", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\ns : Set \u03b1\nst : s \u2286 t\ns_closed : IsClosed s\n\u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7\nhs\u03bc : \u2191\u2191\u03bc s < \u22a4\n\u22a2 snorm (Set.indicator (t \\ s) fun _y => c) p \u03bc \u2264 \u03b4", "state_after": "no goals"}, {"tactic": "convert\n  (h\u03b4 _ _\n      (f_mem.aestronglyMeasurable.sub\n        (aestronglyMeasurable_const.indicator s_closed.measurableSet))\n      ((aestronglyMeasurable_const.indicator s_closed.measurableSet).sub\n        (aestronglyMeasurable_const.indicator ht))\n      I2 I1).le using 2", "annotated_tactic": ["convert\n      (h\u03b4 _ _\n          (f_mem.aestronglyMeasurable.sub\n            (aestronglyMeasurable_const.indicator s_closed.measurableSet))\n          ((aestronglyMeasurable_const.indicator s_closed.measurableSet).<a>sub</a>\n            (aestronglyMeasurable_const.indicator ht))\n          I2 I1).<a>le</a> using 2", [{"full_name": "MeasureTheory.AEStronglyMeasurable.sub", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1322, 3], "def_end_pos": [1322, 14]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf\u271d : \u03b1 \u2192 E\nhf : Mem\u2112p f\u271d p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\ns : Set \u03b1\nst : s \u2286 t\ns_closed : IsClosed s\n\u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7\nhs\u03bc : \u2191\u2191\u03bc s < \u22a4\nI1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b4\nf : \u03b1 \u2192 E\nf_cont : Continuous f\nI2 : snorm (fun x => f x - Set.indicator s (fun _y => c) x) p \u03bc \u2264 \u03b4\nf_bound : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 \u2016c\u2016\nf_mem : Mem\u2112p f p\n\u22a2 snorm (f - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b5", "state_after": "case h.e'_3.h.e'_5\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf\u271d : \u03b1 \u2192 E\nhf : Mem\u2112p f\u271d p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\ns : Set \u03b1\nst : s \u2286 t\ns_closed : IsClosed s\n\u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7\nhs\u03bc : \u2191\u2191\u03bc s < \u22a4\nI1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b4\nf : \u03b1 \u2192 E\nf_cont : Continuous f\nI2 : snorm (fun x => f x - Set.indicator s (fun _y => c) x) p \u03bc \u2264 \u03b4\nf_bound : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 \u2016c\u2016\nf_mem : Mem\u2112p f p\n\u22a2 (f - Set.indicator t fun _y => c) =\n    (f - Set.indicator s fun x => c) + ((Set.indicator s fun x => c) - Set.indicator t fun x => c)"}, {"tactic": "simp only [sub_add_sub_cancel]", "annotated_tactic": ["simp only [<a>sub_add_sub_cancel</a>]", [{"full_name": "sub_add_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [789, 30], "def_end_pos": [789, 48]}]], "state_before": "case h.e'_3.h.e'_5\n\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Measure.WeaklyRegular \u03bc\nhp : p \u2260 \u22a4\nf\u271d : \u03b1 \u2192 E\nhf : Mem\u2112p f\u271d p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5\u271d : \u03b5\u271d \u2260 0\nc : E\nt : Set \u03b1\nht : MeasurableSet t\nht\u03bc : \u2191\u2191\u03bc t < \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : 0 < \u03b4\nh\u03b4 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b4 \u2192 snorm g p \u03bc \u2264 \u03b4 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u03b7 : \u211d\u22650\n\u03b7pos : 0 < \u03b7\nh\u03b7 : \u2200 (s : Set \u03b1), \u2191\u2191\u03bc s \u2264 \u2191\u03b7 \u2192 snorm (Set.indicator s fun _x => c) p \u03bc \u2264 \u03b4\nh\u03b7_pos' : 0 < \u2191\u03b7\ns : Set \u03b1\nst : s \u2286 t\ns_closed : IsClosed s\n\u03bcs : \u2191\u2191\u03bc (t \\ s) < \u2191\u03b7\nhs\u03bc : \u2191\u2191\u03bc s < \u22a4\nI1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p \u03bc \u2264 \u03b4\nf : \u03b1 \u2192 E\nf_cont : Continuous f\nI2 : snorm (fun x => f x - Set.indicator s (fun _y => c) x) p \u03bc \u2264 \u03b4\nf_bound : \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 \u2016c\u2016\nf_mem : Mem\u2112p f p\n\u22a2 (f - Set.indicator t fun _y => c) =\n    (f - Set.indicator s fun x => c) + ((Set.indicator s fun x => c) - Set.indicator t fun x => c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.ofNat_eq_zero", "start": [68, 1], "end": [68, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/GCD/Basic.lean", "full_name": "Nat.coprime_self_sub_right", "start": [236, 1], "end": [237, 46], "traced_tactics": [{"tactic": "rw [Coprime, Coprime, gcd_self_sub_right h]", "annotated_tactic": ["rw [<a>Coprime</a>, <a>Coprime</a>, <a>gcd_self_sub_right</a> h]", [{"full_name": "Nat.Coprime", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [33, 18], "def_end_pos": [33, 25]}, {"full_name": "Nat.Coprime", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [33, 18], "def_end_pos": [33, 25]}, {"full_name": "Nat.gcd_self_sub_right", "def_path": "Mathlib/Data/Nat/GCD/Basic.lean", "def_pos": [113, 9], "def_end_pos": [113, 27]}]], "state_before": "m n : \u2115\nh : m \u2264 n\n\u22a2 Coprime n (n - m) \u2194 Coprime n m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Manifold/ContMDiff.lean", "full_name": "contMDiffAt_const", "start": [1256, 1], "end": [1257, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Subring/Basic.lean", "full_name": "Subring.prod_mono", "start": [1121, 1], "end": [1123, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.diff_diff_cancel_left", "start": [2097, 1], "end": [2098, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.mapsTo_prod_map_diagonal", "start": [400, 1], "end": [401, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector3.lean", "full_name": "VectorAllP.imp", "start": [292, 1], "end": [294, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Perm/Support.lean", "full_name": "Equiv.Perm.support_refl", "start": [313, 1], "end": [314, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/LocallyConstant/Basic.lean", "full_name": "IsLocallyConstant.iff_isOpen_fiber", "start": [104, 1], "end": [105, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Manifold/BumpFunction.lean", "full_name": "SmoothBumpFunction.support_eq_symm_image", "start": [123, 1], "end": [127, 43], "traced_tactics": [{"tactic": "rw [f.support_eq_inter_preimage, \u2190 extChartAt_source I,\n  \u2190 (extChartAt I c).symm_image_target_inter_eq', inter_comm,\n  ball_inter_range_eq_ball_inter_target]", "annotated_tactic": ["rw [f.support_eq_inter_preimage, \u2190 <a>extChartAt_source</a> I,\n    \u2190 (<a>extChartAt</a> I c).<a>symm_image_target_inter_eq'</a>, <a>inter_comm</a>,\n    <a>ball_inter_range_eq_ball_inter_target</a>]", [{"full_name": "extChartAt_source", "def_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "def_pos": [1254, 9], "def_end_pos": [1254, 26]}, {"full_name": "extChartAt", "def_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "def_pos": [1242, 5], "def_end_pos": [1242, 15]}, {"full_name": "LocalEquiv.symm_image_target_inter_eq'", "def_path": "Mathlib/Logic/Equiv/LocalEquiv.lean", "def_pos": [506, 9], "def_end_pos": [506, 36]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "SmoothBumpFunction.ball_inter_range_eq_ball_inter_target", "def_path": "Mathlib/Geometry/Manifold/BumpFunction.lean", "def_pos": [91, 9], "def_end_pos": [91, 46]}]], "state_before": "E : Type uE\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : FiniteDimensional \u211d E\nH : Type uH\ninst\u271d\u00b3 : TopologicalSpace H\nI : ModelWithCorners \u211d E H\nM : Type uM\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : SmoothManifoldWithCorners I M\nc : M\nf : SmoothBumpFunction I c\nx : M\n\u22a2 support \u2191f = \u2191(LocalEquiv.symm (extChartAt I c)) '' (ball (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/Deriv/Polynomial.lean", "full_name": "Polynomial.fderivWithin_aeval", "start": [192, 11], "end": [194, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/FieldTheory/PolynomialGaloisGroup.lean", "full_name": "Polynomial.Gal.card_complex_roots_eq_card_real_add_card_not_gal_inv", "start": [436, 1], "end": [486, 10], "traced_tactics": [{"tactic": "by_cases hp : p = 0", "annotated_tactic": ["by_cases hp : p = 0", []], "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n    Finset.card (Set.toFinset (rootSet p \u211d)) +\n      Finset.card\n        (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))", "state_after": "case pos\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : p = 0\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n    Finset.card (Set.toFinset (rootSet p \u211d)) +\n      Finset.card\n        (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\n\ncase neg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n    Finset.card (Set.toFinset (rootSet p \u211d)) +\n      Finset.card\n        (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))"}, {"tactic": "have inj : Function.Injective (IsScalarTower.toAlgHom \u211a \u211d \u2102) := (algebraMap \u211d \u2102).injective", "annotated_tactic": ["have inj : <a>Function.Injective</a> (<a>IsScalarTower.toAlgHom</a> \u211a \u211d \u2102) := (<a>algebraMap</a> \u211d \u2102).<a>injective</a>", [{"full_name": "Function.Injective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [109, 5], "def_end_pos": [109, 14]}, {"full_name": "IsScalarTower.toAlgHom", "def_path": "Mathlib/Algebra/Algebra/Tower.lean", "def_pos": [140, 5], "def_end_pos": [140, 13]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "RingHom.injective", "def_path": "Mathlib/Algebra/Field/Basic.lean", "def_pos": [255, 19], "def_end_pos": [255, 28]}]], "state_before": "case neg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n    Finset.card (Set.toFinset (rootSet p \u211d)) +\n      Finset.card\n        (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))", "state_after": "case neg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n    Finset.card (Set.toFinset (rootSet p \u211d)) +\n      Finset.card\n        (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))"}, {"tactic": "rw [\u2190 Finset.card_image_of_injective _ Subtype.coe_injective, \u2190\n  Finset.card_image_of_injective _ inj]", "annotated_tactic": ["rw [\u2190 <a>Finset.card_image_of_injective</a> _ <a>Subtype.coe_injective</a>, \u2190\n    <a>Finset.card_image_of_injective</a> _ inj]", [{"full_name": "Finset.card_image_of_injective", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [250, 9], "def_end_pos": [250, 32]}, {"full_name": "Subtype.coe_injective", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [119, 9], "def_end_pos": [119, 22]}, {"full_name": "Finset.card_image_of_injective", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [250, 9], "def_end_pos": [250, 32]}]], "state_before": "case neg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n    Finset.card (Set.toFinset (rootSet p \u211d)) +\n      Finset.card\n        (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))", "state_after": "case neg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n    Finset.card (Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))) +\n      Finset.card\n        (Finset.image (fun a => \u2191a)\n          (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))))"}, {"tactic": "let a : Finset \u2102 := ?_", "annotated_tactic": ["let a : <a>Finset</a> \u2102 := ?_", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [138, 11], "def_end_pos": [138, 17]}]], "state_before": "case neg\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n    Finset.card (Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))) +\n      Finset.card\n        (Finset.image (fun a => \u2191a)\n          (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))))", "state_after": "case neg.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := ?neg.refine_1\u271d\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n    Finset.card (Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))) +\n      Finset.card\n        (Finset.image (fun a => \u2191a)\n          (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))))\n\ncase neg.refine_1\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\n\u22a2 Finset \u2102"}, {"tactic": "let b : Finset \u2102 := ?_", "annotated_tactic": ["let b : <a>Finset</a> \u2102 := ?_", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [138, 11], "def_end_pos": [138, 17]}]], "state_before": "case neg.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := ?neg.refine_1\u271d\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n    Finset.card (Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))) +\n      Finset.card\n        (Finset.image (fun a => \u2191a)\n          (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))))\n\ncase neg.refine_1\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\n\u22a2 Finset \u2102", "state_after": "case neg.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := ?neg.refine_1\u271d\nb : Finset \u2102 := ?neg.refine_2.refine_1\u271d\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n    Finset.card (Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))) +\n      Finset.card\n        (Finset.image (fun a => \u2191a)\n          (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))))\n\ncase neg.refine_2.refine_1\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := ?neg.refine_1\u271d\n\u22a2 Finset \u2102\n\ncase neg.refine_1\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\n\u22a2 Finset \u2102"}, {"tactic": "let c : Finset \u2102 := ?_", "annotated_tactic": ["let c : <a>Finset</a> \u2102 := ?_", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [138, 11], "def_end_pos": [138, 17]}]], "state_before": "case neg.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := ?neg.refine_1\u271d\nb : Finset \u2102 := ?neg.refine_2.refine_1\u271d\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n    Finset.card (Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))) +\n      Finset.card\n        (Finset.image (fun a => \u2191a)\n          (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))))\n\ncase neg.refine_2.refine_1\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := ?neg.refine_1\u271d\n\u22a2 Finset \u2102\n\ncase neg.refine_1\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\n\u22a2 Finset \u2102", "state_after": "case neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := ?neg.refine_1\u271d\nb : Finset \u2102 := ?neg.refine_2.refine_1\u271d\nc : Finset \u2102 := ?neg.refine_2.refine_2.refine_1\u271d\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n    Finset.card (Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))) +\n      Finset.card\n        (Finset.image (fun a => \u2191a)\n          (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))))\n\ncase neg.refine_2.refine_2.refine_1\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := ?neg.refine_1\u271d\nb : Finset \u2102 := ?neg.refine_2.refine_1\u271d\n\u22a2 Finset \u2102\n\ncase neg.refine_2.refine_1\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := ?neg.refine_1\u271d\n\u22a2 Finset \u2102\n\ncase neg.refine_1\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\n\u22a2 Finset \u2102"}, {"tactic": "suffices a.card = b.card + c.card by exact this", "annotated_tactic": ["suffices a.card = b.card + c.card by exact this", []], "state_before": "case neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := ?neg.refine_1\u271d\nb : Finset \u2102 := ?neg.refine_2.refine_1\u271d\nc : Finset \u2102 := ?neg.refine_2.refine_2.refine_1\u271d\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n    Finset.card (Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))) +\n      Finset.card\n        (Finset.image (fun a => \u2191a)\n          (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))))\n\ncase neg.refine_2.refine_2.refine_1\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := ?neg.refine_1\u271d\nb : Finset \u2102 := ?neg.refine_2.refine_1\u271d\n\u22a2 Finset \u2102\n\ncase neg.refine_2.refine_1\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := ?neg.refine_1\u271d\n\u22a2 Finset \u2102\n\ncase neg.refine_1\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\n\u22a2 Finset \u2102", "state_after": "case neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\n\u22a2 Finset.card a = Finset.card b + Finset.card c"}, {"tactic": "have ha : \u2200 z : \u2102, z \u2208 a \u2194 aeval z p = 0 := by\n  intro z; rw [Set.mem_toFinset, mem_rootSet_of_ne hp]", "annotated_tactic": ["have ha : \u2200 z : \u2102, z \u2208 a \u2194 <a>aeval</a> z p = 0 := by\n    intro z; rw [<a>Set.mem_toFinset</a>, <a>mem_rootSet_of_ne</a> hp]", [{"full_name": "Polynomial.aeval", "def_path": "Mathlib/Data/Polynomial/AlgebraMap.lean", "def_pos": [166, 5], "def_end_pos": [166, 10]}, {"full_name": "Set.mem_toFinset", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [629, 9], "def_end_pos": [629, 21]}, {"full_name": "Polynomial.mem_rootSet_of_ne", "def_path": "Mathlib/Data/Polynomial/RingDivision.lean", "def_pos": [1064, 9], "def_end_pos": [1064, 26]}]], "state_before": "case neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\n\u22a2 Finset.card a = Finset.card b + Finset.card c", "state_after": "case neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\n\u22a2 Finset.card a = Finset.card b + Finset.card c"}, {"tactic": "have hc0 :\n  \u2200 w : p.rootSet \u2102, galActionHom p \u2102 (restrict p \u2102 (Complex.conjAe.restrictScalars \u211a)) w = w \u2194\n      w.val.im = 0 := by\n  intro w\n  rw [Subtype.ext_iff, galActionHom_restrict]\n  exact Complex.conj_eq_iff_im", "annotated_tactic": ["have hc0 :\n    \u2200 w : p.rootSet \u2102, <a>galActionHom</a> p \u2102 (<a>restrict</a> p \u2102 (Complex.conjAe.restrictScalars \u211a)) w = w \u2194\n        w.val.im = 0 := by\n    intro w\n    rw [<a>Subtype.ext_iff</a>, <a>galActionHom_restrict</a>]\n    exact <a>Complex.conj_eq_iff_im</a>", [{"full_name": "Polynomial.Gal.galActionHom", "def_path": "Mathlib/FieldTheory/PolynomialGaloisGroup.lean", "def_pos": [217, 5], "def_end_pos": [217, 17]}, {"full_name": "Polynomial.Gal.restrict", "def_path": "Mathlib/FieldTheory/PolynomialGaloisGroup.lean", "def_pos": [141, 5], "def_end_pos": [141, 13]}, {"full_name": "Subtype.ext_iff", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [69, 9], "def_end_pos": [69, 16]}, {"full_name": "Polynomial.Gal.galActionHom_restrict", "def_path": "Mathlib/FieldTheory/PolynomialGaloisGroup.lean", "def_pos": [221, 9], "def_end_pos": [221, 30]}, {"full_name": "Complex.conj_eq_iff_im", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [576, 9], "def_end_pos": [576, 23]}]], "state_before": "case neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\n\u22a2 Finset.card a = Finset.card b + Finset.card c", "state_after": "case neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\n\u22a2 Finset.card a = Finset.card b + Finset.card c"}, {"tactic": "rw [\u2190 Finset.card_disjoint_union]", "annotated_tactic": ["rw [\u2190 <a>Finset.card_disjoint_union</a>]", [{"full_name": "Finset.card_disjoint_union", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [436, 9], "def_end_pos": [436, 28]}]], "state_before": "case neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\n\u22a2 Finset.card a = Finset.card b + Finset.card c", "state_after": "case neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\n\u22a2 Finset.card a = Finset.card (b \u222a c)\n\ncase neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\n\u22a2 Disjoint b c"}, {"tactic": "haveI : IsEmpty (p.rootSet \u2102) := by rw [hp, rootSet_zero]; infer_instance", "annotated_tactic": ["haveI : <a>IsEmpty</a> (p.rootSet \u2102) := by rw [hp, <a>rootSet_zero</a>]; infer_instance", [{"full_name": "IsEmpty", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [26, 7], "def_end_pos": [26, 14]}, {"full_name": "Polynomial.rootSet_zero", "def_path": "Mathlib/Data/Polynomial/RingDivision.lean", "def_pos": [1021, 9], "def_end_pos": [1021, 21]}]], "state_before": "case pos\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : p = 0\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n    Finset.card (Set.toFinset (rootSet p \u211d)) +\n      Finset.card\n        (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))", "state_after": "case pos\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : p = 0\nthis : IsEmpty \u2191(rootSet p \u2102)\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n    Finset.card (Set.toFinset (rootSet p \u211d)) +\n      Finset.card\n        (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))"}, {"tactic": "simp_rw [(galActionHom p \u2102 _).support.eq_empty_of_isEmpty, hp, rootSet_zero,\n  Set.toFinset_empty, Finset.card_empty]", "annotated_tactic": ["simp_rw [(<a>galActionHom</a> p \u2102 _).support.eq_empty_of_isEmpty, hp, <a>rootSet_zero</a>,\n      <a>Set.toFinset_empty</a>, <a>Finset.card_empty</a>]", [{"full_name": "Polynomial.Gal.galActionHom", "def_path": "Mathlib/FieldTheory/PolynomialGaloisGroup.lean", "def_pos": [217, 5], "def_end_pos": [217, 17]}, {"full_name": "Polynomial.rootSet_zero", "def_path": "Mathlib/Data/Polynomial/RingDivision.lean", "def_pos": [1021, 9], "def_end_pos": [1021, 21]}, {"full_name": "Set.toFinset_empty", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [743, 9], "def_end_pos": [743, 23]}, {"full_name": "Finset.card_empty", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [57, 9], "def_end_pos": [57, 19]}]], "state_before": "case pos\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : p = 0\nthis : IsEmpty \u2191(rootSet p \u2102)\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n    Finset.card (Set.toFinset (rootSet p \u211d)) +\n      Finset.card\n        (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))", "state_after": "no goals"}, {"tactic": "rw [hp, rootSet_zero]", "annotated_tactic": ["rw [hp, <a>rootSet_zero</a>]", [{"full_name": "Polynomial.rootSet_zero", "def_path": "Mathlib/Data/Polynomial/RingDivision.lean", "def_pos": [1021, 9], "def_end_pos": [1021, 21]}]], "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : p = 0\n\u22a2 IsEmpty \u2191(rootSet p \u2102)", "state_after": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : p = 0\n\u22a2 IsEmpty \u2191\u2205"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : p = 0\n\u22a2 IsEmpty \u2191\u2205", "state_after": "no goals"}, {"tactic": "exact this", "annotated_tactic": ["exact this", []], "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := ?neg.refine_1\u271d\nb : Finset \u2102 := ?neg.refine_2.refine_1\u271d\nc : Finset \u2102 := ?neg.refine_2.refine_2.refine_1\u271d\nthis : Finset.card a = Finset.card b + Finset.card c\n\u22a2 Finset.card (Set.toFinset (rootSet p \u2102)) =\n    Finset.card (Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))) +\n      Finset.card\n        (Finset.image (fun a => \u2191a)\n          (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))))", "state_after": "no goals"}, {"tactic": "intro z", "annotated_tactic": ["intro z", []], "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\n\u22a2 \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0", "state_after": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nz : \u2102\n\u22a2 z \u2208 a \u2194 \u2191(aeval z) p = 0"}, {"tactic": "rw [Set.mem_toFinset, mem_rootSet_of_ne hp]", "annotated_tactic": ["rw [<a>Set.mem_toFinset</a>, <a>mem_rootSet_of_ne</a> hp]", [{"full_name": "Set.mem_toFinset", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [629, 9], "def_end_pos": [629, 21]}, {"full_name": "Polynomial.mem_rootSet_of_ne", "def_path": "Mathlib/Data/Polynomial/RingDivision.lean", "def_pos": [1064, 9], "def_end_pos": [1064, 26]}]], "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nz : \u2102\n\u22a2 z \u2208 a \u2194 \u2191(aeval z) p = 0", "state_after": "no goals"}, {"tactic": "intro z", "annotated_tactic": ["intro z", []], "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\n\u22a2 \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0", "state_after": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\n\u22a2 z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0"}, {"tactic": "simp_rw [Finset.mem_image, Set.mem_toFinset, mem_rootSet_of_ne hp]", "annotated_tactic": ["simp_rw [<a>Finset.mem_image</a>, <a>Set.mem_toFinset</a>, <a>mem_rootSet_of_ne</a> hp]", [{"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "Set.mem_toFinset", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [629, 9], "def_end_pos": [629, 21]}, {"full_name": "Polynomial.mem_rootSet_of_ne", "def_path": "Mathlib/Data/Polynomial/RingDivision.lean", "def_pos": [1064, 9], "def_end_pos": [1064, 26]}]], "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\n\u22a2 z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0", "state_after": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\n\u22a2 (\u2203 a, \u2191(aeval a) p = 0 \u2227 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a = z) \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\n\u22a2 (\u2203 a, \u2191(aeval a) p = 0 \u2227 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a = z) \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0", "state_after": "case mp\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\n\u22a2 (\u2203 a, \u2191(aeval a) p = 0 \u2227 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a = z) \u2192 \u2191(aeval z) p = 0 \u2227 z.im = 0\n\ncase mpr\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\n\u22a2 \u2191(aeval z) p = 0 \u2227 z.im = 0 \u2192 \u2203 a, \u2191(aeval a) p = 0 \u2227 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a = z"}, {"tactic": "rintro \u27e8w, hw, rfl\u27e9", "annotated_tactic": ["rintro \u27e8w, hw, rfl\u27e9", []], "state_before": "case mp\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\n\u22a2 (\u2203 a, \u2191(aeval a) p = 0 \u2227 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a = z) \u2192 \u2191(aeval z) p = 0 \u2227 z.im = 0", "state_after": "case mp.intro.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nw : \u211d\nhw : \u2191(aeval w) p = 0\n\u22a2 \u2191(aeval (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) w)) p = 0 \u2227 (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) w).im = 0"}, {"tactic": "exact \u27e8by rw [aeval_algHom_apply, hw, AlgHom.map_zero], rfl\u27e9", "annotated_tactic": ["exact \u27e8by rw [<a>aeval_algHom_apply</a>, hw, <a>AlgHom.map_zero</a>], <a>rfl</a>\u27e9", [{"full_name": "Polynomial.aeval_algHom_apply", "def_path": "Mathlib/Data/Polynomial/AlgebraMap.lean", "def_pos": [279, 9], "def_end_pos": [279, 27]}, {"full_name": "AlgHom.map_zero", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [248, 19], "def_end_pos": [248, 27]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case mp.intro.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nw : \u211d\nhw : \u2191(aeval w) p = 0\n\u22a2 \u2191(aeval (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) w)) p = 0 \u2227 (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) w).im = 0", "state_after": "no goals"}, {"tactic": "rw [aeval_algHom_apply, hw, AlgHom.map_zero]", "annotated_tactic": ["rw [<a>aeval_algHom_apply</a>, hw, <a>AlgHom.map_zero</a>]", [{"full_name": "Polynomial.aeval_algHom_apply", "def_path": "Mathlib/Data/Polynomial/AlgebraMap.lean", "def_pos": [279, 9], "def_end_pos": [279, 27]}, {"full_name": "AlgHom.map_zero", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [248, 19], "def_end_pos": [248, 27]}]], "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nw : \u211d\nhw : \u2191(aeval w) p = 0\n\u22a2 \u2191(aeval (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) w)) p = 0", "state_after": "no goals"}, {"tactic": "rintro \u27e8hz1, hz2\u27e9", "annotated_tactic": ["rintro \u27e8hz1, hz2\u27e9", []], "state_before": "case mpr\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\n\u22a2 \u2191(aeval z) p = 0 \u2227 z.im = 0 \u2192 \u2203 a, \u2191(aeval a) p = 0 \u2227 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a = z", "state_after": "case mpr.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im = 0\n\u22a2 \u2203 a, \u2191(aeval a) p = 0 \u2227 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a = z"}, {"tactic": "have key : IsScalarTower.toAlgHom \u211a \u211d \u2102 z.re = z := by ext; rfl; rw [hz2]; rfl", "annotated_tactic": ["have key : <a>IsScalarTower.toAlgHom</a> \u211a \u211d \u2102 z.re = z := by ext; rfl; rw [hz2]; rfl", [{"full_name": "IsScalarTower.toAlgHom", "def_path": "Mathlib/Algebra/Algebra/Tower.lean", "def_pos": [140, 5], "def_end_pos": [140, 13]}]], "state_before": "case mpr.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im = 0\n\u22a2 \u2203 a, \u2191(aeval a) p = 0 \u2227 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a = z", "state_after": "case mpr.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im = 0\nkey : \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re = z\n\u22a2 \u2203 a, \u2191(aeval a) p = 0 \u2227 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a = z"}, {"tactic": "exact \u27e8z.re, inj (by rwa [\u2190 aeval_algHom_apply, key, AlgHom.map_zero]), key\u27e9", "annotated_tactic": ["exact \u27e8z.re, inj (by rwa [\u2190 <a>aeval_algHom_apply</a>, key, <a>AlgHom.map_zero</a>]), key\u27e9", [{"full_name": "Polynomial.aeval_algHom_apply", "def_path": "Mathlib/Data/Polynomial/AlgebraMap.lean", "def_pos": [279, 9], "def_end_pos": [279, 27]}, {"full_name": "AlgHom.map_zero", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [248, 19], "def_end_pos": [248, 27]}]], "state_before": "case mpr.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im = 0\nkey : \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re = z\n\u22a2 \u2203 a, \u2191(aeval a) p = 0 \u2227 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) a = z", "state_after": "no goals"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im = 0\n\u22a2 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re = z", "state_after": "case a\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im = 0\n\u22a2 (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re).re = z.re\n\ncase a\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im = 0\n\u22a2 (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re).im = z.im"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case a\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im = 0\n\u22a2 (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re).re = z.re\n\ncase a\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im = 0\n\u22a2 (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re).im = z.im", "state_after": "case a\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im = 0\n\u22a2 (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re).im = z.im"}, {"tactic": "rw [hz2]", "annotated_tactic": ["rw [hz2]", []], "state_before": "case a\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im = 0\n\u22a2 (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re).im = z.im", "state_after": "case a\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im = 0\n\u22a2 (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re).im = 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case a\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im = 0\n\u22a2 (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re).im = 0", "state_after": "no goals"}, {"tactic": "rwa [\u2190 aeval_algHom_apply, key, AlgHom.map_zero]", "annotated_tactic": ["rwa [\u2190 <a>aeval_algHom_apply</a>, key, <a>AlgHom.map_zero</a>]", [{"full_name": "Polynomial.aeval_algHom_apply", "def_path": "Mathlib/Data/Polynomial/AlgebraMap.lean", "def_pos": [279, 9], "def_end_pos": [279, 27]}, {"full_name": "AlgHom.map_zero", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [248, 19], "def_end_pos": [248, 27]}]], "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im = 0\nkey : \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) z.re = z\n\u22a2 \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) (\u2191(aeval z.re) p) = \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102) 0", "state_after": "no goals"}, {"tactic": "intro w", "annotated_tactic": ["intro w", []], "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\n\u22a2 \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0", "state_after": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nw : \u2191(rootSet p \u2102)\n\u22a2 \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0"}, {"tactic": "rw [Subtype.ext_iff, galActionHom_restrict]", "annotated_tactic": ["rw [<a>Subtype.ext_iff</a>, <a>galActionHom_restrict</a>]", [{"full_name": "Subtype.ext_iff", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [69, 9], "def_end_pos": [69, 16]}, {"full_name": "Polynomial.Gal.galActionHom_restrict", "def_path": "Mathlib/FieldTheory/PolynomialGaloisGroup.lean", "def_pos": [221, 9], "def_end_pos": [221, 30]}]], "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nw : \u2191(rootSet p \u2102)\n\u22a2 \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0", "state_after": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nw : \u2191(rootSet p \u2102)\n\u22a2 \u2191(AlgEquiv.restrictScalars \u211a Complex.conjAe) \u2191w = \u2191w \u2194 (\u2191w).im = 0"}, {"tactic": "exact Complex.conj_eq_iff_im", "annotated_tactic": ["exact <a>Complex.conj_eq_iff_im</a>", [{"full_name": "Complex.conj_eq_iff_im", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [576, 9], "def_end_pos": [576, 23]}]], "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nw : \u2191(rootSet p \u2102)\n\u22a2 \u2191(AlgEquiv.restrictScalars \u211a Complex.conjAe) \u2191w = \u2191w \u2194 (\u2191w).im = 0", "state_after": "no goals"}, {"tactic": "intro z", "annotated_tactic": ["intro z", []], "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\n\u22a2 \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0", "state_after": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nz : \u2102\n\u22a2 z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0"}, {"tactic": "simp_rw [Finset.mem_image]", "annotated_tactic": ["simp_rw [<a>Finset.mem_image</a>]", [{"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}]], "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nz : \u2102\n\u22a2 z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0", "state_after": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nz : \u2102\n\u22a2 (\u2203 a,\n      a \u2208 Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) \u2227\n        \u2191a = z) \u2194\n    \u2191(aeval z) p = 0 \u2227 z.im \u2260 0"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "F : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nz : \u2102\n\u22a2 (\u2203 a,\n      a \u2208 Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) \u2227\n        \u2191a = z) \u2194\n    \u2191(aeval z) p = 0 \u2227 z.im \u2260 0", "state_after": "case mp\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nz : \u2102\n\u22a2 (\u2203 a,\n      a \u2208 Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) \u2227\n        \u2191a = z) \u2192\n    \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\n\ncase mpr\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nz : \u2102\n\u22a2 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0 \u2192\n    \u2203 a,\n      a \u2208 Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) \u2227\n        \u2191a = z"}, {"tactic": "rintro \u27e8w, hw, rfl\u27e9", "annotated_tactic": ["rintro \u27e8w, hw, rfl\u27e9", []], "state_before": "case mp\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nz : \u2102\n\u22a2 (\u2203 a,\n      a \u2208 Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) \u2227\n        \u2191a = z) \u2192\n    \u2191(aeval z) p = 0 \u2227 z.im \u2260 0", "state_after": "case mp.intro.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nw : { x // x \u2208 rootSet p \u2102 }\nhw : w \u2208 Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))\n\u22a2 \u2191(aeval \u2191w) p = 0 \u2227 (\u2191w).im \u2260 0"}, {"tactic": "exact \u27e8(mem_rootSet.mp w.2).2, mt (hc0 w).mpr (Equiv.Perm.mem_support.mp hw)\u27e9", "annotated_tactic": ["exact \u27e8(mem_rootSet.mp w.2).2, <a>mt</a> (hc0 w).<a>mpr</a> (Equiv.Perm.mem_support.mp hw)\u27e9", [{"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "case mp.intro.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nw : { x // x \u2208 rootSet p \u2102 }\nhw : w \u2208 Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe)))\n\u22a2 \u2191(aeval \u2191w) p = 0 \u2227 (\u2191w).im \u2260 0", "state_after": "no goals"}, {"tactic": "rintro \u27e8hz1, hz2\u27e9", "annotated_tactic": ["rintro \u27e8hz1, hz2\u27e9", []], "state_before": "case mpr\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nz : \u2102\n\u22a2 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0 \u2192\n    \u2203 a,\n      a \u2208 Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) \u2227\n        \u2191a = z", "state_after": "case mpr.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im \u2260 0\n\u22a2 \u2203 a,\n    a \u2208 Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) \u2227 \u2191a = z"}, {"tactic": "exact \u27e8\u27e8z, mem_rootSet.mpr \u27e8hp, hz1\u27e9\u27e9, Equiv.Perm.mem_support.mpr (mt (hc0 _).mp hz2), rfl\u27e9", "annotated_tactic": ["exact \u27e8\u27e8z, mem_rootSet.mpr \u27e8hp, hz1\u27e9\u27e9, Equiv.Perm.mem_support.mpr (<a>mt</a> (hc0 _).<a>mp</a> hz2), <a>rfl</a>\u27e9", [{"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}, {"full_name": "Iff.mp", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [90, 3], "def_end_pos": [90, 5]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case mpr.intro\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nz : \u2102\nhz1 : \u2191(aeval z) p = 0\nhz2 : z.im \u2260 0\n\u22a2 \u2203 a,\n    a \u2208 Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) \u2227 \u2191a = z", "state_after": "no goals"}, {"tactic": "apply congr_arg Finset.card", "annotated_tactic": ["apply <a>congr_arg</a> <a>Finset.card</a>", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}]], "state_before": "case neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\n\u22a2 Finset.card a = Finset.card (b \u222a c)", "state_after": "case neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\n\u22a2 a = b \u222a c"}, {"tactic": "simp_rw [Finset.ext_iff, Finset.mem_union, ha, hb, hc]", "annotated_tactic": ["simp_rw [<a>Finset.ext_iff</a>, <a>Finset.mem_union</a>, ha, hb, hc]", [{"full_name": "Finset.ext_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [234, 9], "def_end_pos": [234, 16]}, {"full_name": "Finset.mem_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1373, 9], "def_end_pos": [1373, 18]}]], "state_before": "case neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\n\u22a2 a = b \u222a c", "state_after": "case neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\n\u22a2 \u2200 (a : \u2102), \u2191(aeval a) p = 0 \u2194 \u2191(aeval a) p = 0 \u2227 a.im = 0 \u2228 \u2191(aeval a) p = 0 \u2227 a.im \u2260 0"}, {"tactic": "tauto", "annotated_tactic": ["tauto", []], "state_before": "case neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\n\u22a2 \u2200 (a : \u2102), \u2191(aeval a) p = 0 \u2194 \u2191(aeval a) p = 0 \u2227 a.im = 0 \u2228 \u2191(aeval a) p = 0 \u2227 a.im \u2260 0", "state_after": "no goals"}, {"tactic": "rw [Finset.disjoint_left]", "annotated_tactic": ["rw [<a>Finset.disjoint_left</a>]", [{"full_name": "Finset.disjoint_left", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [939, 9], "def_end_pos": [939, 22]}]], "state_before": "case neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\n\u22a2 Disjoint b c", "state_after": "case neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\n\u22a2 \u2200 \u2983a : \u2102\u2984, a \u2208 b \u2192 \u00aca \u2208 c"}, {"tactic": "intro z", "annotated_tactic": ["intro z", []], "state_before": "case neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\n\u22a2 \u2200 \u2983a : \u2102\u2984, a \u2208 b \u2192 \u00aca \u2208 c", "state_after": "case neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\nz : \u2102\n\u22a2 z \u2208 b \u2192 \u00acz \u2208 c"}, {"tactic": "rw [hb, hc]", "annotated_tactic": ["rw [hb, hc]", []], "state_before": "case neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\nz : \u2102\n\u22a2 z \u2208 b \u2192 \u00acz \u2208 c", "state_after": "case neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\nz : \u2102\n\u22a2 \u2191(aeval z) p = 0 \u2227 z.im = 0 \u2192 \u00ac(\u2191(aeval z) p = 0 \u2227 z.im \u2260 0)"}, {"tactic": "tauto", "annotated_tactic": ["tauto", []], "state_before": "case neg.refine_2.refine_2.refine_2\nF : Type u_1\ninst\u271d\u00b2 : Field F\np\u271d q : F[X]\nE : Type u_2\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra F E\np : \u211a[X]\nhp : \u00acp = 0\ninj : Function.Injective \u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)\na : Finset \u2102 := Set.toFinset (rootSet p \u2102)\nb : Finset \u2102 := Finset.image (\u2191(IsScalarTower.toAlgHom \u211a \u211d \u2102)) (Set.toFinset (rootSet p \u211d))\nc : Finset \u2102 :=\n  Finset.image (fun a => \u2191a)\n    (Equiv.Perm.support (\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))))\nha : \u2200 (z : \u2102), z \u2208 a \u2194 \u2191(aeval z) p = 0\nhb : \u2200 (z : \u2102), z \u2208 b \u2194 \u2191(aeval z) p = 0 \u2227 z.im = 0\nhc0 :\n  \u2200 (w : \u2191(rootSet p \u2102)),\n    \u2191(\u2191(galActionHom p \u2102) (\u2191(restrict p \u2102) (AlgEquiv.restrictScalars \u211a Complex.conjAe))) w = w \u2194 (\u2191w).im = 0\nhc : \u2200 (z : \u2102), z \u2208 c \u2194 \u2191(aeval z) p = 0 \u2227 z.im \u2260 0\nz : \u2102\n\u22a2 \u2191(aeval z) p = 0 \u2227 z.im = 0 \u2192 \u00ac(\u2191(aeval z) p = 0 \u2227 z.im \u2260 0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Filtered/Basic.lean", "full_name": "CategoryTheory.IsFilteredOrEmpty.of_isRightAdjoint", "start": [217, 1], "end": [218, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Monotone.lean", "full_name": "MonotoneOn.Ico", "start": [126, 11], "end": [128, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/QuadraticForm/Basic.lean", "full_name": "QuadraticForm.linMulLin_comp", "start": [611, 1], "end": [613, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Pi/Algebra.lean", "full_name": "Function.extend_inv", "start": [379, 1], "end": [383, 71], "traced_tactics": [{"tactic": "classical\nfunext x\nsimp only [not_exists, extend_def, Pi.inv_apply, apply_dite Inv.inv]", "annotated_tactic": ["classical\n  funext x\n  simp only [<a>not_exists</a>, <a>extend_def</a>, <a>Pi.inv_apply</a>, <a>apply_dite</a> <a>Inv.inv</a>]", [{"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"full_name": "Function.extend_def", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [720, 9], "def_end_pos": [720, 19]}, {"full_name": "Pi.inv_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [171, 9], "def_end_pos": [171, 18]}, {"full_name": "apply_dite", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [734, 9], "def_end_pos": [734, 19]}, {"full_name": "Inv.inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [109, 3], "def_end_pos": [109, 6]}]], "state_before": "I : Type u\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf\u271d : I \u2192 Type v\u2081\ng\u271d : I \u2192 Type v\u2082\nh : I \u2192 Type v\u2083\nx y : (i : I) \u2192 f\u271d i\ni : I\ninst\u271d : Inv \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\ne : \u03b2 \u2192 \u03b3\n\u22a2 extend f g\u207b\u00b9 e\u207b\u00b9 = (extend f g e)\u207b\u00b9", "state_after": "no goals"}, {"tactic": "funext x", "annotated_tactic": ["funext x", []], "state_before": "I : Type u\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf\u271d : I \u2192 Type v\u2081\ng\u271d : I \u2192 Type v\u2082\nh : I \u2192 Type v\u2083\nx y : (i : I) \u2192 f\u271d i\ni : I\ninst\u271d : Inv \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\ne : \u03b2 \u2192 \u03b3\n\u22a2 extend f g\u207b\u00b9 e\u207b\u00b9 = (extend f g e)\u207b\u00b9", "state_after": "case h\nI : Type u\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf\u271d : I \u2192 Type v\u2081\ng\u271d : I \u2192 Type v\u2082\nh : I \u2192 Type v\u2083\nx\u271d y : (i : I) \u2192 f\u271d i\ni : I\ninst\u271d : Inv \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\ne : \u03b2 \u2192 \u03b3\nx : \u03b2\n\u22a2 extend f g\u207b\u00b9 e\u207b\u00b9 x = (extend f g e)\u207b\u00b9 x"}, {"tactic": "simp only [not_exists, extend_def, Pi.inv_apply, apply_dite Inv.inv]", "annotated_tactic": ["simp only [<a>not_exists</a>, <a>extend_def</a>, <a>Pi.inv_apply</a>, <a>apply_dite</a> <a>Inv.inv</a>]", [{"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"full_name": "Function.extend_def", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [720, 9], "def_end_pos": [720, 19]}, {"full_name": "Pi.inv_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [171, 9], "def_end_pos": [171, 18]}, {"full_name": "apply_dite", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [734, 9], "def_end_pos": [734, 19]}, {"full_name": "Inv.inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [109, 3], "def_end_pos": [109, 6]}]], "state_before": "case h\nI : Type u\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf\u271d : I \u2192 Type v\u2081\ng\u271d : I \u2192 Type v\u2082\nh : I \u2192 Type v\u2083\nx\u271d y : (i : I) \u2192 f\u271d i\ni : I\ninst\u271d : Inv \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192 \u03b3\ne : \u03b2 \u2192 \u03b3\nx : \u03b2\n\u22a2 extend f g\u207b\u00b9 e\u207b\u00b9 x = (extend f g e)\u207b\u00b9 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/LocallyConstant/Basic.lean", "full_name": "LocallyConstant.coe_desc", "start": [534, 1], "end": [537, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/W/Basic.lean", "full_name": "WType.depth_pos", "start": [130, 1], "end": [132, 21], "traced_tactics": [{"tactic": "cases t", "annotated_tactic": ["cases t", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d : (a : \u03b1) \u2192 Fintype (\u03b2 a)\nt : WType \u03b2\n\u22a2 0 < depth t", "state_after": "case mk\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d : (a : \u03b1) \u2192 Fintype (\u03b2 a)\na\u271d : \u03b1\nf\u271d : \u03b2 a\u271d \u2192 WType \u03b2\n\u22a2 0 < depth (mk a\u271d f\u271d)"}, {"tactic": "apply Nat.succ_pos", "annotated_tactic": ["apply <a>Nat.succ_pos</a>", [{"full_name": "Nat.succ_pos", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1608, 9], "def_end_pos": [1608, 21]}]], "state_before": "case mk\n\u03b1 : Type u_1\n\u03b2 : \u03b1 \u2192 Type u_2\ninst\u271d : (a : \u03b1) \u2192 Fintype (\u03b2 a)\na\u271d : \u03b1\nf\u271d : \u03b2 a\u271d \u2192 WType \u03b2\n\u22a2 0 < depth (mk a\u271d f\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/NormedSpace/MStructure.lean", "full_name": "IsLprojection.le_def", "start": [217, 1], "end": [219, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Functor/FullyFaithful.lean", "full_name": "CategoryTheory.Faithful.div_faithful", "start": [366, 1], "end": [370, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/HNNExtension.lean", "full_name": "HNNExtension.NormalWord.consRecOn_ofGroup", "start": [311, 1], "end": [317, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean", "full_name": "balancedHull.balanced", "start": [146, 1], "end": [151, 92], "traced_tactics": [{"tactic": "intro a ha", "annotated_tactic": ["intro a ha", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d s : Set E\n\u22a2 Balanced \ud835\udd5c (balancedHull \ud835\udd5c s)", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d s : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\n\u22a2 a \u2022 balancedHull \ud835\udd5c s \u2286 balancedHull \ud835\udd5c s"}, {"tactic": "simp_rw [balancedHull, smul_set_iUnion\u2082, subset_def, mem_iUnion\u2082]", "annotated_tactic": ["simp_rw [<a>balancedHull</a>, <a>smul_set_iUnion\u2082</a>, <a>subset_def</a>, <a>mem_iUnion\u2082</a>]", [{"full_name": "balancedHull", "def_path": "Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean", "def_pos": [67, 5], "def_end_pos": [67, 17]}, {"full_name": "Set.smul_set_iUnion\u2082", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [384, 9], "def_end_pos": [384, 25]}, {"full_name": "Set.subset_def", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [345, 9], "def_end_pos": [345, 19]}, {"full_name": "Set.mem_iUnion\u2082", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [212, 9], "def_end_pos": [212, 20]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d s : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\n\u22a2 a \u2022 balancedHull \ud835\udd5c s \u2286 balancedHull \ud835\udd5c s", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d s : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\n\u22a2 \u2200 (x : E), (\u2203 i j, x \u2208 a \u2022 i \u2022 s) \u2192 \u2203 i j, x \u2208 i \u2022 s"}, {"tactic": "rintro x \u27e8r, hr, hx\u27e9", "annotated_tactic": ["rintro x \u27e8r, hr, hx\u27e9", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d s : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\n\u22a2 \u2200 (x : E), (\u2203 i j, x \u2208 a \u2022 i \u2022 s) \u2192 \u2203 i j, x \u2208 i \u2022 s", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d s : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\nx : E\nr : \ud835\udd5c\nhr : \u2016r\u2016 \u2264 1\nhx : x \u2208 a \u2022 r \u2022 s\n\u22a2 \u2203 i j, x \u2208 i \u2022 s"}, {"tactic": "rw [\u2190 smul_assoc] at hx", "annotated_tactic": ["rw [\u2190 <a>smul_assoc</a>] at hx", [{"full_name": "smul_assoc", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 19]}]], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d s : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\nx : E\nr : \ud835\udd5c\nhr : \u2016r\u2016 \u2264 1\nhx : x \u2208 a \u2022 r \u2022 s\n\u22a2 \u2203 i j, x \u2208 i \u2022 s", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d s : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\nx : E\nr : \ud835\udd5c\nhr : \u2016r\u2016 \u2264 1\nhx\u271d : x \u2208 a \u2022 r \u2022 s\nhx : x \u2208 (a \u2022 r) \u2022 s\n\u22a2 \u2203 i j, x \u2208 i \u2022 s"}, {"tactic": "exact \u27e8a \u2022 r, (SeminormedRing.norm_mul _ _).trans (mul_le_one ha (norm_nonneg r) hr), hx\u27e9", "annotated_tactic": ["exact \u27e8a \u2022 r, (<a>SeminormedRing.norm_mul</a> _ _).<a>trans</a> (<a>mul_le_one</a> ha (<a>norm_nonneg</a> r) hr), hx\u27e9", [{"full_name": "SeminormedRing.norm_mul", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [42, 3], "def_end_pos": [42, 11]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "mul_le_one", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [313, 9], "def_end_pos": [313, 19]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b2 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ns\u271d s : Set E\na : \ud835\udd5c\nha : \u2016a\u2016 \u2264 1\nx : E\nr : \ud835\udd5c\nhr : \u2016r\u2016 \u2264 1\nhx\u271d : x \u2208 a \u2022 r \u2022 s\nhx : x \u2208 (a \u2022 r) \u2022 s\n\u22a2 \u2203 i j, x \u2208 i \u2022 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/NormedSpace/IsROrC.lean", "full_name": "ContinuousLinearMap.op_norm_bound_of_ball_bound", "start": [85, 1], "end": [93, 27], "traced_tactics": [{"tactic": "apply ContinuousLinearMap.op_norm_le_bound", "annotated_tactic": ["apply <a>ContinuousLinearMap.op_norm_le_bound</a>", [{"full_name": "ContinuousLinearMap.op_norm_le_bound", "def_path": "Mathlib/Analysis/NormedSpace/OperatorNorm.lean", "def_pos": [150, 9], "def_end_pos": [150, 25]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nr_pos : 0 < r\nc : \u211d\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nh : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c\n\u22a2 \u2016f\u2016 \u2264 c / r", "state_after": "case hMp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nr_pos : 0 < r\nc : \u211d\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nh : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c\n\u22a2 0 \u2264 c / r\n\ncase hM\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nr_pos : 0 < r\nc : \u211d\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nh : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c\n\u22a2 \u2200 (x : E), \u2016\u2191f x\u2016 \u2264 c / r * \u2016x\u2016"}, {"tactic": "apply LinearMap.bound_of_ball_bound' r_pos", "annotated_tactic": ["apply <a>LinearMap.bound_of_ball_bound'</a> r_pos", [{"full_name": "LinearMap.bound_of_ball_bound'", "def_path": "Mathlib/Analysis/NormedSpace/IsROrC.lean", "def_pos": [80, 9], "def_end_pos": [80, 39]}]], "state_before": "case hM\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nr_pos : 0 < r\nc : \u211d\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nh : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c\n\u22a2 \u2200 (x : E), \u2016\u2191f x\u2016 \u2264 c / r * \u2016x\u2016", "state_after": "case hM.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nr_pos : 0 < r\nc : \u211d\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nh : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c\n\u22a2 \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191\u2191f z\u2016 \u2264 c"}, {"tactic": "exact fun z hz => h z hz", "annotated_tactic": ["exact fun z hz => h z hz", []], "state_before": "case hM.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nr_pos : 0 < r\nc : \u211d\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nh : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c\n\u22a2 \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191\u2191f z\u2016 \u2264 c", "state_after": "no goals"}, {"tactic": "apply div_nonneg _ r_pos.le", "annotated_tactic": ["apply <a>div_nonneg</a> _ r_pos.le", [{"full_name": "div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [94, 9], "def_end_pos": [94, 19]}]], "state_before": "case hMp\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nr_pos : 0 < r\nc : \u211d\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nh : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c\n\u22a2 0 \u2264 c / r", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nr_pos : 0 < r\nc : \u211d\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nh : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c\n\u22a2 0 \u2264 c"}, {"tactic": "exact\n  (norm_nonneg _).trans\n    (h 0 (by simp only [norm_zero, mem_closedBall, dist_zero_left, r_pos.le]))", "annotated_tactic": ["exact\n      (<a>norm_nonneg</a> _).<a>trans</a>\n        (h 0 (by simp only [<a>norm_zero</a>, <a>mem_closedBall</a>, <a>dist_zero_left</a>, r_pos.le]))", [{"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "norm_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [528, 30], "def_end_pos": [528, 39]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}, {"full_name": "dist_zero_left", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [400, 3], "def_end_pos": [400, 14]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nr_pos : 0 < r\nc : \u211d\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nh : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c\n\u22a2 0 \u2264 c", "state_after": "no goals"}, {"tactic": "simp only [norm_zero, mem_closedBall, dist_zero_left, r_pos.le]", "annotated_tactic": ["simp only [<a>norm_zero</a>, <a>mem_closedBall</a>, <a>dist_zero_left</a>, r_pos.le]", [{"full_name": "norm_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [528, 30], "def_end_pos": [528, 39]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [478, 17], "def_end_pos": [478, 31]}, {"full_name": "dist_zero_left", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [400, 3], "def_end_pos": [400, 14]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nr : \u211d\nr_pos : 0 < r\nc : \u211d\nf : E \u2192L[\ud835\udd5c] \ud835\udd5c\nh : \u2200 (z : E), z \u2208 closedBall 0 r \u2192 \u2016\u2191f z\u2016 \u2264 c\n\u22a2 0 \u2208 closedBall 0 r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/GDelta.lean", "full_name": "isG\u03b4_empty", "start": [68, 1], "end": [69, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Ackermann.lean", "full_name": "ack_le_iff_right", "start": [162, 1], "end": [163, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Fin/Tuple/Sort.lean", "full_name": "Tuple.antitone_pair_of_not_sorted", "start": [214, 1], "end": [215, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Matrix/IsDiag.lean", "full_name": "Matrix.isDiag_iff_diagonal_diag", "start": [55, 1], "end": [57, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "iInf_emptyset", "start": [1466, 1], "end": [1466, 74], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b2 \u2192 \u03b1\n\u22a2 \u2a05 x \u2208 \u2205, f x = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "full_name": "List.append_inj", "start": [57, 1], "end": [61, 71], "traced_tactics": [{"tactic": "simp [append_inj (cons.inj h).2 (Nat.succ.inj hl)] at h \u22a2", "annotated_tactic": ["simp [append_inj (cons.inj h).2 (Nat.succ.inj hl)] at h \u22a2", []], "state_before": "\u03b1 : Type u_1\na : \u03b1\ns\u2081 : List \u03b1\nb : \u03b1\ns\u2082 t\u2081 t\u2082 : List \u03b1\nh : a :: s\u2081 ++ t\u2081 = b :: s\u2082 ++ t\u2082\nhl : length (a :: s\u2081) = length (b :: s\u2082)\n\u22a2 a :: s\u2081 = b :: s\u2082 \u2227 t\u2081 = t\u2082", "state_after": "\u03b1 : Type u_1\na : \u03b1\ns\u2081 : List \u03b1\nb : \u03b1\ns\u2082 t\u2081 t\u2082 : List \u03b1\nhl : length (a :: s\u2081) = length (b :: s\u2082)\nh : a = b\n\u22a2 a = b"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "\u03b1 : Type u_1\na : \u03b1\ns\u2081 : List \u03b1\nb : \u03b1\ns\u2082 t\u2081 t\u2082 : List \u03b1\nhl : length (a :: s\u2081) = length (b :: s\u2082)\nh : a = b\n\u22a2 a = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Valuation/Basic.lean", "full_name": "AddValuation.map_add", "start": [678, 1], "end": [679, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/InnerProductSpace/Projection.lean", "full_name": "orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero", "start": [877, 1], "end": [880, 79], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\nhv : v \u2208 K\u15ee\n\u22a2 \u2191(orthogonalProjection K) v = 0", "state_after": "case a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\nhv : v \u2208 K\u15ee\n\u22a2 \u2191(\u2191(orthogonalProjection K) v) = \u21910"}, {"tactic": "convert eq_orthogonalProjection_of_mem_orthogonal (K := K) _ _ <;> simp [hv]", "annotated_tactic": ["convert <a>eq_orthogonalProjection_of_mem_orthogonal</a> (K := K) _ _ <;> simp [hv]", [{"full_name": "eq_orthogonalProjection_of_mem_orthogonal", "def_path": "Mathlib/Analysis/InnerProductSpace/Projection.lean", "def_pos": [532, 9], "def_end_pos": [532, 50]}]], "state_before": "case a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\nhv : v \u2208 K\u15ee\n\u22a2 \u2191(\u2191(orthogonalProjection K) v) = \u21910", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Heyting/Hom.lean", "full_name": "CoheytingHom.coe_id", "start": [429, 1], "end": [430, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Images.lean", "full_name": "CategoryTheory.Limits.image.isoStrongEpiMono_inv_comp_mono", "start": [1032, 1], "end": [1034, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/LocallyFinite.lean", "full_name": "Prod.uIcc_eq", "start": [1004, 1], "end": [1007, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Finsupp.lean", "full_name": "Finsupp.mapRange.linearEquiv_apply", "start": [861, 1], "end": [862, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "full_name": "DifferentiableOn.hasFDerivAt", "start": [559, 1], "end": [561, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Fintype/Basic.lean", "full_name": "Finset.piecewise_univ", "start": [289, 1], "end": [292, 19], "traced_tactics": [{"tactic": "ext i", "annotated_tactic": ["ext i", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : Fintype \u03b1\ns t : Finset \u03b1\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 univ)\n\u03b4 : \u03b1 \u2192 Sort u_4\nf g : (i : \u03b1) \u2192 \u03b4 i\n\u22a2 piecewise univ f g = f", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : Fintype \u03b1\ns t : Finset \u03b1\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 univ)\n\u03b4 : \u03b1 \u2192 Sort u_4\nf g : (i : \u03b1) \u2192 \u03b4 i\ni : \u03b1\n\u22a2 piecewise univ f g i = f i"}, {"tactic": "simp [piecewise]", "annotated_tactic": ["simp [<a>piecewise</a>]", [{"full_name": "Finset.piecewise", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2485, 5], "def_end_pos": [2485, 14]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : Fintype \u03b1\ns t : Finset \u03b1\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 univ)\n\u03b4 : \u03b1 \u2192 Sort u_4\nf g : (i : \u03b1) \u2192 \u03b4 i\ni : \u03b1\n\u22a2 piecewise univ f g i = f i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Init/Logic.lean", "full_name": "decidableEq_inr_neg", "start": [338, 1], "end": [341, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Add.lean", "full_name": "DifferentiableWithinAt.sub_const", "start": [550, 1], "end": [552, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "full_name": "AEMeasurable.indicator", "start": [369, 1], "end": [371, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.tendsto_const_mul_atBot_iff_pos", "start": [1222, 1], "end": [1224, 76], "traced_tactics": [{"tactic": "simp [tendsto_const_mul_atBot_iff, h, h.not_tendsto disjoint_atBot_atTop]", "annotated_tactic": ["simp [<a>tendsto_const_mul_atBot_iff</a>, h, h.not_tendsto <a>disjoint_atBot_atTop</a>]", [{"full_name": "Filter.tendsto_const_mul_atBot_iff", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1194, 9], "def_end_pos": [1194, 36]}, {"full_name": "Filter.disjoint_atBot_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [117, 9], "def_end_pos": [117, 29]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : LinearOrderedField \u03b1\nl : Filter \u03b2\nf : \u03b2 \u2192 \u03b1\nr : \u03b1\ninst\u271d : NeBot l\nh : Tendsto f l atBot\n\u22a2 Tendsto (fun x => r * f x) l atBot \u2194 0 < r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.AEStronglyMeasurable.indicator", "start": [1524, 11], "end": [1526, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Covering/Differentiation.lean", "full_name": "VitaliFamily.ae_eventually_measure_pos", "start": [100, 1], "end": [116, 48], "traced_tactics": [{"tactic": "set s := {x | \u00ac\u2200\u1da0 a in v.filterAt x, 0 < \u03bc a} with hs", "annotated_tactic": ["set s := {x | \u00ac\u2200\u1da0 a in v.filterAt x, 0 < \u03bc a} with hs", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a"}, {"tactic": "simp (config := { zeta := false }) only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs", "annotated_tactic": ["simp (config := { zeta := <a>false</a> }) only [<a>not_lt</a>, <a>not_eventually</a>, <a>nonpos_iff_eq_zero</a>] at hs", [{"full_name": "Bool.false", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [547, 5], "def_end_pos": [547, 10]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "Filter.not_eventually", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1326, 9], "def_end_pos": [1326, 23]}, {"full_name": "nonpos_iff_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [237, 3], "def_end_pos": [237, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a"}, {"tactic": "change \u03bc s = 0", "annotated_tactic": ["change \u03bc s = 0", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\n\u22a2 \u2191\u2191\u03bc s = 0"}, {"tactic": "let f : \u03b1 \u2192 Set (Set \u03b1) := fun _ => {a | \u03bc a = 0}", "annotated_tactic": ["let f : \u03b1 \u2192 <a>Set</a> (<a>Set</a> \u03b1) := fun _ => {a | \u03bc a = 0}", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\n\u22a2 \u2191\u2191\u03bc s = 0", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\n\u22a2 \u2191\u2191\u03bc s = 0"}, {"tactic": "have h : v.FineSubfamilyOn f s := by\n  intro x hx \u03b5 \u03b5pos\n  rw [hs] at hx\n  simp only [frequently_filterAt_iff, exists_prop, gt_iff_lt, mem_setOf_eq] at hx\n  rcases hx \u03b5 \u03b5pos with \u27e8a, a_sets, ax, \u03bca\u27e9\n  exact \u27e8a, \u27e8a_sets, \u03bca\u27e9, ax\u27e9", "annotated_tactic": ["have h : v.FineSubfamilyOn f s := by\n    intro x hx \u03b5 \u03b5pos\n    rw [hs] at hx\n    simp only [<a>frequently_filterAt_iff</a>, <a>exists_prop</a>, <a>gt_iff_lt</a>, <a>mem_setOf_eq</a>] at hx\n    rcases hx \u03b5 \u03b5pos with \u27e8a, a_sets, ax, \u03bca\u27e9\n    exact \u27e8a, \u27e8a_sets, \u03bca\u27e9, ax\u27e9", [{"full_name": "VitaliFamily.frequently_filterAt_iff", "def_path": "Mathlib/MeasureTheory/Covering/VitaliFamily.lean", "def_pos": [275, 9], "def_end_pos": [275, 32]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "gt_iff_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 18]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\n\u22a2 \u2191\u2191\u03bc s = 0", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nh : FineSubfamilyOn v f s\n\u22a2 \u2191\u2191\u03bc s = 0"}, {"tactic": "refine' le_antisymm _ bot_le", "annotated_tactic": ["refine' <a>le_antisymm</a> _ <a>bot_le</a>", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [256, 9], "def_end_pos": [256, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nh : FineSubfamilyOn v f s\n\u22a2 \u2191\u2191\u03bc s = 0", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nh : FineSubfamilyOn v f s\n\u22a2 \u2191\u2191\u03bc s \u2264 0"}, {"tactic": "calc\n  \u03bc s \u2264 \u2211' x : h.index, \u03bc (h.covering x) := h.measure_le_tsum\n  _ = \u2211' x : h.index, 0 := by congr; ext1 x; exact h.covering_mem x.2\n  _ = 0 := by simp only [tsum_zero, add_zero]", "annotated_tactic": ["calc\n    \u03bc s \u2264 \u2211' x : h.index, \u03bc (h.covering x) := h.measure_le_tsum\n    _ = \u2211' x : h.index, 0 := by congr; ext1 x; exact h.covering_mem x.2\n    _ = 0 := by simp only [<a>tsum_zero</a>, <a>add_zero</a>]", [{"full_name": "tsum_zero", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [489, 9], "def_end_pos": [489, 18]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nh : FineSubfamilyOn v f s\n\u22a2 \u2191\u2191\u03bc s \u2264 0", "state_after": "no goals"}, {"tactic": "intro x hx \u03b5 \u03b5pos", "annotated_tactic": ["intro x hx \u03b5 \u03b5pos", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\n\u22a2 FineSubfamilyOn v f s", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nx : \u03b1\nhx : x \u2208 s\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5"}, {"tactic": "rw [hs] at hx", "annotated_tactic": ["rw [hs] at hx", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nx : \u03b1\nhx : x \u2208 s\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nx : \u03b1\nhx : x \u2208 {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5"}, {"tactic": "simp only [frequently_filterAt_iff, exists_prop, gt_iff_lt, mem_setOf_eq] at hx", "annotated_tactic": ["simp only [<a>frequently_filterAt_iff</a>, <a>exists_prop</a>, <a>gt_iff_lt</a>, <a>mem_setOf_eq</a>] at hx", [{"full_name": "VitaliFamily.frequently_filterAt_iff", "def_path": "Mathlib/MeasureTheory/Covering/VitaliFamily.lean", "def_pos": [275, 9], "def_end_pos": [275, 32]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "gt_iff_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 18]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nx : \u03b1\nhx : x \u2208 {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nx : \u03b1\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nhx : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 a, a \u2208 setsAt v x \u2227 a \u2286 closedBall x \u03b5 \u2227 \u2191\u2191\u03bc a = 0\n\u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5"}, {"tactic": "rcases hx \u03b5 \u03b5pos with \u27e8a, a_sets, ax, \u03bca\u27e9", "annotated_tactic": ["rcases hx \u03b5 \u03b5pos with \u27e8a, a_sets, ax, \u03bca\u27e9", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nx : \u03b1\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nhx : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 a, a \u2208 setsAt v x \u2227 a \u2286 closedBall x \u03b5 \u2227 \u2191\u2191\u03bc a = 0\n\u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nx : \u03b1\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nhx : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 a, a \u2208 setsAt v x \u2227 a \u2286 closedBall x \u03b5 \u2227 \u2191\u2191\u03bc a = 0\na : Set \u03b1\na_sets : a \u2208 setsAt v x\nax : a \u2286 closedBall x \u03b5\n\u03bca : \u2191\u2191\u03bc a = 0\n\u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5"}, {"tactic": "exact \u27e8a, \u27e8a_sets, \u03bca\u27e9, ax\u27e9", "annotated_tactic": ["exact \u27e8a, \u27e8a_sets, \u03bca\u27e9, ax\u27e9", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nx : \u03b1\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nhx : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 a, a \u2208 setsAt v x \u2227 a \u2286 closedBall x \u03b5 \u2227 \u2191\u2191\u03bc a = 0\na : Set \u03b1\na_sets : a \u2208 setsAt v x\nax : a \u2286 closedBall x \u03b5\n\u03bca : \u2191\u2191\u03bc a = 0\n\u22a2 \u2203 a, a \u2208 setsAt v x \u2229 f x \u2227 a \u2286 closedBall x \u03b5", "state_after": "no goals"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nh : FineSubfamilyOn v f s\n\u22a2 \u2211' (x : \u2191(FineSubfamilyOn.index h)), \u2191\u2191\u03bc (FineSubfamilyOn.covering h \u2191x) = \u2211' (x : \u2191(FineSubfamilyOn.index h)), 0", "state_after": "case e_f\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nh : FineSubfamilyOn v f s\n\u22a2 (fun x => \u2191\u2191\u03bc (FineSubfamilyOn.covering h \u2191x)) = fun x => 0"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "case e_f\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nh : FineSubfamilyOn v f s\n\u22a2 (fun x => \u2191\u2191\u03bc (FineSubfamilyOn.covering h \u2191x)) = fun x => 0", "state_after": "case e_f.h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nh : FineSubfamilyOn v f s\nx : \u2191(FineSubfamilyOn.index h)\n\u22a2 \u2191\u2191\u03bc (FineSubfamilyOn.covering h \u2191x) = 0"}, {"tactic": "exact h.covering_mem x.2", "annotated_tactic": ["exact h.covering_mem x.2", []], "state_before": "case e_f.h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nh : FineSubfamilyOn v f s\nx : \u2191(FineSubfamilyOn.index h)\n\u22a2 \u2191\u2191\u03bc (FineSubfamilyOn.covering h \u2191x) = 0", "state_after": "no goals"}, {"tactic": "simp only [tsum_zero, add_zero]", "annotated_tactic": ["simp only [<a>tsum_zero</a>, <a>add_zero</a>]", [{"full_name": "tsum_zero", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [489, 9], "def_end_pos": [489, 18]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : MetricSpace \u03b1\nm0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nv : VitaliFamily \u03bc\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : SecondCountableTopology \u03b1\ns : Set \u03b1 := {x | \u00ac\u2200\u1da0 (a : Set \u03b1) in filterAt v x, 0 < \u2191\u2191\u03bc a}\nhs : s = {x | \u2203\u1da0 (x : Set \u03b1) in filterAt v x, \u2191\u2191\u03bc x = 0}\nf : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03bc a = 0}\nh : FineSubfamilyOn v f s\n\u22a2 \u2211' (x : \u2191(FineSubfamilyOn.index h)), 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/Log.lean", "full_name": "Nat.log_div_mul_self", "start": [216, 1], "end": [222, 59], "traced_tactics": [{"tactic": "cases' le_or_lt b 1 with hb hb", "annotated_tactic": ["cases' <a>le_or_lt</a> b 1 with hb hb", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [340, 9], "def_end_pos": [340, 17]}]], "state_before": "b n : \u2115\n\u22a2 log b (n / b * b) = log b n", "state_after": "case inl\nb n : \u2115\nhb : b \u2264 1\n\u22a2 log b (n / b * b) = log b n\n\ncase inr\nb n : \u2115\nhb : 1 < b\n\u22a2 log b (n / b * b) = log b n"}, {"tactic": "cases' lt_or_le n b with h h", "annotated_tactic": ["cases' <a>lt_or_le</a> n b with h h", [{"full_name": "lt_or_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [336, 9], "def_end_pos": [336, 17]}]], "state_before": "case inr\nb n : \u2115\nhb : 1 < b\n\u22a2 log b (n / b * b) = log b n", "state_after": "case inr.inl\nb n : \u2115\nhb : 1 < b\nh : n < b\n\u22a2 log b (n / b * b) = log b n\n\ncase inr.inr\nb n : \u2115\nhb : 1 < b\nh : b \u2264 n\n\u22a2 log b (n / b * b) = log b n"}, {"tactic": "rw [log_mul_base hb (Nat.div_pos h (zero_le_one.trans_lt hb)).ne', log_div_base,\n  tsub_add_cancel_of_le (succ_le_iff.2 <| log_pos hb h)]", "annotated_tactic": ["rw [<a>log_mul_base</a> hb (<a>Nat.div_pos</a> h (zero_le_one.trans_lt hb)).<a>ne'</a>, <a>log_div_base</a>,\n    <a>tsub_add_cancel_of_le</a> (<a>succ_le_iff</a>.2 <| <a>log_pos</a> hb h)]", [{"full_name": "Nat.log_mul_base", "def_path": "Mathlib/Data/Nat/Log.lean", "def_pos": [169, 9], "def_end_pos": [169, 21]}, {"full_name": "Nat.div_pos", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [659, 19], "def_end_pos": [659, 26]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"full_name": "Nat.log_div_base", "def_path": "Mathlib/Data/Nat/Log.lean", "def_pos": [207, 9], "def_end_pos": [207, 21]}, {"full_name": "tsub_add_cancel_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [30, 9], "def_end_pos": [30, 30]}, {"full_name": "Nat.succ_le_iff", "def_path": "Mathlib/Data/Nat/Basic.lean", "def_pos": [211, 9], "def_end_pos": [211, 20]}, {"full_name": "Nat.log_pos", "def_path": "Mathlib/Data/Nat/Log.lean", "def_pos": [57, 9], "def_end_pos": [57, 16]}]], "state_before": "case inr.inr\nb n : \u2115\nhb : 1 < b\nh : b \u2264 n\n\u22a2 log b (n / b * b) = log b n", "state_after": "no goals"}, {"tactic": "rw [log_of_left_le_one hb, log_of_left_le_one hb]", "annotated_tactic": ["rw [<a>log_of_left_le_one</a> hb, <a>log_of_left_le_one</a> hb]", [{"full_name": "Nat.log_of_left_le_one", "def_path": "Mathlib/Data/Nat/Log.lean", "def_pos": [48, 9], "def_end_pos": [48, 27]}, {"full_name": "Nat.log_of_left_le_one", "def_path": "Mathlib/Data/Nat/Log.lean", "def_pos": [48, 9], "def_end_pos": [48, 27]}]], "state_before": "case inl\nb n : \u2115\nhb : b \u2264 1\n\u22a2 log b (n / b * b) = log b n", "state_after": "no goals"}, {"tactic": "rw [div_eq_of_lt h, zero_mul, log_zero_right, log_of_lt h]", "annotated_tactic": ["rw [<a>div_eq_of_lt</a> h, <a>zero_mul</a>, <a>log_zero_right</a>, <a>log_of_lt</a> h]", [{"full_name": "Nat.div_eq_of_lt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [590, 9], "def_end_pos": [590, 21]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "Nat.log_zero_right", "def_path": "Mathlib/Data/Nat/Log.lean", "def_pos": [72, 9], "def_end_pos": [72, 23]}, {"full_name": "Nat.log_of_lt", "def_path": "Mathlib/Data/Nat/Log.lean", "def_pos": [44, 9], "def_end_pos": [44, 18]}]], "state_before": "case inr.inl\nb n : \u2115\nhb : 1 < b\nh : n < b\n\u22a2 log b (n / b * b) = log b n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/OrderOfElement.lean", "full_name": "Commute.orderOf_mul_dvd_lcm", "start": [389, 1], "end": [391, 83], "traced_tactics": [{"tactic": "rw [orderOf, \u2190 comp_mul_left]", "annotated_tactic": ["rw [<a>orderOf</a>, \u2190 <a>comp_mul_left</a>]", [{"full_name": "orderOf", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [136, 19], "def_end_pos": [136, 26]}, {"full_name": "comp_mul_left", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [34, 9], "def_end_pos": [34, 22]}]], "state_before": "G : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\n\u22a2 orderOf (x * y) \u2223 lcm (orderOf x) (orderOf y)", "state_after": "G : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\n\u22a2 minimalPeriod ((fun x_1 => x * x_1) \u2218 fun x => y * x) 1 \u2223 lcm (orderOf x) (orderOf y)"}, {"tactic": "exact Function.Commute.minimalPeriod_of_comp_dvd_lcm h.function_commute_mul_left", "annotated_tactic": ["exact <a>Function.Commute.minimalPeriod_of_comp_dvd_lcm</a> h.function_commute_mul_left", [{"full_name": "Function.Commute.minimalPeriod_of_comp_dvd_lcm", "def_path": "Mathlib/Dynamics/PeriodicPts.lean", "def_pos": [426, 9], "def_end_pos": [426, 46]}]], "state_before": "G : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\n\u22a2 minimalPeriod ((fun x_1 => x * x_1) \u2218 fun x => y * x) 1 \u2223 lcm (orderOf x) (orderOf y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Trace.lean", "full_name": "PowerBasis.trace_gen_eq_sum_roots", "start": [246, 1], "end": [253, 13], "traced_tactics": [{"tactic": "rw [PowerBasis.trace_gen_eq_nextCoeff_minpoly, RingHom.map_neg, \u2190\n  nextCoeff_map (algebraMap K F).injective,\n  sum_roots_eq_nextCoeff_of_monic_of_split ((minpoly.monic (PowerBasis.isIntegral_gen _)).map _)\n    ((splits_id_iff_splits _).2 hf),\n  neg_neg]", "annotated_tactic": ["rw [<a>PowerBasis.trace_gen_eq_nextCoeff_minpoly</a>, <a>RingHom.map_neg</a>, \u2190\n    <a>nextCoeff_map</a> (<a>algebraMap</a> K F).<a>injective</a>,\n    <a>sum_roots_eq_nextCoeff_of_monic_of_split</a> ((<a>minpoly.monic</a> (<a>PowerBasis.isIntegral_gen</a> _)).<a>map</a> _)\n      ((<a>splits_id_iff_splits</a> _).2 hf),\n    <a>neg_neg</a>]", [{"full_name": 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"PowerBasis.isIntegral_gen", "def_path": "Mathlib/RingTheory/PowerBasis.lean", "def_pos": [207, 9], "def_end_pos": [207, 23]}, {"full_name": "Polynomial.Monic.map", "def_path": "Mathlib/Data/Polynomial/Monic.lean", "def_pos": [65, 9], "def_end_pos": [65, 18]}, {"full_name": "Polynomial.splits_id_iff_splits", "def_path": "Mathlib/Data/Polynomial/Splits.lean", "def_pos": [171, 9], "def_end_pos": [171, 29]}, {"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [799, 3], "def_end_pos": [799, 14]}]], "state_before": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2074 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K S\ninst\u271d\u00b9 : Algebra K F\ninst\u271d : Nontrivial S\npb : PowerBasis K S\nhf : Splits (algebraMap K F) (minpoly K pb.gen)\n\u22a2 \u2191(algebraMap K F) (\u2191(Algebra.trace K S) pb.gen) = Multiset.sum (aroots (minpoly K pb.gen) F)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/ModEq.lean", "full_name": "Nat.ModEq.add", "start": [144, 11], "end": [146, 30], "traced_tactics": [{"tactic": "rw [modEq_iff_dvd, Int.ofNat_add, Int.ofNat_add, add_sub_add_comm]", "annotated_tactic": ["rw [<a>modEq_iff_dvd</a>, <a>Int.ofNat_add</a>, <a>Int.ofNat_add</a>, <a>add_sub_add_comm</a>]", [{"full_name": "Nat.modEq_iff_dvd", "def_path": "Mathlib/Data/Nat/ModEq.lean", "def_pos": [88, 9], "def_end_pos": [88, 22]}, {"full_name": "Int.ofNat_add", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [35, 9], "def_end_pos": [35, 18]}, {"full_name": "Int.ofNat_add", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [35, 9], "def_end_pos": [35, 18]}, {"full_name": "add_sub_add_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [586, 3], "def_end_pos": [586, 14]}]], "state_before": "m n a b c d : \u2115\nh\u2081 : a \u2261 b [MOD n]\nh\u2082 : c \u2261 d [MOD n]\n\u22a2 a + c \u2261 b + d [MOD n]", "state_after": "m n a b c d : \u2115\nh\u2081 : a \u2261 b [MOD n]\nh\u2082 : c \u2261 d [MOD n]\n\u22a2 \u2191n \u2223 \u2191b - \u2191a + (\u2191d - \u2191c)"}, {"tactic": "exact dvd_add h\u2081.dvd h\u2082.dvd", "annotated_tactic": ["exact <a>dvd_add</a> h\u2081.dvd h\u2082.dvd", [{"full_name": "dvd_add", "def_path": "Mathlib/Algebra/Ring/Divisibility/Basic.lean", "def_pos": [27, 9], "def_end_pos": [27, 16]}]], "state_before": "m n a b c d : \u2115\nh\u2081 : a \u2261 b [MOD n]\nh\u2082 : c \u2261 d [MOD n]\n\u22a2 \u2191n \u2223 \u2191b - \u2191a + (\u2191d - \u2191c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean", "full_name": "span_gramSchmidt_Iio", "start": [165, 1], "end": [170, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/SimpleGraph/Subgraph.lean", "full_name": "SimpleGraph.Subgraph.edgeSet_sSup", "start": [548, 1], "end": [551, 7], "traced_tactics": [{"tactic": "ext e", "annotated_tactic": ["ext e", []], "state_before": "\u03b9 : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG\u2081 G\u2082 : Subgraph G\na b : V\ns : Set (Subgraph G)\n\u22a2 edgeSet (sSup s) = \u22c3 G' \u2208 s, edgeSet G'", "state_after": "case h\n\u03b9 : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG\u2081 G\u2082 : Subgraph G\na b : V\ns : Set (Subgraph G)\ne : Sym2 V\n\u22a2 e \u2208 edgeSet (sSup s) \u2194 e \u2208 \u22c3 G' \u2208 s, edgeSet G'"}, {"tactic": "induction e using Sym2.ind", "annotated_tactic": ["induction e using <a>Sym2.ind</a>", [{"full_name": "Sym2.ind", "def_path": "Mathlib/Data/Sym/Sym2.lean", "def_pos": [110, 19], "def_end_pos": [110, 22]}]], "state_before": "case h\n\u03b9 : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG\u2081 G\u2082 : Subgraph G\na b : V\ns : Set (Subgraph G)\ne : Sym2 V\n\u22a2 e \u2208 edgeSet (sSup s) \u2194 e \u2208 \u22c3 G' \u2208 s, edgeSet G'", "state_after": "case h.h\n\u03b9 : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG\u2081 G\u2082 : Subgraph G\na b : V\ns : Set (Subgraph G)\nx\u271d y\u271d : V\n\u22a2 Quotient.mk (Sym2.Rel.setoid V) (x\u271d, y\u271d) \u2208 edgeSet (sSup s) \u2194\n    Quotient.mk (Sym2.Rel.setoid V) (x\u271d, y\u271d) \u2208 \u22c3 G' \u2208 s, edgeSet G'"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.h\n\u03b9 : Sort 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"ZMod.cast_eq_val", "start": [169, 1], "end": [172, 6], "traced_tactics": [{"tactic": "cases n", "annotated_tactic": ["cases n", []], "state_before": "n : \u2115\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\ninst\u271d : NeZero n\na : ZMod n\n\u22a2 \u2191a = \u2191(val a)", "state_after": "case zero\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\ninst\u271d : NeZero Nat.zero\na : ZMod Nat.zero\n\u22a2 \u2191a = \u2191(val a)\n\ncase succ\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\nn\u271d : \u2115\ninst\u271d : NeZero (Nat.succ n\u271d)\na : ZMod (Nat.succ n\u271d)\n\u22a2 \u2191a = \u2191(val a)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case succ\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\nn\u271d : \u2115\ninst\u271d : NeZero (Nat.succ n\u271d)\na : ZMod (Nat.succ n\u271d)\n\u22a2 \u2191a = \u2191(val a)", "state_after": "no goals"}, {"tactic": "cases NeZero.ne 0 rfl", "annotated_tactic": ["cases <a>NeZero.ne</a> 0 <a>rfl</a>", [{"full_name": "NeZero.ne", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [30, 9], "def_end_pos": [30, 18]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case zero\nR : Type u_1\ninst\u271d\u00b9 : AddGroupWithOne R\ninst\u271d : NeZero Nat.zero\na : ZMod Nat.zero\n\u22a2 \u2191a = \u2191(val a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Finiteness.lean", "full_name": "Ideal.exists_radical_pow_le_of_fg", "start": [495, 1], "end": [513, 29], "traced_tactics": [{"tactic": "have := le_refl I.radical", "annotated_tactic": ["have := <a>le_refl</a> I.radical", [{"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}]], "state_before": "R\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\n\u22a2 \u2203 n, radical I ^ n \u2264 I", "state_after": "R\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\nthis : radical I \u2264 radical I\n\u22a2 \u2203 n, radical I ^ n \u2264 I"}, {"tactic": "revert this", "annotated_tactic": ["revert this", []], "state_before": "R\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\nthis : radical I \u2264 radical I\n\u22a2 \u2203 n, radical I ^ n \u2264 I", "state_after": "R\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\n\u22a2 radical I \u2264 radical I \u2192 \u2203 n, radical I ^ n \u2264 I"}, {"tactic": "refine' Submodule.fg_induction _ _ (fun J => J \u2264 I.radical \u2192 \u2203 n : \u2115, J ^ n \u2264 I) _ _ _ h", "annotated_tactic": ["refine' <a>Submodule.fg_induction</a> _ _ (fun J => J \u2264 I.radical \u2192 \u2203 n : \u2115, J ^ n \u2264 I) _ _ _ h", [{"full_name": "Submodule.fg_induction", "def_path": "Mathlib/RingTheory/Finiteness.lean", "def_pos": [336, 9], "def_end_pos": [336, 21]}]], "state_before": "R\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\n\u22a2 radical I \u2264 radical I \u2192 \u2203 n, radical I ^ n \u2264 I", "state_after": "case refine'_1\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\n\u22a2 \u2200 (x : R), (fun J => J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I) (Submodule.span R {x})\n\ncase refine'_2\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\n\u22a2 \u2200 (M\u2081 M\u2082 : Submodule R R),\n    (fun J => J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I) M\u2081 \u2192\n      (fun J => J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I) M\u2082 \u2192 (fun J => J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I) (M\u2081 \u2294 M\u2082)"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case refine'_1\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\n\u22a2 \u2200 (x : R), (fun J => J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I) (Submodule.span R {x})", "state_after": "case refine'_1\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\nx : R\nhx : Submodule.span R {x} \u2264 radical I\n\u22a2 \u2203 n, Submodule.span R {x} ^ n \u2264 I"}, {"tactic": "obtain \u27e8n, hn\u27e9 := hx (subset_span (Set.mem_singleton x))", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 := hx (<a>subset_span</a> (<a>Set.mem_singleton</a> x))", [{"full_name": "Ideal.subset_span", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [140, 9], "def_end_pos": [140, 20]}, {"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1289, 9], "def_end_pos": [1289, 22]}]], "state_before": "case refine'_1\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\nx : R\nhx : Submodule.span R {x} \u2264 radical I\n\u22a2 \u2203 n, Submodule.span R {x} ^ n \u2264 I", "state_after": "case refine'_1.intro\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\nx : R\nhx : Submodule.span R {x} \u2264 radical I\nn : \u2115\nhn : x ^ n \u2208 I\n\u22a2 \u2203 n, Submodule.span R {x} ^ n \u2264 I"}, {"tactic": "exact \u27e8n, by rwa [\u2190 Ideal.span, span_singleton_pow, span_le, Set.singleton_subset_iff]\u27e9", "annotated_tactic": ["exact \u27e8n, by rwa [\u2190 <a>Ideal.span</a>, <a>span_singleton_pow</a>, <a>span_le</a>, <a>Set.singleton_subset_iff</a>]\u27e9", [{"full_name": "Ideal.span", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [109, 5], "def_end_pos": [109, 9]}, {"full_name": "Ideal.span_singleton_pow", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [533, 9], "def_end_pos": [533, 27]}, {"full_name": "Ideal.span_le", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [144, 9], "def_end_pos": [144, 16]}, {"full_name": "Set.singleton_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1330, 9], "def_end_pos": [1330, 29]}]], "state_before": "case refine'_1.intro\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\nx : R\nhx : Submodule.span R {x} \u2264 radical I\nn : \u2115\nhn : x ^ n \u2208 I\n\u22a2 \u2203 n, Submodule.span R {x} ^ n \u2264 I", "state_after": "no goals"}, {"tactic": "rwa [\u2190 Ideal.span, span_singleton_pow, span_le, Set.singleton_subset_iff]", "annotated_tactic": ["rwa [\u2190 <a>Ideal.span</a>, <a>span_singleton_pow</a>, <a>span_le</a>, <a>Set.singleton_subset_iff</a>]", [{"full_name": "Ideal.span", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [109, 5], "def_end_pos": [109, 9]}, {"full_name": "Ideal.span_singleton_pow", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [533, 9], "def_end_pos": [533, 27]}, {"full_name": "Ideal.span_le", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [144, 9], "def_end_pos": [144, 16]}, {"full_name": "Set.singleton_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1330, 9], "def_end_pos": [1330, 29]}]], "state_before": "R\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\nx : R\nhx : Submodule.span R {x} \u2264 radical I\nn : \u2115\nhn : x ^ n \u2208 I\n\u22a2 Submodule.span R {x} ^ n \u2264 I", "state_after": "no goals"}, {"tactic": "intro J K hJ hK hJK", "annotated_tactic": ["intro J K hJ hK hJK", []], "state_before": "case refine'_2\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\n\u22a2 \u2200 (M\u2081 M\u2082 : Submodule R R),\n    (fun J => J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I) M\u2081 \u2192\n      (fun J => J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I) M\u2082 \u2192 (fun J => J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I) (M\u2081 \u2294 M\u2082)", "state_after": "case refine'_2\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\nJ K : Submodule R R\nhJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I\nhK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I\nhJK : J \u2294 K \u2264 radical I\n\u22a2 \u2203 n, (J \u2294 K) ^ n \u2264 I"}, {"tactic": "obtain \u27e8n, hn\u27e9 := hJ fun x hx => hJK <| Ideal.mem_sup_left hx", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 := hJ fun x hx => hJK <| <a>Ideal.mem_sup_left</a> hx", [{"full_name": "Ideal.mem_sup_left", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [406, 9], "def_end_pos": [406, 21]}]], "state_before": "case refine'_2\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\nJ K : Submodule R R\nhJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I\nhK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I\nhJK : J \u2294 K \u2264 radical I\n\u22a2 \u2203 n, (J \u2294 K) ^ n \u2264 I", "state_after": "case refine'_2.intro\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\nJ K : Submodule R R\nhJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I\nhK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I\nhJK : J \u2294 K \u2264 radical I\nn : \u2115\nhn : J ^ n \u2264 I\n\u22a2 \u2203 n, (J \u2294 K) ^ n \u2264 I"}, {"tactic": "obtain \u27e8m, hm\u27e9 := hK fun x hx => hJK <| Ideal.mem_sup_right hx", "annotated_tactic": ["obtain \u27e8m, hm\u27e9 := hK fun x hx => hJK <| <a>Ideal.mem_sup_right</a> hx", [{"full_name": "Ideal.mem_sup_right", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [410, 9], "def_end_pos": [410, 22]}]], "state_before": "case refine'_2.intro\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\nJ K : Submodule R R\nhJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I\nhK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I\nhJK : J \u2294 K \u2264 radical I\nn : \u2115\nhn : J ^ n \u2264 I\n\u22a2 \u2203 n, (J \u2294 K) ^ n \u2264 I", "state_after": "case refine'_2.intro.intro\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\nJ K : Submodule R R\nhJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I\nhK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I\nhJK : J \u2294 K \u2264 radical I\nn : \u2115\nhn : J ^ n \u2264 I\nm : \u2115\nhm : K ^ m \u2264 I\n\u22a2 \u2203 n, (J \u2294 K) ^ n \u2264 I"}, {"tactic": "use n + m", "annotated_tactic": ["use n + m", []], "state_before": "case refine'_2.intro.intro\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\nJ K : Submodule R R\nhJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I\nhK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I\nhJK : J \u2294 K \u2264 radical I\nn : \u2115\nhn : J ^ n \u2264 I\nm : \u2115\nhm : K ^ m \u2264 I\n\u22a2 \u2203 n, (J \u2294 K) ^ n \u2264 I", "state_after": "case h\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\nJ K : Submodule R R\nhJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I\nhK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I\nhJK : J \u2294 K \u2264 radical I\nn : \u2115\nhn : J ^ n \u2264 I\nm : \u2115\nhm : K ^ m \u2264 I\n\u22a2 (J \u2294 K) ^ (n + m) \u2264 I"}, {"tactic": "rw [\u2190 Ideal.add_eq_sup, add_pow, Ideal.sum_eq_sup, Finset.sup_le_iff]", "annotated_tactic": ["rw [\u2190 <a>Ideal.add_eq_sup</a>, <a>add_pow</a>, <a>Ideal.sum_eq_sup</a>, <a>Finset.sup_le_iff</a>]", [{"full_name": "Ideal.add_eq_sup", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [415, 9], "def_end_pos": [415, 19]}, {"full_name": "add_pow", "def_path": "Mathlib/Data/Nat/Choose/Sum.lean", "def_pos": [83, 9], "def_end_pos": [83, 16]}, {"full_name": "Ideal.sum_eq_sup", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [425, 9], "def_end_pos": [425, 19]}, {"full_name": "Finset.sup_le_iff", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [99, 19], "def_end_pos": [99, 29]}]], "state_before": "case h\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\nJ K : Submodule R R\nhJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I\nhK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I\nhJK : J \u2294 K \u2264 radical I\nn : \u2115\nhn : J ^ n \u2264 I\nm : \u2115\nhm : K ^ m \u2264 I\n\u22a2 (J \u2294 K) ^ (n + m) \u2264 I", "state_after": "case h\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\nJ K : Submodule R R\nhJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I\nhK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I\nhJK : J \u2294 K \u2264 radical I\nn : \u2115\nhn : J ^ n \u2264 I\nm : \u2115\nhm : K ^ m \u2264 I\n\u22a2 \u2200 (b : \u2115), b \u2208 Finset.range (n + m + 1) \u2192 J ^ b * K ^ (n + m - b) * \u2191(Nat.choose (n + m) b) \u2264 I"}, {"tactic": "refine' fun i _ => Ideal.mul_le_right.trans _", "annotated_tactic": ["refine' fun i _ => Ideal.mul_le_right.trans _", []], "state_before": "case h\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\nJ K : Submodule R R\nhJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I\nhK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I\nhJK : J \u2294 K \u2264 radical I\nn : \u2115\nhn : J ^ n \u2264 I\nm : \u2115\nhm : K ^ m \u2264 I\n\u22a2 \u2200 (b : \u2115), b \u2208 Finset.range (n + m + 1) \u2192 J ^ b * K ^ (n + m - b) * \u2191(Nat.choose (n + m) b) \u2264 I", "state_after": "case h\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\nJ K : Submodule R R\nhJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I\nhK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I\nhJK : J \u2294 K \u2264 radical I\nn : \u2115\nhn : J ^ n \u2264 I\nm : \u2115\nhm : K ^ m \u2264 I\ni : \u2115\nx\u271d : i \u2208 Finset.range (n + m + 1)\n\u22a2 J ^ i * K ^ (n + m - i) \u2264 I"}, {"tactic": "obtain h | h := le_or_lt n i", "annotated_tactic": ["obtain h | h := <a>le_or_lt</a> n i", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [340, 9], "def_end_pos": [340, 17]}]], "state_before": "case h\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh : FG (radical I)\nJ K : Submodule R R\nhJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I\nhK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I\nhJK : J \u2294 K \u2264 radical I\nn : \u2115\nhn : J ^ n \u2264 I\nm : \u2115\nhm : K ^ m \u2264 I\ni : \u2115\nx\u271d : i \u2208 Finset.range (n + m + 1)\n\u22a2 J ^ i * K ^ (n + m - i) \u2264 I", "state_after": "case h.inl\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh\u271d : FG (radical I)\nJ K : Submodule R R\nhJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I\nhK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I\nhJK : J \u2294 K \u2264 radical I\nn : \u2115\nhn : J ^ n \u2264 I\nm : \u2115\nhm : K ^ m \u2264 I\ni : \u2115\nx\u271d : i \u2208 Finset.range (n + m + 1)\nh : n \u2264 i\n\u22a2 J ^ i * K ^ (n + m - i) \u2264 I\n\ncase h.inr\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh\u271d : FG (radical I)\nJ K : Submodule R R\nhJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I\nhK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I\nhJK : J \u2294 K \u2264 radical I\nn : \u2115\nhn : J ^ n \u2264 I\nm : \u2115\nhm : K ^ m \u2264 I\ni : \u2115\nx\u271d : i \u2208 Finset.range (n + m + 1)\nh : i < n\n\u22a2 J ^ i * K ^ (n + m - i) \u2264 I"}, {"tactic": "apply Ideal.mul_le_right.trans ((Ideal.pow_le_pow h).trans hn)", "annotated_tactic": ["apply Ideal.mul_le_right.trans ((<a>Ideal.pow_le_pow</a> h).<a>trans</a> hn)", [{"full_name": "Ideal.pow_le_pow", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [793, 9], "def_end_pos": [793, 19]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case h.inl\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh\u271d : FG (radical I)\nJ K : Submodule R R\nhJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I\nhK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I\nhJK : J \u2294 K \u2264 radical I\nn : \u2115\nhn : J ^ n \u2264 I\nm : \u2115\nhm : K ^ m \u2264 I\ni : \u2115\nx\u271d : i \u2208 Finset.range (n + m + 1)\nh : n \u2264 i\n\u22a2 J ^ i * K ^ (n + m - i) \u2264 I", "state_after": "no goals"}, {"tactic": "apply Ideal.mul_le_left.trans", "annotated_tactic": ["apply Ideal.mul_le_left.trans", []], "state_before": "case h.inr\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh\u271d : FG (radical I)\nJ K : Submodule R R\nhJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I\nhK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I\nhJK : J \u2294 K \u2264 radical I\nn : \u2115\nhn : J ^ n \u2264 I\nm : \u2115\nhm : K ^ m \u2264 I\ni : \u2115\nx\u271d : i \u2208 Finset.range (n + m + 1)\nh : i < n\n\u22a2 J ^ i * K ^ (n + m - i) \u2264 I", "state_after": "case h.inr\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh\u271d : FG (radical I)\nJ K : Submodule R R\nhJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I\nhK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I\nhJK : J \u2294 K \u2264 radical I\nn : \u2115\nhn : J ^ n \u2264 I\nm : \u2115\nhm : K ^ m \u2264 I\ni : \u2115\nx\u271d : i \u2208 Finset.range (n + m + 1)\nh : i < n\n\u22a2 K ^ (n + m - i) \u2264 I"}, {"tactic": "refine' (Ideal.pow_le_pow _).trans hm", "annotated_tactic": ["refine' (<a>Ideal.pow_le_pow</a> _).<a>trans</a> hm", [{"full_name": "Ideal.pow_le_pow", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [793, 9], "def_end_pos": [793, 19]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case h.inr\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh\u271d : FG (radical I)\nJ K : Submodule R R\nhJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I\nhK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I\nhJK : J \u2294 K \u2264 radical I\nn : \u2115\nhn : J ^ n \u2264 I\nm : \u2115\nhm : K ^ m \u2264 I\ni : \u2115\nx\u271d : i \u2208 Finset.range (n + m + 1)\nh : i < n\n\u22a2 K ^ (n + m - i) \u2264 I", "state_after": "case h.inr\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh\u271d : FG (radical I)\nJ K : Submodule R R\nhJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I\nhK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I\nhJK : J \u2294 K \u2264 radical I\nn : \u2115\nhn : J ^ n \u2264 I\nm : \u2115\nhm : K ^ m \u2264 I\ni : \u2115\nx\u271d : i \u2208 Finset.range (n + m + 1)\nh : i < n\n\u22a2 m \u2264 n + m - i"}, {"tactic": "rw [add_comm, Nat.add_sub_assoc h.le]", "annotated_tactic": ["rw [<a>add_comm</a>, <a>Nat.add_sub_assoc</a> h.le]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "Nat.add_sub_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [602, 19], "def_end_pos": [602, 32]}]], "state_before": "case h.inr\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh\u271d : FG (radical I)\nJ K : Submodule R R\nhJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I\nhK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I\nhJK : J \u2294 K \u2264 radical I\nn : \u2115\nhn : J ^ n \u2264 I\nm : \u2115\nhm : K ^ m \u2264 I\ni : \u2115\nx\u271d : i \u2208 Finset.range (n + m + 1)\nh : i < n\n\u22a2 m \u2264 n + m - i", "state_after": "case h.inr\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh\u271d : FG (radical I)\nJ K : Submodule R R\nhJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I\nhK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I\nhJK : J \u2294 K \u2264 radical I\nn : \u2115\nhn : J ^ n \u2264 I\nm : \u2115\nhm : K ^ m \u2264 I\ni : \u2115\nx\u271d : i \u2208 Finset.range (n + m + 1)\nh : i < n\n\u22a2 m \u2264 m + (n - i)"}, {"tactic": "apply Nat.le_add_right", "annotated_tactic": ["apply <a>Nat.le_add_right</a>", [{"full_name": "Nat.le_add_right", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 21]}]], "state_before": "case h.inr\nR\u271d : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\u271d\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R\u271d M\nR : Type u_3\ninst\u271d : CommSemiring R\nI : Ideal R\nh\u271d : FG (radical I)\nJ K : Submodule R R\nhJ : J \u2264 radical I \u2192 \u2203 n, J ^ n \u2264 I\nhK : K \u2264 radical I \u2192 \u2203 n, K ^ n \u2264 I\nhJK : J \u2294 K \u2264 radical I\nn : \u2115\nhn : J ^ n \u2264 I\nm : \u2115\nhm : K ^ m \u2264 I\ni : \u2115\nx\u271d : i \u2208 Finset.range (n + m + 1)\nh : i < n\n\u22a2 m \u2264 m + (n - i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "full_name": "MulAction.fixed_eq_iInter_fixedBy", "start": [122, 1], "end": [124, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Instances/Nat.lean", "full_name": "Nat.pairwise_one_le_dist", "start": [35, 1], "end": [36, 52], "traced_tactics": [{"tactic": "exact_mod_cast hne", "annotated_tactic": ["exact_mod_cast hne", []], "state_before": "m n : \u2115\nhne : m \u2260 n\n\u22a2 \u2191m \u2260 \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Semicontinuous.lean", "full_name": "upperSemicontinuousOn_const", "start": [720, 1], "end": [721, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Pointwise.lean", "full_name": "Filter.isUnit_iff", "start": [770, 1], "end": [778, 20], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\n\u03b5 : Type u_6\ninst\u271d : DivisionMonoid \u03b1\nf g : Filter \u03b1\n\u22a2 IsUnit f \u2194 \u2203 a, f = pure a \u2227 IsUnit a", "state_after": "case mp\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\n\u03b5 : Type u_6\ninst\u271d : DivisionMonoid \u03b1\nf g : Filter \u03b1\n\u22a2 IsUnit f \u2192 \u2203 a, f = pure a \u2227 IsUnit a\n\ncase mpr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\n\u03b5 : Type u_6\ninst\u271d : DivisionMonoid \u03b1\nf g : Filter \u03b1\n\u22a2 (\u2203 a, f = pure a \u2227 IsUnit a) \u2192 IsUnit f"}, {"tactic": "rintro \u27e8u, rfl\u27e9", "annotated_tactic": ["rintro \u27e8u, rfl\u27e9", []], "state_before": "case mp\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\n\u03b5 : Type u_6\ninst\u271d : DivisionMonoid \u03b1\nf g : Filter \u03b1\n\u22a2 IsUnit f \u2192 \u2203 a, f = pure a \u2227 IsUnit a", "state_after": "case mp.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\n\u03b5 : Type u_6\ninst\u271d : DivisionMonoid \u03b1\ng : Filter \u03b1\nu : (Filter \u03b1)\u02e3\n\u22a2 \u2203 a, \u2191u = pure a \u2227 IsUnit a"}, {"tactic": "obtain \u27e8a, b, ha, hb, h\u27e9 := Filter.mul_eq_one_iff.1 u.mul_inv", "annotated_tactic": ["obtain \u27e8a, b, ha, hb, h\u27e9 := <a>Filter.mul_eq_one_iff</a>.1 u.mul_inv", [{"full_name": "Filter.mul_eq_one_iff", "def_path": "Mathlib/Order/Filter/Pointwise.lean", "def_pos": [741, 19], "def_end_pos": [741, 33]}]], "state_before": "case mp.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\n\u03b5 : Type u_6\ninst\u271d : DivisionMonoid \u03b1\ng : Filter \u03b1\nu : (Filter \u03b1)\u02e3\n\u22a2 \u2203 a, \u2191u = pure a \u2227 IsUnit a", "state_after": "case mp.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\n\u03b5 : Type u_6\ninst\u271d : DivisionMonoid \u03b1\ng : Filter \u03b1\nu : (Filter \u03b1)\u02e3\na b : \u03b1\nha : \u2191u = pure a\nhb : \u2191u\u207b\u00b9 = pure b\nh : a * b = 1\n\u22a2 \u2203 a, \u2191u = pure a \u2227 IsUnit a"}, {"tactic": "refine' \u27e8a, ha, \u27e8a, b, h, pure_injective _\u27e9, rfl\u27e9", "annotated_tactic": ["refine' \u27e8a, ha, \u27e8a, b, h, <a>pure_injective</a> _\u27e9, <a>rfl</a>\u27e9", [{"full_name": "Filter.pure_injective", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2675, 9], "def_end_pos": [2675, 23]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case mp.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\n\u03b5 : Type u_6\ninst\u271d : DivisionMonoid \u03b1\ng : Filter \u03b1\nu : (Filter \u03b1)\u02e3\na b : \u03b1\nha : \u2191u = pure a\nhb : \u2191u\u207b\u00b9 = pure b\nh : a * b = 1\n\u22a2 \u2203 a, \u2191u = pure a \u2227 IsUnit a", "state_after": "case mp.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\n\u03b5 : Type u_6\ninst\u271d : DivisionMonoid \u03b1\ng : Filter \u03b1\nu : (Filter \u03b1)\u02e3\na b : \u03b1\nha : \u2191u = pure a\nhb : \u2191u\u207b\u00b9 = pure b\nh : a * b = 1\n\u22a2 pure (b * a) = pure 1"}, {"tactic": "rw [\u2190 pure_mul_pure, \u2190 ha, \u2190 hb]", "annotated_tactic": ["rw [\u2190 <a>pure_mul_pure</a>, \u2190 ha, \u2190 hb]", [{"full_name": "Filter.pure_mul_pure", "def_path": "Mathlib/Order/Filter/Pointwise.lean", "def_pos": [362, 9], "def_end_pos": [362, 22]}]], "state_before": "case mp.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\n\u03b5 : Type u_6\ninst\u271d : DivisionMonoid \u03b1\ng : Filter \u03b1\nu : (Filter \u03b1)\u02e3\na b : \u03b1\nha : \u2191u = pure a\nhb : \u2191u\u207b\u00b9 = pure b\nh : a * b = 1\n\u22a2 pure (b * a) = pure 1", "state_after": "case mp.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\n\u03b5 : Type u_6\ninst\u271d : DivisionMonoid \u03b1\ng : Filter \u03b1\nu : (Filter \u03b1)\u02e3\na b : \u03b1\nha : \u2191u = pure a\nhb : \u2191u\u207b\u00b9 = pure b\nh : a * b = 1\n\u22a2 \u2191u\u207b\u00b9 * \u2191u = pure 1"}, {"tactic": "exact u.inv_mul", "annotated_tactic": ["exact u.inv_mul", []], "state_before": "case mp.intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\n\u03b5 : Type u_6\ninst\u271d : DivisionMonoid \u03b1\ng : Filter \u03b1\nu : (Filter \u03b1)\u02e3\na b : \u03b1\nha : \u2191u = pure a\nhb : \u2191u\u207b\u00b9 = pure b\nh : a * b = 1\n\u22a2 \u2191u\u207b\u00b9 * \u2191u = pure 1", "state_after": "no goals"}, {"tactic": "rintro \u27e8a, rfl, ha\u27e9", "annotated_tactic": ["rintro \u27e8a, rfl, ha\u27e9", []], "state_before": "case mpr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\n\u03b5 : Type u_6\ninst\u271d : DivisionMonoid \u03b1\nf g : Filter \u03b1\n\u22a2 (\u2203 a, f = pure a \u2227 IsUnit a) \u2192 IsUnit f", "state_after": "case mpr.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\n\u03b5 : Type u_6\ninst\u271d : DivisionMonoid \u03b1\ng : Filter \u03b1\na : \u03b1\nha : IsUnit a\n\u22a2 IsUnit (pure a)"}, {"tactic": "exact ha.filter", "annotated_tactic": ["exact ha.filter", []], "state_before": "case mpr.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\n\u03b5 : Type u_6\ninst\u271d : DivisionMonoid \u03b1\ng : Filter \u03b1\na : \u03b1\nha : IsUnit a\n\u22a2 IsUnit (pure a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Convergence.lean", "full_name": "MeasureTheory.Submartingale.ae_tendsto_limitProcess", "start": [216, 1], "end": [238, 57], "traced_tactics": [{"tactic": "classical\nsuffices\n    \u2203 g, StronglyMeasurable[\u2a06 n, \u2131 n] g \u2227 \u2200\u1d50 \u03c9 \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9)) by\n  rw [limitProcess, dif_pos this]\n  exact (Classical.choose_spec this).2\nset g' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) then h.choose else 0\nhave hle : \u2a06 n, \u2131 n \u2264 m0 := sSup_le fun m \u27e8n, hn\u27e9 => hn \u25b8 \u2131.le _\nhave hg' : \u2200\u1d50 \u03c9 \u2202\u03bc.trim hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9)) := by\n  filter_upwards [hf.exists_ae_trim_tendsto_of_bdd hbdd] with \u03c9 h\u03c9\n  simp_rw [dif_pos h\u03c9]\n  exact h\u03c9.choose_spec\nhave hg'm : @AEStronglyMeasurable _ _ _ (\u2a06 n, \u2131 n) g' (\u03bc.trim hle) :=\n  (@aemeasurable_of_tendsto_metrizable_ae' _ _ (\u2a06 n, \u2131 n) _ _ _ _ _ _ _\n    (fun n => ((hf.stronglyMeasurable n).measurable.mono (le_sSup \u27e8n, rfl\u27e9 : \u2131 n \u2264 \u2a06 n, \u2131 n)\n      le_rfl).aemeasurable) hg').aestronglyMeasurable\nobtain \u27e8g, hgm, hae\u27e9 := hg'm\nhave hg : \u2200\u1d50 \u03c9 \u2202\u03bc.trim hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9)) := by\n  filter_upwards [hae, hg'] with \u03c9 h\u03c9 hg'\u03c9\n  exact h\u03c9 \u25b8 hg'\u03c9\nexact \u27e8g, hgm, measure_eq_zero_of_trim_eq_zero hle hg\u27e9", "annotated_tactic": ["classical\n  suffices\n      \u2203 g, StronglyMeasurable[\u2a06 n, \u2131 n] g \u2227 \u2200\u1d50 \u03c9 \u2202\u03bc, <a>Tendsto</a> (fun n => f n \u03c9) <a>atTop</a> (\ud835\udcdd (g \u03c9)) by\n    rw [<a>limitProcess</a>, <a>dif_pos</a> this]\n    exact (<a>Classical.choose_spec</a> this).2\n  set g' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, <a>Tendsto</a> (fun n => f n \u03c9) <a>atTop</a> (\ud835\udcdd c) then h.choose else 0\n  have hle : \u2a06 n, \u2131 n \u2264 m0 := <a>sSup_le</a> fun m \u27e8n, hn\u27e9 => hn \u25b8 \u2131.le _\n  have hg' : \u2200\u1d50 \u03c9 \u2202\u03bc.trim hle, <a>Tendsto</a> (fun n => f n \u03c9) <a>atTop</a> (\ud835\udcdd (g' \u03c9)) := by\n    filter_upwards [hf.exists_ae_trim_tendsto_of_bdd hbdd] with \u03c9 h\u03c9\n    simp_rw [<a>dif_pos</a> h\u03c9]\n    exact h\u03c9.choose_spec\n  have hg'm : @<a>AEStronglyMeasurable</a> _ _ _ (\u2a06 n, \u2131 n) g' (\u03bc.trim hle) :=\n    (@<a>aemeasurable_of_tendsto_metrizable_ae'</a> _ _ (\u2a06 n, \u2131 n) _ _ _ _ _ _ _\n      (fun n => ((hf.stronglyMeasurable n).measurable.mono (<a>le_sSup</a> \u27e8n, <a>rfl</a>\u27e9 : \u2131 n \u2264 \u2a06 n, \u2131 n)\n        <a>le_rfl</a>).<a>aemeasurable</a>) hg').<a>aestronglyMeasurable</a>\n  obtain \u27e8g, hgm, hae\u27e9 := hg'm\n  have hg : \u2200\u1d50 \u03c9 \u2202\u03bc.trim hle, <a>Tendsto</a> (fun n => f n \u03c9) <a>atTop</a> (\ud835\udcdd (g \u03c9)) := by\n    filter_upwards [hae, hg'] with \u03c9 h\u03c9 hg'\u03c9\n    exact h\u03c9 \u25b8 hg'\u03c9\n  exact \u27e8g, hgm, <a>measure_eq_zero_of_trim_eq_zero</a> hle hg\u27e9", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "MeasureTheory.Filtration.limitProcess", "def_path": "Mathlib/Probability/Process/Filtration.lean", "def_pos": [321, 19], "def_end_pos": [321, 31]}, {"full_name": "dif_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [807, 9], "def_end_pos": [807, 16]}, {"full_name": "Classical.choose_spec", "def_path": "lake-packages/lean4/src/lean/Init/Classical.lean", "def_pos": [22, 9], "def_end_pos": [22, 20]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "sSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [200, 9], "def_end_pos": [200, 16]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "dif_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [807, 9], "def_end_pos": [807, 16]}, {"full_name": "MeasureTheory.AEStronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [93, 5], "def_end_pos": [93, 25]}, {"full_name": "aemeasurable_of_tendsto_metrizable_ae'", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean", "def_pos": [116, 9], "def_end_pos": [116, 47]}, {"full_name": "le_sSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [196, 9], "def_end_pos": [196, 16]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "Measurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [713, 9], "def_end_pos": [713, 32]}, {"full_name": "AEMeasurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1451, 9], "def_end_pos": [1451, 49]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "MeasureTheory.measure_eq_zero_of_trim_eq_zero", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [62, 9], "def_end_pos": [62, 40]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (limitProcess f \u2131 \u03bc \u03c9))", "state_after": "no goals"}, {"tactic": "suffices\n    \u2203 g, StronglyMeasurable[\u2a06 n, \u2131 n] g \u2227 \u2200\u1d50 \u03c9 \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9)) by\n  rw [limitProcess, dif_pos this]\n  exact (Classical.choose_spec this).2", "annotated_tactic": ["suffices\n      \u2203 g, StronglyMeasurable[\u2a06 n, \u2131 n] g \u2227 \u2200\u1d50 \u03c9 \u2202\u03bc, <a>Tendsto</a> (fun n => f n \u03c9) <a>atTop</a> (\ud835\udcdd (g \u03c9)) by\n    rw [<a>limitProcess</a>, <a>dif_pos</a> this]\n    exact (<a>Classical.choose_spec</a> this).2", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "MeasureTheory.Filtration.limitProcess", "def_path": "Mathlib/Probability/Process/Filtration.lean", "def_pos": [321, 19], "def_end_pos": [321, 31]}, {"full_name": "dif_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [807, 9], "def_end_pos": [807, 16]}, {"full_name": "Classical.choose_spec", "def_path": "lake-packages/lean4/src/lean/Init/Classical.lean", "def_pos": [22, 9], "def_end_pos": [22, 20]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (limitProcess f \u2131 \u03bc \u03c9))", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\n\u22a2 \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9))"}, {"tactic": "set g' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) then h.choose else 0", "annotated_tactic": ["set g' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, <a>Tendsto</a> (fun n => f n \u03c9) <a>atTop</a> (\ud835\udcdd c) then h.choose else 0", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\n\u22a2 \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9))", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\ng' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) then Exists.choose h else 0\n\u22a2 \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9))"}, {"tactic": "have hle : \u2a06 n, \u2131 n \u2264 m0 := sSup_le fun m \u27e8n, hn\u27e9 => hn \u25b8 \u2131.le _", "annotated_tactic": ["have hle : \u2a06 n, \u2131 n \u2264 m0 := <a>sSup_le</a> fun m \u27e8n, hn\u27e9 => hn \u25b8 \u2131.le _", [{"full_name": "sSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [200, 9], "def_end_pos": [200, 16]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\ng' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) then Exists.choose h else 0\n\u22a2 \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9))", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\ng' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) then Exists.choose h else 0\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\n\u22a2 \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9))"}, {"tactic": "have hg' : \u2200\u1d50 \u03c9 \u2202\u03bc.trim hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9)) := by\n  filter_upwards [hf.exists_ae_trim_tendsto_of_bdd hbdd] with \u03c9 h\u03c9\n  simp_rw [dif_pos h\u03c9]\n  exact h\u03c9.choose_spec", "annotated_tactic": ["have hg' : \u2200\u1d50 \u03c9 \u2202\u03bc.trim hle, <a>Tendsto</a> (fun n => f n \u03c9) <a>atTop</a> (\ud835\udcdd (g' \u03c9)) := by\n    filter_upwards [hf.exists_ae_trim_tendsto_of_bdd hbdd] with \u03c9 h\u03c9\n    simp_rw [<a>dif_pos</a> h\u03c9]\n    exact h\u03c9.choose_spec", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "dif_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [807, 9], "def_end_pos": [807, 16]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\ng' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) then Exists.choose h else 0\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\n\u22a2 \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9))", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\ng' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) then Exists.choose h else 0\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhg' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9))\n\u22a2 \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9))"}, {"tactic": "have hg'm : @AEStronglyMeasurable _ _ _ (\u2a06 n, \u2131 n) g' (\u03bc.trim hle) :=\n  (@aemeasurable_of_tendsto_metrizable_ae' _ _ (\u2a06 n, \u2131 n) _ _ _ _ _ _ _\n    (fun n => ((hf.stronglyMeasurable n).measurable.mono (le_sSup \u27e8n, rfl\u27e9 : \u2131 n \u2264 \u2a06 n, \u2131 n)\n      le_rfl).aemeasurable) hg').aestronglyMeasurable", "annotated_tactic": ["have hg'm : @<a>AEStronglyMeasurable</a> _ _ _ (\u2a06 n, \u2131 n) g' (\u03bc.trim hle) :=\n    (@<a>aemeasurable_of_tendsto_metrizable_ae'</a> _ _ (\u2a06 n, \u2131 n) _ _ _ _ _ _ _\n      (fun n => ((hf.stronglyMeasurable n).measurable.mono (<a>le_sSup</a> \u27e8n, <a>rfl</a>\u27e9 : \u2131 n \u2264 \u2a06 n, \u2131 n)\n        <a>le_rfl</a>).<a>aemeasurable</a>) hg').<a>aestronglyMeasurable</a>", [{"full_name": "MeasureTheory.AEStronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [93, 5], "def_end_pos": [93, 25]}, {"full_name": "aemeasurable_of_tendsto_metrizable_ae'", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean", "def_pos": [116, 9], "def_end_pos": [116, 47]}, {"full_name": "le_sSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [196, 9], "def_end_pos": [196, 16]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "Measurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [713, 9], "def_end_pos": [713, 32]}, {"full_name": "AEMeasurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1451, 9], "def_end_pos": [1451, 49]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\ng' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) then Exists.choose h else 0\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhg' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9))\n\u22a2 \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9))", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\ng' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) then Exists.choose h else 0\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhg' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9))\nhg'm : AEStronglyMeasurable g' (Measure.trim \u03bc hle)\n\u22a2 \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9))"}, {"tactic": "obtain \u27e8g, hgm, hae\u27e9 := hg'm", "annotated_tactic": ["obtain \u27e8g, hgm, hae\u27e9 := hg'm", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\ng' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) then Exists.choose h else 0\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhg' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9))\nhg'm : AEStronglyMeasurable g' (Measure.trim \u03bc hle)\n\u22a2 \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9))", "state_after": "case intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\ng' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) then Exists.choose h else 0\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhg' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9))\ng : \u03a9 \u2192 \u211d\nhgm : StronglyMeasurable g\nhae : g' =\u1d50[Measure.trim \u03bc hle] g\n\u22a2 \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9))"}, {"tactic": "have hg : \u2200\u1d50 \u03c9 \u2202\u03bc.trim hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9)) := by\n  filter_upwards [hae, hg'] with \u03c9 h\u03c9 hg'\u03c9\n  exact h\u03c9 \u25b8 hg'\u03c9", "annotated_tactic": ["have hg : \u2200\u1d50 \u03c9 \u2202\u03bc.trim hle, <a>Tendsto</a> (fun n => f n \u03c9) <a>atTop</a> (\ud835\udcdd (g \u03c9)) := by\n    filter_upwards [hae, hg'] with \u03c9 h\u03c9 hg'\u03c9\n    exact h\u03c9 \u25b8 hg'\u03c9", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}]], "state_before": "case intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\ng' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) then Exists.choose h else 0\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhg' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9))\ng : \u03a9 \u2192 \u211d\nhgm : StronglyMeasurable g\nhae : g' =\u1d50[Measure.trim \u03bc hle] g\n\u22a2 \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9))", "state_after": "case intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\ng' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) then Exists.choose h else 0\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhg' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9))\ng : \u03a9 \u2192 \u211d\nhgm : StronglyMeasurable g\nhae : g' =\u1d50[Measure.trim \u03bc hle] g\nhg : \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9))\n\u22a2 \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9))"}, {"tactic": "exact \u27e8g, hgm, measure_eq_zero_of_trim_eq_zero hle hg\u27e9", "annotated_tactic": ["exact \u27e8g, hgm, <a>measure_eq_zero_of_trim_eq_zero</a> hle hg\u27e9", [{"full_name": "MeasureTheory.measure_eq_zero_of_trim_eq_zero", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [62, 9], "def_end_pos": [62, 40]}]], "state_before": "case intro.intro\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\ng' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) then Exists.choose h else 0\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhg' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9))\ng : \u03a9 \u2192 \u211d\nhgm : StronglyMeasurable g\nhae : g' =\u1d50[Measure.trim \u03bc hle] g\nhg : \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9))\n\u22a2 \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9))", "state_after": "no goals"}, {"tactic": "rw [limitProcess, dif_pos this]", "annotated_tactic": ["rw [<a>limitProcess</a>, <a>dif_pos</a> this]", [{"full_name": "MeasureTheory.Filtration.limitProcess", "def_path": "Mathlib/Probability/Process/Filtration.lean", "def_pos": [321, 19], "def_end_pos": [321, 31]}, {"full_name": "dif_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [807, 9], "def_end_pos": [807, 16]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nthis : \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9))\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (limitProcess f \u2131 \u03bc \u03c9))", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nthis : \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9))\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (Classical.choose this \u03c9))"}, {"tactic": "exact (Classical.choose_spec this).2", "annotated_tactic": ["exact (<a>Classical.choose_spec</a> this).2", [{"full_name": "Classical.choose_spec", "def_path": "lake-packages/lean4/src/lean/Init/Classical.lean", "def_pos": [22, 9], "def_end_pos": [22, 20]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\nthis : \u2203 g, StronglyMeasurable g \u2227 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9))\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (Classical.choose this \u03c9))", "state_after": "no goals"}, {"tactic": "filter_upwards [hf.exists_ae_trim_tendsto_of_bdd hbdd] with \u03c9 h\u03c9", "annotated_tactic": ["filter_upwards [hf.exists_ae_trim_tendsto_of_bdd hbdd] with \u03c9 h\u03c9", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\ng' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) then Exists.choose h else 0\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9))", "state_after": "case h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\ng' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) then Exists.choose h else 0\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\n\u03c9 : \u03a9\nh\u03c9 : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9))"}, {"tactic": "simp_rw [dif_pos h\u03c9]", "annotated_tactic": ["simp_rw [<a>dif_pos</a> h\u03c9]", [{"full_name": "dif_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [807, 9], "def_end_pos": [807, 16]}]], "state_before": "case h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\ng' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) then Exists.choose h else 0\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\n\u03c9 : \u03a9\nh\u03c9 : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9))", "state_after": "case h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\ng' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) then Exists.choose h else 0\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\n\u03c9 : \u03a9\nh\u03c9 : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (Exists.choose h\u03c9))"}, {"tactic": "exact h\u03c9.choose_spec", "annotated_tactic": ["exact h\u03c9.choose_spec", []], "state_before": "case h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\ng' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) then Exists.choose h else 0\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\n\u03c9 : \u03a9\nh\u03c9 : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (Exists.choose h\u03c9))", "state_after": "no goals"}, {"tactic": "filter_upwards [hae, hg'] with \u03c9 h\u03c9 hg'\u03c9", "annotated_tactic": ["filter_upwards [hae, hg'] with \u03c9 h\u03c9 hg'\u03c9", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\ng' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) then Exists.choose h else 0\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhg' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9))\ng : \u03a9 \u2192 \u211d\nhgm : StronglyMeasurable g\nhae : g' =\u1d50[Measure.trim \u03bc hle] g\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9))", "state_after": "case h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\ng' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) then Exists.choose h else 0\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhg' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9))\ng : \u03a9 \u2192 \u211d\nhgm : StronglyMeasurable g\nhae : g' =\u1d50[Measure.trim \u03bc hle] g\n\u03c9 : \u03a9\nh\u03c9 : g' \u03c9 = g \u03c9\nhg'\u03c9 : Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9))\n\u22a2 Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9))"}, {"tactic": "exact h\u03c9 \u25b8 hg'\u03c9", "annotated_tactic": ["exact h\u03c9 \u25b8 hg'\u03c9", []], "state_before": "case h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200 (n : \u2115), snorm (f n) 1 \u03bc \u2264 \u2191R\ng' : \u03a9 \u2192 \u211d := fun \u03c9 => if h : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c) then Exists.choose h else 0\nhle : \u2a06 n, \u2191\u2131 n \u2264 m0\nhg' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202Measure.trim \u03bc hle, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9))\ng : \u03a9 \u2192 \u211d\nhgm : StronglyMeasurable g\nhae : g' =\u1d50[Measure.trim \u03bc hle] g\n\u03c9 : \u03a9\nh\u03c9 : g' \u03c9 = g \u03c9\nhg'\u03c9 : Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g' \u03c9))\n\u22a2 Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (g \u03c9))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.singleton_ne_empty", "start": [716, 1], "end": [717, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.some_nthLe_eq", "start": [1439, 1], "end": [1440, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/LineDeriv/Basic.lean", "full_name": "HasLineDerivWithinAt.mono_of_mem", "start": [235, 1], "end": [240, 50], "traced_tactics": [{"tactic": "apply HasDerivWithinAt.mono_of_mem h", "annotated_tactic": ["apply <a>HasDerivWithinAt.mono_of_mem</a> h", [{"full_name": "HasDerivWithinAt.mono_of_mem", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [377, 9], "def_end_pos": [377, 37]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 : E \u2192 F\nf' : F\ns t : Set E\nx v : E\nL : E \u2192L[\ud835\udd5c] F\nh : HasLineDerivWithinAt \ud835\udd5c f f' t x v\nhst : t \u2208 \ud835\udcdd[s] x\n\u22a2 HasLineDerivWithinAt \ud835\udd5c f f' s x v", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 : E \u2192 F\nf' : F\ns t : Set E\nx v : E\nL : E \u2192L[\ud835\udd5c] F\nh : HasLineDerivWithinAt \ud835\udd5c f f' t x v\nhst : t \u2208 \ud835\udcdd[s] x\n\u22a2 (fun t => x + t \u2022 v) \u207b\u00b9' t \u2208 \ud835\udcdd[(fun t => x + t \u2022 v) \u207b\u00b9' s] 0"}, {"tactic": "apply ContinuousWithinAt.preimage_mem_nhdsWithin'' _ hst (by simp)", "annotated_tactic": ["apply <a>ContinuousWithinAt.preimage_mem_nhdsWithin''</a> _ hst (by simp)", [{"full_name": "ContinuousWithinAt.preimage_mem_nhdsWithin''", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [996, 9], "def_end_pos": [996, 53]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 : E \u2192 F\nf' : F\ns t : Set E\nx v : E\nL : E \u2192L[\ud835\udd5c] F\nh : HasLineDerivWithinAt \ud835\udd5c f f' t x v\nhst : t \u2208 \ud835\udcdd[s] x\n\u22a2 (fun t => x + t \u2022 v) \u207b\u00b9' t \u2208 \ud835\udcdd[(fun t => x + t \u2022 v) \u207b\u00b9' s] 0", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 : E \u2192 F\nf' : F\ns t : Set E\nx v : E\nL : E \u2192L[\ud835\udd5c] F\nh : HasLineDerivWithinAt \ud835\udd5c f f' t x v\nhst : t \u2208 \ud835\udcdd[s] x\n\u22a2 ContinuousWithinAt (fun t => x + t \u2022 v) ((fun t => x + t \u2022 v) \u207b\u00b9' s) 0"}, {"tactic": "apply Continuous.continuousWithinAt", "annotated_tactic": ["apply <a>Continuous.continuousWithinAt</a>", [{"full_name": "Continuous.continuousWithinAt", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [952, 9], "def_end_pos": [952, 38]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 : E \u2192 F\nf' : F\ns t : Set E\nx v : E\nL : E \u2192L[\ud835\udd5c] F\nh : HasLineDerivWithinAt \ud835\udd5c f f' t x v\nhst : t \u2208 \ud835\udcdd[s] x\n\u22a2 ContinuousWithinAt (fun t => x + t \u2022 v) ((fun t => x + t \u2022 v) \u207b\u00b9' s) 0", "state_after": "case h\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf f\u2080 f\u2081 : E \u2192 F\nf' : F\ns t : Set E\nx v : E\nL : E \u2192L[\ud835\udd5c] F\nh : HasLineDerivWithinAt \ud835\udd5c f f' t x v\nhst : t \u2208 \ud835\udcdd[s] x\n\u22a2 Continuous fun t => x + t \u2022 v"}, {"tactic": "continuity", "annotated_tactic": ["continuity", []], "state_before": "case h\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup 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<a>and_true_iff</a>]", [{"full_name": "Finsupp.single_le_iff", "def_path": "Mathlib/Data/Finsupp/Order.lean", "def_pos": [189, 9], "def_end_pos": [189, 22]}, {"full_name": "Nat.one_le_iff_ne_zero", "def_path": "Mathlib/Data/Nat/Order/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 27]}, {"full_name": "Finsupp.single_apply_ne_zero", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [385, 9], "def_end_pos": [385, 29]}, {"full_name": "and_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [145, 9], "def_end_pos": [145, 21]}]], "state_before": "\u03c3 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Nontrivial R\ni j : \u03c3\n\u22a2 ((fun\u2080 | i => 1) \u2264 fun\u2080 | j => 1) \u2194 i = j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Topology.lean", "full_name": 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: B \u2192+* C\nh : FiniteType (RingHom.comp g f)\nx\u271d : Algebra A B := toAlgebra f\n\u22a2 FiniteType g", "state_after": "A : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : CommRing B\ninst\u271d : CommRing C\nf : A \u2192+* B\ng : B \u2192+* C\nh : FiniteType (RingHom.comp g f)\nx\u271d\u00b9 : Algebra A B := toAlgebra f\nx\u271d : Algebra B C := toAlgebra g\n\u22a2 FiniteType g"}, {"tactic": "let _ := (g.comp f).toAlgebra", "annotated_tactic": ["let _ := (g.comp f).<a>toAlgebra</a>", [{"full_name": "RingHom.toAlgebra", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [267, 5], "def_end_pos": [267, 22]}]], "state_before": "A : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : CommRing B\ninst\u271d : CommRing C\nf : A \u2192+* B\ng : B \u2192+* C\nh : FiniteType (RingHom.comp g f)\nx\u271d\u00b9 : Algebra A B := toAlgebra f\nx\u271d : Algebra B C := toAlgebra g\n\u22a2 FiniteType g", 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CommRing C\nf : A \u2192+* B\ng : B \u2192+* C\nh : FiniteType (RingHom.comp g f)\nx\u271d\u00b2 : Algebra A B := toAlgebra f\nx\u271d\u00b9 : Algebra B C := toAlgebra g\nx\u271d : Algebra A C := toAlgebra (RingHom.comp g f)\n\u22a2 FiniteType g", "state_after": "A : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : CommRing B\ninst\u271d : CommRing C\nf : A \u2192+* B\ng : B \u2192+* C\nh : FiniteType (RingHom.comp g f)\nx\u271d\u00b3 : Algebra A B := toAlgebra f\nx\u271d\u00b2 : Algebra B C := toAlgebra g\nx\u271d\u00b9 : Algebra A C := toAlgebra (RingHom.comp g f)\nx\u271d : IsScalarTower A B C := RestrictScalars.isScalarTower A B C\n\u22a2 FiniteType g"}, {"tactic": "let _ : Algebra.FiniteType A C := h", "annotated_tactic": ["let _ : <a>Algebra.FiniteType</a> A C := h", [{"full_name": "Algebra.FiniteType", "def_path": "Mathlib/RingTheory/FiniteType.lean", "def_pos": [40, 7], "def_end_pos": [40, 25]}]], "state_before": "A : Type u_1\nB : Type 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"annotated_tactic": ["exact <a>Algebra.FiniteType.of_restrictScalars_finiteType</a> A B C", [{"full_name": "Algebra.FiniteType.of_restrictScalars_finiteType", "def_path": "Mathlib/RingTheory/FiniteType.lean", "def_pos": [104, 9], "def_end_pos": [104, 38]}]], "state_before": "A : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : CommRing B\ninst\u271d : CommRing C\nf : A \u2192+* B\ng : B \u2192+* C\nh : FiniteType (RingHom.comp g f)\nx\u271d\u2074 : Algebra A B := toAlgebra f\nx\u271d\u00b3 : Algebra B C := toAlgebra g\nx\u271d\u00b2 : Algebra A C := toAlgebra (RingHom.comp g f)\nx\u271d\u00b9 : IsScalarTower A B C := RestrictScalars.isScalarTower A B C\nx\u271d : Algebra.FiniteType A C := h\n\u22a2 FiniteType g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "Monotone.map_ciSup_of_continuousAt", "start": [2908, 1], "end": [2910, 91], "traced_tactics": [{"tactic": "rw [iSup, Mf.map_csSup_of_continuousAt Cf (range_nonempty _) H, \u2190 range_comp, iSup]", "annotated_tactic": ["rw [<a>iSup</a>, Mf.map_csSup_of_continuousAt Cf (<a>range_nonempty</a> _) H, \u2190 <a>range_comp</a>, <a>iSup</a>]", [{"full_name": "iSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [78, 5], "def_end_pos": [78, 9]}, {"full_name": "Set.range_nonempty", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [771, 9], "def_end_pos": [771, 23]}, {"full_name": "Set.range_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [744, 9], "def_end_pos": [744, 19]}, {"full_name": "iSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [78, 5], "def_end_pos": [78, 9]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : OrderTopology \u03b1\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : Nonempty \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b1\nCf : ContinuousAt f (\u2a06 i, g i)\nMf : Monotone f\nH : BddAbove (range g)\n\u22a2 f (\u2a06 i, g i) = \u2a06 i, f (g i)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : OrderTopology \u03b1\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : Nonempty \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b1\nCf : ContinuousAt f (\u2a06 i, g i)\nMf : Monotone f\nH : BddAbove (range g)\n\u22a2 sSup (range (f \u2218 fun i => g i)) = sSup (range fun i => f (g i))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2076 : ConditionallyCompleteLinearOrder \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : OrderTopology \u03b1\ninst\u271d\u00b3 : ConditionallyCompleteLinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderClosedTopology \u03b2\ninst\u271d : Nonempty \u03b3\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b1\nCf : ContinuousAt f (\u2a06 i, g i)\nMf : Monotone f\nH : BddAbove (range g)\n\u22a2 sSup (range (f \u2218 fun i => g i)) = sSup (range fun i => f (g i))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Infsep.lean", "full_name": "Set.Finite.einfsep_pos", "start": [308, 1], "end": [310, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Module/LinearMap.lean", "full_name": "LinearMap.ext_ring", "start": [496, 1], "end": [497, 80], "traced_tactics": [{"tactic": "rw [\u2190 mul_one x, \u2190 smul_eq_mul, f.map_smul\u209b\u2097, g.map_smul\u209b\u2097, h]", "annotated_tactic": ["rw [\u2190 <a>mul_one</a> x, \u2190 <a>smul_eq_mul</a>, f.map_smul\u209b\u2097, g.map_smul\u209b\u2097, h]", [{"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}]], "state_before": "R : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nk : Type u_5\nS : Type u_6\nS\u2083 : Type u_7\nT : Type u_8\nM : Type u_9\nM\u2081 : Type u_10\nM\u2082 : Type u_11\nM\u2083 : Type u_12\nN\u2081 : Type u_13\nN\u2082 : Type u_14\nN\u2083 : Type u_15\n\u03b9 : Type u_16\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring S\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2081\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : AddCommMonoid N\u2081\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : AddCommMonoid N\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module S M\u2083\n\u03c3 : R \u2192+* S\nf\u2097 g\u2097 : M \u2192\u2097[R] M\u2082\nf\u271d g\u271d : M \u2192\u209b\u2097[\u03c3] M\u2083\nf g : R \u2192\u209b\u2097[\u03c3] M\u2083\nh : \u2191f 1 = \u2191g 1\nx : R\n\u22a2 \u2191f x = \u2191g x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Ideal/Basic.lean", "full_name": "Ideal.mem_span_singleton'", "start": [170, 1], "end": [171, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Ioo_ae_eq_Ioc'", "start": [2821, 1], "end": [2822, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/HNNExtension.lean", "full_name": "HNNExtension.NormalWord.prod_smul_empty", "start": [557, 1], "end": [574, 11], "traced_tactics": [{"tactic": "simp [ofGroup, ReducedWord.prod, of_smul_eq_smul, group_smul_def]", "annotated_tactic": ["simp [<a>ofGroup</a>, <a>ReducedWord.prod</a>, <a>of_smul_eq_smul</a>, <a>group_smul_def</a>]", [{"full_name": "HNNExtension.NormalWord.ofGroup", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [245, 5], "def_end_pos": [245, 12]}, {"full_name": "HNNExtension.NormalWord.ReducedWord.prod", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [214, 5], "def_end_pos": [214, 21]}, {"full_name": "HNNExtension.NormalWord.of_smul_eq_smul", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [496, 9], "def_end_pos": [496, 24]}, {"full_name": "HNNExtension.NormalWord.group_smul_def", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [258, 9], "def_end_pos": [258, 23]}]], "state_before": "case ofGroup\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\ng\u271d : G\n\u22a2 ReducedWord.prod \u03c6 (ofGroup g\u271d).toReducedWord \u2022 empty = ofGroup g\u271d", "state_after": "no goals"}, {"tactic": "rw [prod_cons, \u2190 mul_assoc, mul_smul, ih, mul_smul, t_pow_smul_eq_unitsSMul,\n  of_smul_eq_smul, unitsSMul]", "annotated_tactic": ["rw [<a>prod_cons</a>, \u2190 <a>mul_assoc</a>, <a>mul_smul</a>, ih, <a>mul_smul</a>, <a>t_pow_smul_eq_unitsSMul</a>,\n      <a>of_smul_eq_smul</a>, <a>unitsSMul</a>]", [{"full_name": "HNNExtension.NormalWord.prod_cons", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [510, 9], "def_end_pos": [510, 18]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "MulAction.mul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [112, 3], "def_end_pos": [112, 11]}, {"full_name": "MulAction.mul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [112, 3], "def_end_pos": [112, 11]}, {"full_name": "HNNExtension.NormalWord.t_pow_smul_eq_unitsSMul", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [504, 9], "def_end_pos": [504, 32]}, {"full_name": "HNNExtension.NormalWord.of_smul_eq_smul", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [496, 9], "def_end_pos": [496, 24]}, {"full_name": "HNNExtension.NormalWord.unitsSMul", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [368, 19], "def_end_pos": [368, 28]}]], "state_before": "case cons\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\ng : G\nu : \u2124\u02e3\nw : NormalWord d\nh1 : w.head \u2208 TransversalPair.set d u\nh2 : \u2200 (u' : \u2124\u02e3), u' \u2208 Option.map Prod.fst (List.head? w.toList) \u2192 w.head \u2208 toSubgroup A B u \u2192 u = u'\nih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w\n\u22a2 ReducedWord.prod \u03c6 (cons g u w h1 h2).toReducedWord \u2022 empty = cons g u w h1 h2", "state_after": "case cons\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\ng : G\nu : \u2124\u02e3\nw : NormalWord d\nh1 : w.head \u2208 TransversalPair.set d u\nh2 : \u2200 (u' : \u2124\u02e3), u' \u2208 Option.map Prod.fst (List.head? w.toList) \u2192 w.head \u2208 toSubgroup A B u \u2192 u = u'\nih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w\n\u22a2 (g \u2022\n      if h : Cancels u w then unitsSMulWithCancel \u03c6 u w h\n      else\n        let g' := unitsSMulGroup \u03c6 d u w.head;\n        cons (\u2191g'.1) u ((\u2191g'.2 * w.head\u207b\u00b9) \u2022 w)\n          (_ : \u2191(unitsSMulGroup \u03c6 d u w.head).2 * w.head\u207b\u00b9 * w.head \u2208 TransversalPair.set d u)\n          (_ :\n            \u2200 (u' : \u2124\u02e3),\n              u' \u2208 Option.map Prod.fst (List.head? ((\u2191(unitsSMulGroup \u03c6 d u w.head).2 * w.head\u207b\u00b9) \u2022 w).toList) \u2192\n                ((\u2191(unitsSMulGroup \u03c6 d u w.head).2 * w.head\u207b\u00b9) \u2022 w).head \u2208 toSubgroup A B u \u2192 u = u')) =\n    cons g u w h1 h2"}, {"tactic": "rw [dif_neg (not_cancels_of_cons_hyp u w h2)]", "annotated_tactic": ["rw [<a>dif_neg</a> (<a>not_cancels_of_cons_hyp</a> u w h2)]", [{"full_name": "dif_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [812, 9], "def_end_pos": [812, 16]}, {"full_name": "HNNExtension.NormalWord.not_cancels_of_cons_hyp", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [390, 9], "def_end_pos": [390, 32]}]], "state_before": "case cons\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\ng : G\nu : \u2124\u02e3\nw : NormalWord d\nh1 : w.head \u2208 TransversalPair.set d u\nh2 : \u2200 (u' : \u2124\u02e3), u' \u2208 Option.map Prod.fst (List.head? w.toList) \u2192 w.head \u2208 toSubgroup A B u \u2192 u = u'\nih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w\n\u22a2 (g \u2022\n      if h : Cancels u w then unitsSMulWithCancel \u03c6 u w h\n      else\n        let g' := unitsSMulGroup \u03c6 d u w.head;\n        cons (\u2191g'.1) u ((\u2191g'.2 * w.head\u207b\u00b9) \u2022 w)\n          (_ : \u2191(unitsSMulGroup \u03c6 d u w.head).2 * w.head\u207b\u00b9 * w.head \u2208 TransversalPair.set d u)\n          (_ :\n            \u2200 (u' : \u2124\u02e3),\n              u' \u2208 Option.map Prod.fst (List.head? ((\u2191(unitsSMulGroup \u03c6 d u w.head).2 * w.head\u207b\u00b9) \u2022 w).toList) \u2192\n                ((\u2191(unitsSMulGroup \u03c6 d u w.head).2 * w.head\u207b\u00b9) \u2022 w).head \u2208 toSubgroup A B u \u2192 u = u')) =\n    cons g u w h1 h2", "state_after": "case cons\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\ng : G\nu : \u2124\u02e3\nw : NormalWord d\nh1 : w.head \u2208 TransversalPair.set d u\nh2 : \u2200 (u' : \u2124\u02e3), u' \u2208 Option.map Prod.fst (List.head? w.toList) \u2192 w.head \u2208 toSubgroup A B u \u2192 u = u'\nih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w\n\u22a2 (g \u2022\n      let g' := unitsSMulGroup \u03c6 d u w.head;\n      cons (\u2191g'.1) u ((\u2191g'.2 * w.head\u207b\u00b9) \u2022 w)\n        (_ : \u2191(unitsSMulGroup \u03c6 d u w.head).2 * w.head\u207b\u00b9 * w.head \u2208 TransversalPair.set d u)\n        (_ :\n          \u2200 (u' : \u2124\u02e3),\n            u' \u2208 Option.map Prod.fst (List.head? ((\u2191(unitsSMulGroup \u03c6 d u w.head).2 * w.head\u207b\u00b9) \u2022 w).toList) \u2192\n              ((\u2191(unitsSMulGroup \u03c6 d u w.head).2 * w.head\u207b\u00b9) \u2022 w).head \u2208 toSubgroup A B u \u2192 u = u')) =\n    cons g u w h1 h2"}, {"tactic": "simp only [unitsSMulGroup]", "annotated_tactic": ["simp only [<a>unitsSMulGroup</a>]", [{"full_name": "HNNExtension.NormalWord.unitsSMulGroup", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [344, 19], "def_end_pos": [344, 33]}]], "state_before": "case cons\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\ng : G\nu : \u2124\u02e3\nw : NormalWord d\nh1 : w.head \u2208 TransversalPair.set d u\nh2 : \u2200 (u' : \u2124\u02e3), u' \u2208 Option.map Prod.fst (List.head? w.toList) \u2192 w.head \u2208 toSubgroup A B u \u2192 u = u'\nih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w\n\u22a2 (g \u2022\n      let g' := unitsSMulGroup \u03c6 d u w.head;\n      cons (\u2191g'.1) u ((\u2191g'.2 * w.head\u207b\u00b9) \u2022 w)\n        (_ : \u2191(unitsSMulGroup \u03c6 d u w.head).2 * w.head\u207b\u00b9 * w.head \u2208 TransversalPair.set d u)\n        (_ :\n          \u2200 (u' : \u2124\u02e3),\n            u' \u2208 Option.map Prod.fst (List.head? ((\u2191(unitsSMulGroup \u03c6 d u w.head).2 * w.head\u207b\u00b9) \u2022 w).toList) \u2192\n              ((\u2191(unitsSMulGroup \u03c6 d u w.head).2 * w.head\u207b\u00b9) \u2022 w).head \u2208 toSubgroup A B u \u2192 u = u')) =\n    cons g u w h1 h2", "state_after": "case cons\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\ng : G\nu : \u2124\u02e3\nw : NormalWord d\nh1 : w.head \u2208 TransversalPair.set d u\nh2 : \u2200 (u' : \u2124\u02e3), u' \u2208 Option.map Prod.fst (List.head? w.toList) \u2192 w.head \u2208 toSubgroup A B u \u2192 u = u'\nih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w\n\u22a2 g \u2022\n      cons\n        (\u2191(\u2191(toSubgroupEquiv \u03c6 u)\n            (\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).1))\n        u\n        ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2 *\n            w.head\u207b\u00b9) \u2022\n          w)\n        (_ :\n          ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2 *\n                  w.head\u207b\u00b9) \u2022\n                w).head \u2208\n            TransversalPair.set d u)\n        (_ :\n          \u2200 (u' : \u2124\u02e3),\n            u' \u2208\n                Option.map Prod.fst\n                  (List.head?\n                    ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u)))\n                                w.head).2 *\n                          w.head\u207b\u00b9) \u2022\n                        w).toList) \u2192\n              ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2 *\n                        w.head\u207b\u00b9) \u2022\n                      w).head \u2208\n                  toSubgroup A B u \u2192\n                u = u') =\n    cons g u w h1 h2"}, {"tactic": "erw [(d.compl _).equiv_fst_eq_one_of_mem_of_one_mem (one_mem _) h1]", "annotated_tactic": ["erw [(d.compl _).<a>equiv_fst_eq_one_of_mem_of_one_mem</a> (<a>one_mem</a> _) h1]", [{"full_name": "Subgroup.IsComplement.equiv_fst_eq_one_of_mem_of_one_mem", "def_path": "Mathlib/GroupTheory/Complement.lean", "def_pos": [415, 9], "def_end_pos": [415, 43]}, {"full_name": "OneMemClass.one_mem", "def_path": "Mathlib/GroupTheory/Submonoid/Basic.lean", "def_pos": [73, 3], "def_end_pos": [73, 10]}]], "state_before": "case cons\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\ng : G\nu : \u2124\u02e3\nw : NormalWord d\nh1 : w.head \u2208 TransversalPair.set d u\nh2 : \u2200 (u' : \u2124\u02e3), u' \u2208 Option.map Prod.fst (List.head? w.toList) \u2192 w.head \u2208 toSubgroup A B u \u2192 u = u'\nih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w\n\u22a2 g \u2022\n      cons\n        (\u2191(\u2191(toSubgroupEquiv \u03c6 u)\n            (\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).1))\n        u\n        ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2 *\n            w.head\u207b\u00b9) \u2022\n          w)\n        (_ :\n          ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2 *\n                  w.head\u207b\u00b9) \u2022\n                w).head \u2208\n            TransversalPair.set d u)\n        (_ :\n          \u2200 (u' : \u2124\u02e3),\n            u' \u2208\n                Option.map Prod.fst\n                  (List.head?\n                    ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u)))\n                                w.head).2 *\n                          w.head\u207b\u00b9) \u2022\n                        w).toList) \u2192\n              ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2 *\n                        w.head\u207b\u00b9) \u2022\n                      w).head \u2208\n                  toSubgroup A B u \u2192\n                u = u') =\n    cons g u w h1 h2", "state_after": "case cons\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\ng : G\nu : \u2124\u02e3\nw : NormalWord d\nh1 : w.head \u2208 TransversalPair.set d u\nh2 : \u2200 (u' : \u2124\u02e3), u' \u2208 Option.map Prod.fst (List.head? w.toList) \u2192 w.head \u2208 toSubgroup A B u \u2192 u = u'\nih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w\n\u22a2 g \u2022\n      cons (\u2191(\u2191(toSubgroupEquiv \u03c6 u) { val := 1, property := (_ : 1 \u2208 toSubgroup A B u) })) u\n        ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2 *\n            w.head\u207b\u00b9) \u2022\n          w)\n        (_ :\n          ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2 *\n                  w.head\u207b\u00b9) \u2022\n                w).head \u2208\n            TransversalPair.set d u)\n        (_ :\n          \u2200 (u' : \u2124\u02e3),\n            u' \u2208\n                Option.map Prod.fst\n                  (List.head?\n                    ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u)))\n                                w.head).2 *\n                          w.head\u207b\u00b9) \u2022\n                        w).toList) \u2192\n              ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2 *\n                        w.head\u207b\u00b9) \u2022\n                      w).head \u2208\n                  toSubgroup A B u \u2192\n                u = u') =\n    cons g u w h1 h2"}, {"tactic": "ext <;> simp", "annotated_tactic": ["ext <;> simp", []], "state_before": "case cons\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\ng : G\nu : \u2124\u02e3\nw : NormalWord d\nh1 : w.head \u2208 TransversalPair.set d u\nh2 : \u2200 (u' : \u2124\u02e3), u' \u2208 Option.map Prod.fst (List.head? w.toList) \u2192 w.head \u2208 toSubgroup A B u \u2192 u = u'\nih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w\n\u22a2 g \u2022\n      cons (\u2191(\u2191(toSubgroupEquiv \u03c6 u) { val := 1, property := (_ : 1 \u2208 toSubgroup A B u) })) u\n        ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2 *\n            w.head\u207b\u00b9) \u2022\n          w)\n        (_ :\n          ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2 *\n                  w.head\u207b\u00b9) \u2022\n                w).head \u2208\n            TransversalPair.set d u)\n        (_ :\n          \u2200 (u' : \u2124\u02e3),\n            u' \u2208\n                Option.map Prod.fst\n                  (List.head?\n                    ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u)))\n                                w.head).2 *\n                          w.head\u207b\u00b9) \u2022\n                        w).toList) \u2192\n              ((\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2 *\n                        w.head\u207b\u00b9) \u2022\n                      w).head \u2208\n                  toSubgroup A B u \u2192\n                u = u') =\n    cons g u w h1 h2", "state_after": "case cons.h2.a.a\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\ng : G\nu : \u2124\u02e3\nw : NormalWord d\nh1 : w.head \u2208 TransversalPair.set d u\nh2 : \u2200 (u' : \u2124\u02e3), u' \u2208 Option.map Prod.fst (List.head? w.toList) \u2192 w.head \u2208 toSubgroup A B u \u2192 u = u'\nih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w\nn\u271d : \u2115\na\u271d : \u2124\u02e3 \u00d7 G\n\u22a2 List.get?\n        ((u, \u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2) ::\n          w.toList)\n        n\u271d =\n      some a\u271d \u2194\n    List.get? ((u, w.head) :: w.toList) n\u271d = some a\u271d"}, {"tactic": "erw [(d.compl _).equiv_snd_eq_inv_mul]", "annotated_tactic": ["erw [(d.compl _).<a>equiv_snd_eq_inv_mul</a>]", [{"full_name": "Subgroup.IsComplement.equiv_snd_eq_inv_mul", "def_path": "Mathlib/GroupTheory/Complement.lean", "def_pos": [353, 9], "def_end_pos": [353, 29]}]], "state_before": "case cons.h2.a.a\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\ng : G\nu : \u2124\u02e3\nw : NormalWord d\nh1 : w.head \u2208 TransversalPair.set d u\nh2 : \u2200 (u' : \u2124\u02e3), u' \u2208 Option.map Prod.fst (List.head? w.toList) \u2192 w.head \u2208 toSubgroup A B u \u2192 u = u'\nih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w\nn\u271d : \u2115\na\u271d : \u2124\u02e3 \u00d7 G\n\u22a2 List.get?\n        ((u, \u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).2) ::\n          w.toList)\n        n\u271d =\n      some a\u271d \u2194\n    List.get? ((u, w.head) :: w.toList) n\u271d = some a\u271d", "state_after": "case cons.h2.a.a\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\ng : G\nu : \u2124\u02e3\nw : NormalWord d\nh1 : w.head \u2208 TransversalPair.set d u\nh2 : \u2200 (u' : \u2124\u02e3), u' \u2208 Option.map Prod.fst (List.head? w.toList) \u2192 w.head \u2208 toSubgroup A B u \u2192 u = u'\nih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w\nn\u271d : \u2115\na\u271d : \u2124\u02e3 \u00d7 G\n\u22a2 List.get?\n        ((u,\n            (\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).1)\u207b\u00b9 *\n              w.head) ::\n          w.toList)\n        n\u271d =\n      some a\u271d \u2194\n    List.get? ((u, w.head) :: w.toList) n\u271d = some a\u271d"}, {"tactic": "erw [(d.compl _).equiv_fst_eq_one_of_mem_of_one_mem (one_mem _) h1]", "annotated_tactic": ["erw [(d.compl _).<a>equiv_fst_eq_one_of_mem_of_one_mem</a> (<a>one_mem</a> _) h1]", [{"full_name": "Subgroup.IsComplement.equiv_fst_eq_one_of_mem_of_one_mem", "def_path": "Mathlib/GroupTheory/Complement.lean", "def_pos": [415, 9], "def_end_pos": [415, 43]}, {"full_name": "OneMemClass.one_mem", "def_path": "Mathlib/GroupTheory/Submonoid/Basic.lean", "def_pos": [73, 3], "def_end_pos": [73, 10]}]], "state_before": "case cons.h2.a.a\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\ng : G\nu : \u2124\u02e3\nw : NormalWord d\nh1 : w.head \u2208 TransversalPair.set d u\nh2 : \u2200 (u' : \u2124\u02e3), u' \u2208 Option.map Prod.fst (List.head? w.toList) \u2192 w.head \u2208 toSubgroup A B u \u2192 u = u'\nih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w\nn\u271d : \u2115\na\u271d : \u2124\u02e3 \u00d7 G\n\u22a2 List.get?\n        ((u,\n            (\u2191(\u2191(IsComplement.equiv (_ : IsComplement (\u2191(toSubgroup A B u)) (TransversalPair.set d u))) w.head).1)\u207b\u00b9 *\n              w.head) ::\n          w.toList)\n        n\u271d =\n      some a\u271d \u2194\n    List.get? ((u, w.head) :: w.toList) n\u271d = some a\u271d", "state_after": "case cons.h2.a.a\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\ng : G\nu : \u2124\u02e3\nw : NormalWord d\nh1 : w.head \u2208 TransversalPair.set d u\nh2 : \u2200 (u' : \u2124\u02e3), u' \u2208 Option.map Prod.fst (List.head? w.toList) \u2192 w.head \u2208 toSubgroup A B u \u2192 u = u'\nih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w\nn\u271d : \u2115\na\u271d : \u2124\u02e3 \u00d7 G\n\u22a2 List.get? ((u, (\u2191{ val := 1, property := (_ : 1 \u2208 toSubgroup A B u) })\u207b\u00b9 * w.head) :: w.toList) n\u271d = some a\u271d \u2194\n    List.get? ((u, w.head) :: w.toList) n\u271d = some a\u271d"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case cons.h2.a.a\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\ng : G\nu : \u2124\u02e3\nw : NormalWord d\nh1 : w.head \u2208 TransversalPair.set d u\nh2 : \u2200 (u' : \u2124\u02e3), u' \u2208 Option.map Prod.fst (List.head? w.toList) \u2192 w.head \u2208 toSubgroup A B u \u2192 u = u'\nih : ReducedWord.prod \u03c6 w.toReducedWord \u2022 empty = w\nn\u271d : \u2115\na\u271d : \u2124\u02e3 \u00d7 G\n\u22a2 List.get? ((u, (\u2191{ val := 1, property := (_ : 1 \u2208 toSubgroup A B u) })\u207b\u00b9 * w.head) :: w.toList) n\u271d = some a\u271d \u2194\n    List.get? ((u, w.head) :: w.toList) n\u271d = some a\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Equivalence.lean", "full_name": "CategoryTheory.Equivalence.functor_map_inj_iff", "start": [742, 1], "end": [744, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Analytic/Basic.lean", "full_name": "HasFPowerSeriesOnBall.add", "start": [646, 1], "end": [650, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/MapLemmas.lean", "full_name": "Vector.mapAccumr\u2082_mapAccumr_right", "start": [76, 1], "end": [84, 80], "traced_tactics": [{"tactic": "induction xs, ys using Vector.revInductionOn\u2082 generalizing s\u2081 s\u2082 <;> simp_all", "annotated_tactic": ["induction xs, ys using <a>Vector.revInductionOn\u2082</a> generalizing s\u2081 s\u2082 <;> simp_all", [{"full_name": "Vector.revInductionOn\u2082", "def_path": "Mathlib/Data/Vector/Snoc.lean", "def_pos": [95, 5], "def_end_pos": [95, 20]}]], "state_before": "\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b3 \u03c3\u2081 \u03b6 \u03c3\u2082 : Type\ns\u2082 : \u03c3\u2082\ns\u2081 : \u03c3\u2081\nf\u2081 : \u03b1 \u2192 \u03b3 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b6\nf\u2082 : \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3\n\u22a2 mapAccumr\u2082 f\u2081 xs (mapAccumr f\u2082 ys s\u2082).2 s\u2081 =\n    let m :=\n      mapAccumr\u2082\n        (fun x y s =>\n          let r\u2082 := f\u2082 y s.2;\n          let r\u2081 := f\u2081 x r\u2082.2 s.1;\n          ((r\u2081.1, r\u2082.1), r\u2081.2))\n        xs ys (s\u2081, s\u2082);\n    (m.1.1, m.2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Generator.lean", "full_name": "CategoryTheory.IsDetector.isSeparator", "start": [437, 1], "end": [438, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.mem_sep_iff", "start": [1433, 1], "end": [1434, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "full_name": "PiNat.exists_disjoint_cylinder", "start": [487, 1], "end": [500, 26], "traced_tactics": [{"tactic": "rcases eq_empty_or_nonempty s with (rfl | hne)", "annotated_tactic": ["rcases <a>eq_empty_or_nonempty</a> s with (rfl | hne)", [{"full_name": "Set.eq_empty_or_nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [635, 9], "def_end_pos": [635, 29]}]], "state_before": "E : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\n\u22a2 \u2203 n, Disjoint s (cylinder x n)", "state_after": "case inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx : (n : \u2115) \u2192 E n\nhs : IsClosed \u2205\nhx : \u00acx \u2208 \u2205\n\u22a2 \u2203 n, Disjoint \u2205 (cylinder x n)\n\ncase inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhne : Set.Nonempty s\n\u22a2 \u2203 n, Disjoint s (cylinder x n)"}, {"tactic": "have A : 0 < infDist x s := (hs.not_mem_iff_infDist_pos hne).1 hx", "annotated_tactic": ["have A : 0 < <a>infDist</a> x s := (hs.not_mem_iff_infDist_pos hne).1 hx", [{"full_name": "Metric.infDist", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [465, 5], "def_end_pos": [465, 12]}]], "state_before": "case inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhne : Set.Nonempty s\n\u22a2 \u2203 n, Disjoint s (cylinder x n)", "state_after": "case inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhne : Set.Nonempty s\nA : 0 < infDist x s\n\u22a2 \u2203 n, Disjoint s (cylinder x n)"}, {"tactic": "obtain \u27e8n, hn\u27e9 : \u2203 n, (1 / 2 : \u211d) ^ n < infDist x s := exists_pow_lt_of_lt_one A one_half_lt_one", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 : \u2203 n, (1 / 2 : \u211d) ^ n < <a>infDist</a> x s := <a>exists_pow_lt_of_lt_one</a> A <a>one_half_lt_one</a>", [{"full_name": "Metric.infDist", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [465, 5], "def_end_pos": [465, 12]}, {"full_name": "exists_pow_lt_of_lt_one", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [232, 9], "def_end_pos": [232, 32]}, {"full_name": "one_half_lt_one", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [531, 9], "def_end_pos": [531, 24]}]], "state_before": "case inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhne : Set.Nonempty s\nA : 0 < infDist x s\n\u22a2 \u2203 n, Disjoint s (cylinder x n)", "state_after": "case inr.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : \u2115\nhn : (1 / 2) ^ n < infDist x s\n\u22a2 \u2203 n, Disjoint s (cylinder x n)"}, {"tactic": "refine' \u27e8n, disjoint_left.2 fun y ys hy => ?_\u27e9", "annotated_tactic": ["refine' \u27e8n, <a>disjoint_left</a>.2 fun y ys hy => ?_\u27e9", [{"full_name": "Set.disjoint_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1546, 9], "def_end_pos": [1546, 22]}]], "state_before": "case inr.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : \u2115\nhn : (1 / 2) ^ n < infDist x s\n\u22a2 \u2203 n, Disjoint s (cylinder x n)", "state_after": "case inr.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : \u2115\nhn : (1 / 2) ^ n < infDist x s\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : y \u2208 cylinder x n\n\u22a2 False"}, {"tactic": "apply lt_irrefl (infDist x s)", "annotated_tactic": ["apply <a>lt_irrefl</a> (<a>infDist</a> x s)", [{"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}, {"full_name": "Metric.infDist", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [465, 5], "def_end_pos": [465, 12]}]], "state_before": "case inr.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : \u2115\nhn : (1 / 2) ^ n < infDist x s\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : y \u2208 cylinder x n\n\u22a2 False", "state_after": "case inr.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : \u2115\nhn : (1 / 2) ^ n < infDist x s\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : y \u2208 cylinder x n\n\u22a2 infDist x s < infDist x s"}, {"tactic": "calc\n  infDist x s \u2264 dist x y := infDist_le_dist_of_mem ys\n  _ \u2264 (1 / 2) ^ n := by\n    rw [mem_cylinder_comm] at hy\n    exact mem_cylinder_iff_dist_le.1 hy\n  _ < infDist x s := hn", "annotated_tactic": ["calc\n    <a>infDist</a> x s \u2264 <a>dist</a> x y := <a>infDist_le_dist_of_mem</a> ys\n    _ \u2264 (1 / 2) ^ n := by\n      rw [<a>mem_cylinder_comm</a>] at hy\n      exact <a>mem_cylinder_iff_dist_le</a>.1 hy\n    _ < <a>infDist</a> x s := hn", [{"full_name": "Metric.infDist", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [465, 5], "def_end_pos": [465, 12]}, {"full_name": "Dist.dist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [94, 3], "def_end_pos": [94, 7]}, {"full_name": "Metric.infDist_le_dist_of_mem", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [506, 9], "def_end_pos": [506, 31]}, {"full_name": "PiNat.mem_cylinder_comm", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [149, 9], "def_end_pos": [149, 26]}, {"full_name": "PiNat.mem_cylinder_iff_dist_le", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [314, 9], "def_end_pos": [314, 33]}, {"full_name": "Metric.infDist", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [465, 5], "def_end_pos": [465, 12]}]], "state_before": "case inr.intro\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : \u2115\nhn : (1 / 2) ^ n < infDist x s\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : y \u2208 cylinder x n\n\u22a2 infDist x s < infDist x s", "state_after": "no goals"}, {"tactic": "exact \u27e80, by simp\u27e9", "annotated_tactic": ["exact \u27e80, by simp\u27e9", []], "state_before": "case inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx : (n : \u2115) \u2192 E n\nhs : IsClosed \u2205\nhx : \u00acx \u2208 \u2205\n\u22a2 \u2203 n, Disjoint \u2205 (cylinder x n)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "E : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx : (n : \u2115) \u2192 E n\nhs : IsClosed \u2205\nhx : \u00acx \u2208 \u2205\n\u22a2 Disjoint \u2205 (cylinder x 0)", "state_after": "no goals"}, {"tactic": "rw [mem_cylinder_comm] at hy", "annotated_tactic": ["rw [<a>mem_cylinder_comm</a>] at hy", [{"full_name": "PiNat.mem_cylinder_comm", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [149, 9], "def_end_pos": [149, 26]}]], "state_before": "E : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : \u2115\nhn : (1 / 2) ^ n < infDist x s\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : y \u2208 cylinder x n\n\u22a2 dist x y \u2264 (1 / 2) ^ n", "state_after": "E : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : \u2115\nhn : (1 / 2) ^ n < infDist x s\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : x \u2208 cylinder y n\n\u22a2 dist x y \u2264 (1 / 2) ^ n"}, {"tactic": "exact mem_cylinder_iff_dist_le.1 hy", "annotated_tactic": ["exact <a>mem_cylinder_iff_dist_le</a>.1 hy", [{"full_name": "PiNat.mem_cylinder_iff_dist_le", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [314, 9], "def_end_pos": [314, 33]}]], "state_before": "E : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nx : (n : \u2115) \u2192 E n\nhx : \u00acx \u2208 s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : \u2115\nhn : (1 / 2) ^ n < infDist x s\ny : (n : \u2115) \u2192 E n\nys : y \u2208 s\nhy : x \u2208 cylinder y n\n\u22a2 dist x y \u2264 (1 / 2) ^ n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/Sigma.lean", "full_name": "List.mem_keys_of_mem_keys_kerase", "start": [423, 1], "end": [425, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Matrix/Adjugate.lean", "full_name": "Matrix.adjugate_mul", "start": [295, 1], "end": [299, 83], "traced_tactics": [{"tactic": "rw [\u2190 adjugate_transpose, \u2190 transpose_mul, transpose_transpose]", "annotated_tactic": ["rw [\u2190 <a>adjugate_transpose</a>, \u2190 <a>transpose_mul</a>, <a>transpose_transpose</a>]", [{"full_name": "Matrix.adjugate_transpose", "def_path": "Mathlib/LinearAlgebra/Matrix/Adjugate.lean", "def_pos": [219, 9], "def_end_pos": [219, 27]}, {"full_name": "Matrix.transpose_mul", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [2006, 9], "def_end_pos": [2006, 22]}, {"full_name": "Matrix.transpose_transpose", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [1961, 9], "def_end_pos": [1961, 28]}]], "state_before": "m : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\n\u22a2 adjugate A * A = (A\u1d40 * adjugate A\u1d40)\u1d40", "state_after": "no goals"}, {"tactic": "rw [mul_adjugate A\u1d40, det_transpose, transpose_smul, transpose_one]", "annotated_tactic": ["rw [<a>mul_adjugate</a> A\u1d40, <a>det_transpose</a>, <a>transpose_smul</a>, <a>transpose_one</a>]", [{"full_name": "Matrix.mul_adjugate", "def_path": "Mathlib/LinearAlgebra/Matrix/Adjugate.lean", "def_pos": [289, 9], "def_end_pos": [289, 21]}, {"full_name": "Matrix.det_transpose", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [228, 9], "def_end_pos": [228, 22]}, {"full_name": "Matrix.transpose_smul", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [2013, 9], "def_end_pos": [2013, 23]}, {"full_name": "Matrix.transpose_one", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [1981, 9], "def_end_pos": [1981, 22]}]], "state_before": "m : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\n\u22a2 (A\u1d40 * adjugate A\u1d40)\u1d40 = det A \u2022 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/NormedSpace/CompactOperator.lean", "full_name": "IsCompactOperator.clm_comp", "start": [268, 1], "end": [271, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/LucasPrimality.lean", "full_name": "lucas_primality", "start": [42, 1], "end": [63, 55], "traced_tactics": [{"tactic": "have h0 : p \u2260 0 := by\n  rintro \u27e8\u27e9\n  exact hd 2 Nat.prime_two (dvd_zero _) (pow_zero _)", "annotated_tactic": ["have h0 : p \u2260 0 := by\n    rintro \u27e8\u27e9\n    exact hd 2 <a>Nat.prime_two</a> (<a>dvd_zero</a> _) (<a>pow_zero</a> _)", [{"full_name": "Nat.prime_two", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [169, 9], "def_end_pos": [169, 18]}, {"full_name": "dvd_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Divisibility.lean", "def_pos": [40, 9], "def_end_pos": [40, 17]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [639, 9], "def_end_pos": [639, 17]}]], "state_before": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\n\u22a2 Nat.Prime p", "state_after": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\n\u22a2 Nat.Prime p"}, {"tactic": "have h1 : p \u2260 1 := by\n  rintro \u27e8\u27e9\n  exact hd 2 Nat.prime_two (dvd_zero _) (pow_zero _)", "annotated_tactic": ["have h1 : p \u2260 1 := by\n    rintro \u27e8\u27e9\n    exact hd 2 <a>Nat.prime_two</a> (<a>dvd_zero</a> _) (<a>pow_zero</a> _)", [{"full_name": "Nat.prime_two", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [169, 9], "def_end_pos": [169, 18]}, {"full_name": "dvd_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Divisibility.lean", "def_pos": [40, 9], "def_end_pos": [40, 17]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [639, 9], "def_end_pos": [639, 17]}]], "state_before": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\n\u22a2 Nat.Prime p", "state_after": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\nh1 : p \u2260 1\n\u22a2 Nat.Prime p"}, {"tactic": "have hp1 : 1 < p := lt_of_le_of_ne h0.bot_lt h1.symm", "annotated_tactic": ["have hp1 : 1 < p := <a>lt_of_le_of_ne</a> h0.bot_lt h1.symm", [{"full_name": "lt_of_le_of_ne", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [196, 9], "def_end_pos": [196, 23]}]], "state_before": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\nh1 : p \u2260 1\n\u22a2 Nat.Prime p", "state_after": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\nh1 : p \u2260 1\nhp1 : 1 < p\n\u22a2 Nat.Prime p"}, {"tactic": "have order_of_a : orderOf a = p - 1 := by\n  apply orderOf_eq_of_pow_and_pow_div_prime _ ha hd\n  exact tsub_pos_of_lt hp1", "annotated_tactic": ["have order_of_a : <a>orderOf</a> a = p - 1 := by\n    apply <a>orderOf_eq_of_pow_and_pow_div_prime</a> _ ha hd\n    exact <a>tsub_pos_of_lt</a> hp1", [{"full_name": "orderOf", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [136, 19], "def_end_pos": [136, 26]}, {"full_name": "orderOf_eq_of_pow_and_pow_div_prime", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [311, 9], "def_end_pos": [311, 44]}, {"full_name": "tsub_pos_of_lt", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [353, 9], "def_end_pos": [353, 23]}]], "state_before": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\nh1 : p \u2260 1\nhp1 : 1 < p\n\u22a2 Nat.Prime p", "state_after": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\nh1 : p \u2260 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\n\u22a2 Nat.Prime p"}, {"tactic": "haveI : NeZero p := \u27e8h0\u27e9", "annotated_tactic": ["haveI : <a>NeZero</a> p := \u27e8h0\u27e9", [{"full_name": "NeZero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [25, 7], "def_end_pos": [25, 13]}]], "state_before": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\nh1 : p \u2260 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\n\u22a2 Nat.Prime p", "state_after": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\nh1 : p \u2260 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\nthis : NeZero p\n\u22a2 Nat.Prime p"}, {"tactic": "rw [Nat.prime_iff_card_units]", "annotated_tactic": ["rw [<a>Nat.prime_iff_card_units</a>]", [{"full_name": "Nat.prime_iff_card_units", "def_path": "Mathlib/Data/Nat/Totient.lean", "def_pos": [251, 9], "def_end_pos": [251, 29]}]], "state_before": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\nh1 : p \u2260 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\nthis : NeZero p\n\u22a2 Nat.Prime p", "state_after": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\nh1 : p \u2260 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\nthis : NeZero p\n\u22a2 Fintype.card (ZMod p)\u02e3 = p - 1"}, {"tactic": "refine' le_antisymm (Nat.card_units_zmod_lt_sub_one hp1) _", "annotated_tactic": ["refine' <a>le_antisymm</a> (<a>Nat.card_units_zmod_lt_sub_one</a> hp1) _", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "Nat.card_units_zmod_lt_sub_one", "def_path": "Mathlib/Data/Nat/Totient.lean", "def_pos": [244, 9], "def_end_pos": [244, 35]}]], "state_before": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\nh1 : p \u2260 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\nthis : NeZero p\n\u22a2 Fintype.card (ZMod p)\u02e3 = p - 1", "state_after": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\nh1 : p \u2260 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\nthis : NeZero p\n\u22a2 p - 1 \u2264 Fintype.card (ZMod p)\u02e3"}, {"tactic": "have hp' : p - 2 + 1 = p - 1 := tsub_add_eq_add_tsub hp1", "annotated_tactic": ["have hp' : p - 2 + 1 = p - 1 := <a>tsub_add_eq_add_tsub</a> hp1", [{"full_name": "tsub_add_eq_add_tsub", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [218, 9], "def_end_pos": [218, 29]}]], "state_before": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\nh1 : p \u2260 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\nthis : NeZero p\n\u22a2 p - 1 \u2264 Fintype.card (ZMod p)\u02e3", "state_after": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\nh1 : p \u2260 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\nthis : NeZero p\nhp' : p - 2 + 1 = p - 1\n\u22a2 p - 1 \u2264 Fintype.card (ZMod p)\u02e3"}, {"tactic": "let a' : (ZMod p)\u02e3 := Units.mkOfMulEqOne a (a ^ (p - 2)) (by rw [\u2190 pow_succ, hp', ha])", "annotated_tactic": ["let a' : (<a>ZMod</a> p)\u02e3 := <a>Units.mkOfMulEqOne</a> a (a ^ (p - 2)) (by rw [\u2190 <a>pow_succ</a>, hp', ha])", [{"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [94, 5], "def_end_pos": [94, 9]}, {"full_name": "Units.mkOfMulEqOne", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [415, 5], "def_end_pos": [415, 23]}, {"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [645, 9], "def_end_pos": [645, 17]}]], "state_before": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\nh1 : p \u2260 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\nthis : NeZero p\nhp' : p - 2 + 1 = p - 1\n\u22a2 p - 1 \u2264 Fintype.card (ZMod p)\u02e3", "state_after": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\nh1 : p \u2260 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\nthis : NeZero p\nhp' : p - 2 + 1 = p - 1\na' : (ZMod p)\u02e3 := Units.mkOfMulEqOne a (a ^ (p - 2)) (_ : a * a ^ (p - 2) = 1)\n\u22a2 p - 1 \u2264 Fintype.card (ZMod p)\u02e3"}, {"tactic": "calc\n  p - 1 = orderOf a := order_of_a.symm\n  _ = orderOf a' := (orderOf_injective (Units.coeHom (ZMod p)) Units.ext a')\n  _ \u2264 Fintype.card (ZMod p)\u02e3 := orderOf_le_card_univ", "annotated_tactic": ["calc\n    p - 1 = <a>orderOf</a> a := order_of_a.symm\n    _ = <a>orderOf</a> a' := (<a>orderOf_injective</a> (<a>Units.coeHom</a> (<a>ZMod</a> p)) <a>Units.ext</a> a')\n    _ \u2264 <a>Fintype.card</a> (<a>ZMod</a> p)\u02e3 := <a>orderOf_le_card_univ</a>", [{"full_name": "orderOf", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [136, 19], "def_end_pos": [136, 26]}, {"full_name": "orderOf", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [136, 19], "def_end_pos": [136, 26]}, {"full_name": "orderOf_injective", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [340, 9], "def_end_pos": [340, 26]}, {"full_name": "Units.coeHom", "def_path": "Mathlib/Algebra/Hom/Units.lean", "def_pos": [105, 5], "def_end_pos": [105, 11]}, {"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [94, 5], "def_end_pos": [94, 9]}, {"full_name": "Units.ext", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [139, 9], "def_end_pos": [139, 12]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [94, 5], "def_end_pos": [94, 9]}, {"full_name": "orderOf_le_card_univ", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [683, 9], "def_end_pos": [683, 29]}]], "state_before": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\nh1 : p \u2260 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\nthis : NeZero p\nhp' : p - 2 + 1 = p - 1\na' : (ZMod p)\u02e3 := Units.mkOfMulEqOne a (a ^ (p - 2)) (_ : a * a ^ (p - 2) = 1)\n\u22a2 p - 1 \u2264 Fintype.card (ZMod p)\u02e3", "state_after": "no goals"}, {"tactic": "rintro \u27e8\u27e9", "annotated_tactic": ["rintro \u27e8\u27e9", []], "state_before": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\n\u22a2 p \u2260 0", "state_after": "case refl\na : ZMod 0\nha : a ^ (0 - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 0 - 1 \u2192 a ^ ((0 - 1) / q) \u2260 1\n\u22a2 False"}, {"tactic": "exact hd 2 Nat.prime_two (dvd_zero _) (pow_zero _)", "annotated_tactic": ["exact hd 2 <a>Nat.prime_two</a> (<a>dvd_zero</a> _) (<a>pow_zero</a> _)", [{"full_name": "Nat.prime_two", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [169, 9], "def_end_pos": [169, 18]}, {"full_name": "dvd_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Divisibility.lean", "def_pos": [40, 9], "def_end_pos": [40, 17]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [639, 9], "def_end_pos": [639, 17]}]], "state_before": "case refl\na : ZMod 0\nha : a ^ (0 - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 0 - 1 \u2192 a ^ ((0 - 1) / q) \u2260 1\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rintro \u27e8\u27e9", "annotated_tactic": ["rintro \u27e8\u27e9", []], "state_before": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\n\u22a2 p \u2260 1", "state_after": "case refl\na : ZMod 1\nha : a ^ (1 - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 1 - 1 \u2192 a ^ ((1 - 1) / q) \u2260 1\nh0 : 1 \u2260 0\n\u22a2 False"}, {"tactic": "exact hd 2 Nat.prime_two (dvd_zero _) (pow_zero _)", "annotated_tactic": ["exact hd 2 <a>Nat.prime_two</a> (<a>dvd_zero</a> _) (<a>pow_zero</a> _)", [{"full_name": "Nat.prime_two", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [169, 9], "def_end_pos": [169, 18]}, {"full_name": "dvd_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Divisibility.lean", "def_pos": [40, 9], "def_end_pos": [40, 17]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [639, 9], "def_end_pos": [639, 17]}]], "state_before": "case refl\na : ZMod 1\nha : a ^ (1 - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 1 - 1 \u2192 a ^ ((1 - 1) / q) \u2260 1\nh0 : 1 \u2260 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "apply orderOf_eq_of_pow_and_pow_div_prime _ ha hd", "annotated_tactic": ["apply <a>orderOf_eq_of_pow_and_pow_div_prime</a> _ ha hd", [{"full_name": "orderOf_eq_of_pow_and_pow_div_prime", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [311, 9], "def_end_pos": [311, 44]}]], "state_before": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\nh1 : p \u2260 1\nhp1 : 1 < p\n\u22a2 orderOf a = p - 1", "state_after": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\nh1 : p \u2260 1\nhp1 : 1 < p\n\u22a2 0 < p - 1"}, {"tactic": "exact tsub_pos_of_lt hp1", "annotated_tactic": ["exact <a>tsub_pos_of_lt</a> hp1", [{"full_name": "tsub_pos_of_lt", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [353, 9], "def_end_pos": [353, 23]}]], "state_before": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\nh1 : p \u2260 1\nhp1 : 1 < p\n\u22a2 0 < p - 1", "state_after": "no goals"}, {"tactic": "rw [\u2190 pow_succ, hp', ha]", "annotated_tactic": ["rw [\u2190 <a>pow_succ</a>, hp', ha]", [{"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [645, 9], "def_end_pos": [645, 17]}]], "state_before": "p : \u2115\na : ZMod p\nha : a ^ (p - 1) = 1\nhd : \u2200 (q : \u2115), Nat.Prime q \u2192 q \u2223 p - 1 \u2192 a ^ ((p - 1) / q) \u2260 1\nh0 : p \u2260 0\nh1 : p \u2260 1\nhp1 : 1 < p\norder_of_a : orderOf a = p - 1\nthis : NeZero p\nhp' : p - 2 + 1 = p - 1\n\u22a2 a * a ^ (p - 2) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Manifold/Instances/Real.lean", "full_name": "EuclideanHalfSpace.ext", "start": [87, 1], "end": [89, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Star/NonUnitalSubalgebra.lean", "full_name": "NonUnitalStarAlgebra.map_top", "start": [780, 1], "end": [781, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean", "full_name": "InnerProductGeometry.angle_add_eq_arctan_of_inner_eq_zero", "start": [99, 1], "end": [105, 101], "traced_tactics": [{"tactic": "rw [angle_add_eq_arcsin_of_inner_eq_zero h (Or.inl h0), Real.arctan_eq_arcsin, \u2190\n  div_mul_eq_div_div, norm_add_eq_sqrt_iff_real_inner_eq_zero.2 h]", "annotated_tactic": ["rw [<a>angle_add_eq_arcsin_of_inner_eq_zero</a> h (<a>Or.inl</a> h0), <a>Real.arctan_eq_arcsin</a>, \u2190\n    <a>div_mul_eq_div_div</a>, <a>norm_add_eq_sqrt_iff_real_inner_eq_zero</a>.2 h]", [{"full_name": "InnerProductGeometry.angle_add_eq_arcsin_of_inner_eq_zero", "def_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean", "def_pos": [81, 9], "def_end_pos": [81, 45]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "Real.arctan_eq_arcsin", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean", "def_pos": [161, 9], "def_end_pos": [161, 25]}, {"full_name": "div_mul_eq_div_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [547, 9], "def_end_pos": [547, 27]}, {"full_name": "norm_add_eq_sqrt_iff_real_inner_eq_zero", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [1462, 9], "def_end_pos": [1462, 48]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : inner x y = 0\nh0 : x \u2260 0\n\u22a2 angle x (x + y) = Real.arctan (\u2016y\u2016 / \u2016x\u2016)", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : inner x y = 0\nh0 : x \u2260 0\n\u22a2 Real.arcsin (\u2016y\u2016 / Real.sqrt (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016)) = Real.arcsin (\u2016y\u2016 / (\u2016x\u2016 * Real.sqrt (\u21911 + (\u2016y\u2016 / \u2016x\u2016) ^ 2)))"}, {"tactic": "nth_rw 3 [\u2190 Real.sqrt_sq (norm_nonneg x)]", "annotated_tactic": ["nth_rw 3 [\u2190 <a>Real.sqrt_sq</a> (<a>norm_nonneg</a> x)]", [{"full_name": "Real.sqrt_sq", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [238, 9], "def_end_pos": [238, 16]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : inner x y = 0\nh0 : x \u2260 0\n\u22a2 Real.arcsin (\u2016y\u2016 / Real.sqrt (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016)) = Real.arcsin (\u2016y\u2016 / (\u2016x\u2016 * Real.sqrt (\u21911 + (\u2016y\u2016 / \u2016x\u2016) ^ 2)))", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : inner x y = 0\nh0 : x \u2260 0\n\u22a2 Real.arcsin (\u2016y\u2016 / Real.sqrt (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016)) =\n    Real.arcsin (\u2016y\u2016 / (Real.sqrt (\u2016x\u2016 ^ 2) * Real.sqrt (\u21911 + (\u2016y\u2016 / \u2016x\u2016) ^ 2)))"}, {"tactic": "rw_mod_cast [\u2190 Real.sqrt_mul (sq_nonneg _), div_pow, pow_two, pow_two, mul_add, mul_one, mul_div,\n  mul_comm (\u2016x\u2016 * \u2016x\u2016), \u2190 mul_div, div_self (mul_self_pos.2 (norm_ne_zero_iff.2 h0)).ne', mul_one]", "annotated_tactic": ["rw_mod_cast [\u2190 <a>Real.sqrt_mul</a> (<a>sq_nonneg</a> _), <a>div_pow</a>, <a>pow_two</a>, <a>pow_two</a>, <a>mul_add</a>, <a>mul_one</a>, <a>mul_div</a>,\n    <a>mul_comm</a> (\u2016x\u2016 * \u2016x\u2016), \u2190 <a>mul_div</a>, <a>div_self</a> (<a>mul_self_pos</a>.2 (<a>norm_ne_zero_iff</a>.2 h0)).<a>ne'</a>, <a>mul_one</a>]", [{"full_name": "Real.sqrt_mul", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [394, 9], "def_end_pos": [394, 17]}, {"full_name": "sq_nonneg", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [645, 9], "def_end_pos": [645, 18]}, {"full_name": "div_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [391, 9], "def_end_pos": [391, 16]}, {"full_name": "pow_two", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [102, 9], "def_end_pos": [102, 16]}, {"full_name": "pow_two", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [102, 9], "def_end_pos": [102, 16]}, {"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "mul_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [324, 9], "def_end_pos": [324, 16]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "mul_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [324, 9], "def_end_pos": [324, 16]}, {"full_name": "div_self", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [29, 9], "def_end_pos": [29, 17]}, {"full_name": "mul_self_pos", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [1171, 9], "def_end_pos": [1171, 21]}, {"full_name": "norm_ne_zero_iff", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [2024, 15], "def_end_pos": [2024, 31]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : inner x y = 0\nh0 : x \u2260 0\n\u22a2 Real.arcsin (\u2016y\u2016 / Real.sqrt (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016)) =\n    Real.arcsin (\u2016y\u2016 / (Real.sqrt (\u2016x\u2016 ^ 2) * Real.sqrt (\u21911 + (\u2016y\u2016 / \u2016x\u2016) ^ 2)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Finite.lean", "full_name": "Set.finite_range_findGreatest", "start": [1569, 1], "end": [1571, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Convex/Between.lean", "full_name": "wbtw_smul_vadd_smul_vadd_of_nonneg_of_le", "start": [763, 1], "end": [767, 76], "traced_tactics": [{"tactic": "refine' \u27e8r\u2081 / r\u2082, \u27e8div_nonneg hr\u2081 (hr\u2081.trans hr\u2082), div_le_one_of_le hr\u2082 (hr\u2081.trans hr\u2082)\u27e9, _\u27e9", "annotated_tactic": ["refine' \u27e8r\u2081 / r\u2082, \u27e8<a>div_nonneg</a> hr\u2081 (hr\u2081.trans hr\u2082), <a>div_le_one_of_le</a> hr\u2082 (hr\u2081.trans hr\u2082)\u27e9, _\u27e9", [{"full_name": "div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [94, 9], "def_end_pos": [94, 19]}, {"full_name": "div_le_one_of_le", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [242, 9], "def_end_pos": [242, 25]}]], "state_before": "R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : 0 \u2264 r\u2081\nhr\u2082 : r\u2081 \u2264 r\u2082\n\u22a2 Wbtw R x (r\u2081 \u2022 v +\u1d65 x) (r\u2082 \u2022 v +\u1d65 x)", "state_after": "R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : 0 \u2264 r\u2081\nhr\u2082 : r\u2081 \u2264 r\u2082\n\u22a2 \u2191(lineMap x (r\u2082 \u2022 v +\u1d65 x)) (r\u2081 / r\u2082) = r\u2081 \u2022 v +\u1d65 x"}, {"tactic": "by_cases h : r\u2081 = 0", "annotated_tactic": ["by_cases h : r\u2081 = 0", []], "state_before": "R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : 0 \u2264 r\u2081\nhr\u2082 : r\u2081 \u2264 r\u2082\n\u22a2 \u2191(lineMap x (r\u2082 \u2022 v +\u1d65 x)) (r\u2081 / r\u2082) = r\u2081 \u2022 v +\u1d65 x", "state_after": "case pos\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : 0 \u2264 r\u2081\nhr\u2082 : r\u2081 \u2264 r\u2082\nh : r\u2081 = 0\n\u22a2 \u2191(lineMap x (r\u2082 \u2022 v +\u1d65 x)) (r\u2081 / r\u2082) = r\u2081 \u2022 v +\u1d65 x\n\ncase neg\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : 0 \u2264 r\u2081\nhr\u2082 : r\u2081 \u2264 r\u2082\nh : \u00acr\u2081 = 0\n\u22a2 \u2191(lineMap x (r\u2082 \u2022 v +\u1d65 x)) (r\u2081 / r\u2082) = r\u2081 \u2022 v +\u1d65 x"}, {"tactic": "simp [lineMap_apply, smul_smul, ((hr\u2081.lt_of_ne' h).trans_le hr\u2082).ne.symm]", "annotated_tactic": ["simp [<a>lineMap_apply</a>, <a>smul_smul</a>, ((hr\u2081.lt_of_ne' h).<a>trans_le</a> hr\u2082).ne.symm]", [{"full_name": "AffineMap.lineMap_apply", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean", "def_pos": [514, 9], "def_end_pos": [514, 22]}, {"full_name": "smul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [484, 9], "def_end_pos": [484, 18]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}]], "state_before": "case neg\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : 0 \u2264 r\u2081\nhr\u2082 : r\u2081 \u2264 r\u2082\nh : \u00acr\u2081 = 0\n\u22a2 \u2191(lineMap x (r\u2082 \u2022 v +\u1d65 x)) (r\u2081 / r\u2082) = r\u2081 \u2022 v +\u1d65 x", "state_after": "no goals"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "case pos\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx : P\nv : V\nr\u2081 r\u2082 : R\nhr\u2081 : 0 \u2264 r\u2081\nhr\u2082 : r\u2081 \u2264 r\u2082\nh : r\u2081 = 0\n\u22a2 \u2191(lineMap x (r\u2082 \u2022 v +\u1d65 x)) (r\u2081 / r\u2082) = r\u2081 \u2022 v +\u1d65 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "Filter.Tendsto.zpow", "start": [524, 1], "end": [526, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean", "full_name": "PrimeSpectrum.basicOpen_pow", "start": [804, 1], "end": [805, 78], "traced_tactics": [{"tactic": "simpa using zeroLocus_singleton_pow f n hn", "annotated_tactic": ["simpa using <a>zeroLocus_singleton_pow</a> f n hn", [{"full_name": "PrimeSpectrum.zeroLocus_singleton_pow", "def_path": "Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean", "def_pos": [378, 9], "def_end_pos": [378, 32]}]], "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R\nn : \u2115\nhn : 0 < n\n\u22a2 \u2191(basicOpen (f ^ n)) = \u2191(basicOpen f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Perm/Basic.lean", "full_name": "Equiv.mul_swap_eq_swap_mul", "start": [530, 1], "end": [531, 70], "traced_tactics": [{"tactic": "rw [swap_mul_eq_mul_swap, Perm.inv_apply_self, Perm.inv_apply_self]", "annotated_tactic": ["rw [<a>swap_mul_eq_mul_swap</a>, <a>Perm.inv_apply_self</a>, <a>Perm.inv_apply_self</a>]", [{"full_name": "Equiv.swap_mul_eq_mul_swap", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [523, 9], "def_end_pos": [523, 29]}, {"full_name": "Equiv.Perm.inv_apply_self", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [76, 9], "def_end_pos": [76, 23]}, {"full_name": "Equiv.Perm.inv_apply_self", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [76, 9], "def_end_pos": [76, 23]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nf : Perm \u03b1\nx y : \u03b1\n\u22a2 f * swap x y = swap (\u2191f x) (\u2191f y) * f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "MeasurableEmbedding.comap_eq", "start": [1726, 9], "end": [1728, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Analytic/Basic.lean", "full_name": "HasFPowerSeriesOnBall.analyticAt", "start": [528, 1], "end": [529, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Real/CauSeqCompletion.lean", "full_name": "CauSeq.lim_le", "start": [446, 1], "end": [447, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.div_mul_cancel", "start": [701, 11], "end": [702, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Complex/Basic.lean", "full_name": "Complex.nnnorm_int", "start": [193, 1], "end": [194, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/VonNeumannAlgebra/Basic.lean", "full_name": "VonNeumannAlgebra.commutant_commutant", "start": [151, 1], "end": [152, 35], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "H : Type u\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u2102 H\ninst\u271d : CompleteSpace H\nS : VonNeumannAlgebra H\n\u22a2 \u2191(commutant (commutant S)) = \u2191S", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Subsemigroup/Operations.lean", "full_name": "Subsemigroup.map_map", "start": [252, 1], "end": [253, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "full_name": "HasFDerivWithinAt.mono_of_mem", "start": [374, 1], "end": [376, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.measure_union_add_inter'", "start": [147, 1], "end": [149, 70], "traced_tactics": [{"tactic": "rw [union_comm, inter_comm, measure_union_add_inter t hs, add_comm]", "annotated_tactic": ["rw [<a>union_comm</a>, <a>inter_comm</a>, <a>measure_union_add_inter</a> t hs, <a>add_comm</a>]", [{"full_name": "Set.union_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [786, 9], "def_end_pos": [786, 19]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "MeasureTheory.measure_union_add_inter", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [140, 9], "def_end_pos": [140, 32]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t\u271d : Set \u03b1\nhs : MeasurableSet s\nt : Set \u03b1\n\u22a2 \u2191\u2191\u03bc (s \u222a t) + \u2191\u2191\u03bc (s \u2229 t) = \u2191\u2191\u03bc s + \u2191\u2191\u03bc t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "full_name": "Polynomial.cyclotomic'_eq_X_pow_sub_one_div", "start": [182, 1], "end": [196, 53], "traced_tactics": [{"tactic": "rw [\u2190 prod_cyclotomic'_eq_X_pow_sub_one hpos h, \u2190 Nat.cons_self_properDivisors hpos.ne',\n  Finset.prod_cons]", "annotated_tactic": ["rw [\u2190 <a>prod_cyclotomic'_eq_X_pow_sub_one</a> hpos h, \u2190 <a>Nat.cons_self_properDivisors</a> hpos.ne',\n    <a>Finset.prod_cons</a>]", [{"full_name": "Polynomial.prod_cyclotomic'_eq_X_pow_sub_one", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [169, 9], "def_end_pos": [169, 42]}, {"full_name": "Nat.cons_self_properDivisors", "def_path": "Mathlib/NumberTheory/Divisors.lean", "def_pos": [88, 9], "def_end_pos": [88, 33]}, {"full_name": "Finset.prod_cons", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [311, 9], "def_end_pos": [311, 18]}]], "state_before": "K\u271d : Type u_1\ninst\u271d\u00b2 : Field K\u271d\nK : Type u_2\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u03b6 : K\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\n\u22a2 cyclotomic' n K = (X ^ n - 1) /\u2098 \u220f i in Nat.properDivisors n, cyclotomic' i K", "state_after": "K\u271d : Type u_1\ninst\u271d\u00b2 : Field K\u271d\nK : Type u_2\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u03b6 : K\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\n\u22a2 cyclotomic' n K =\n    (cyclotomic' n K * \u220f x in Nat.properDivisors n, cyclotomic' x K) /\u2098 \u220f i in Nat.properDivisors n, cyclotomic' i K"}, {"tactic": "have prod_monic : (\u220f i in Nat.properDivisors n, cyclotomic' i K).Monic := by\n  apply monic_prod_of_monic\n  intro i _\n  exact cyclotomic'.monic i K", "annotated_tactic": ["have prod_monic : (\u220f i in <a>Nat.properDivisors</a> n, <a>cyclotomic'</a> i K).<a>Monic</a> := by\n    apply <a>monic_prod_of_monic</a>\n    intro i _\n    exact <a>cyclotomic'.monic</a> i K", [{"full_name": "Nat.properDivisors", "def_path": "Mathlib/NumberTheory/Divisors.lean", "def_pos": [47, 5], "def_end_pos": [47, 19]}, {"full_name": "Polynomial.cyclotomic'", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [68, 5], "def_end_pos": [68, 16]}, {"full_name": "Polynomial.Monic", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [75, 5], "def_end_pos": [75, 10]}, {"full_name": "Polynomial.monic_prod_of_monic", "def_path": "Mathlib/Data/Polynomial/Monic.lean", "def_pos": [272, 9], "def_end_pos": [272, 28]}, {"full_name": "Polynomial.cyclotomic'.monic", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 26]}]], "state_before": "K\u271d : Type u_1\ninst\u271d\u00b2 : Field K\u271d\nK : Type u_2\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u03b6 : K\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\n\u22a2 cyclotomic' n K =\n    (cyclotomic' n K * \u220f x in Nat.properDivisors n, cyclotomic' x K) /\u2098 \u220f i in Nat.properDivisors n, cyclotomic' i K", "state_after": "K\u271d : Type u_1\ninst\u271d\u00b2 : Field K\u271d\nK : Type u_2\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u03b6 : K\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\nprod_monic : Monic (\u220f i in Nat.properDivisors n, cyclotomic' i K)\n\u22a2 cyclotomic' n K =\n    (cyclotomic' n K * \u220f x in Nat.properDivisors n, cyclotomic' x K) /\u2098 \u220f i in Nat.properDivisors n, cyclotomic' i K"}, {"tactic": "rw [(div_modByMonic_unique (cyclotomic' n K) 0 prod_monic _).1]", "annotated_tactic": ["rw [(<a>div_modByMonic_unique</a> (<a>cyclotomic'</a> n K) 0 prod_monic _).1]", [{"full_name": "Polynomial.div_modByMonic_unique", "def_path": "Mathlib/Data/Polynomial/Div.lean", "def_pos": [358, 9], "def_end_pos": [358, 30]}, {"full_name": "Polynomial.cyclotomic'", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [68, 5], "def_end_pos": [68, 16]}]], "state_before": "K\u271d : Type u_1\ninst\u271d\u00b2 : Field K\u271d\nK : Type u_2\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u03b6 : K\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\nprod_monic : Monic (\u220f i in Nat.properDivisors n, cyclotomic' i K)\n\u22a2 cyclotomic' n K =\n    (cyclotomic' n K * \u220f x in Nat.properDivisors n, cyclotomic' x K) /\u2098 \u220f i in Nat.properDivisors n, cyclotomic' i K", "state_after": "K\u271d : Type u_1\ninst\u271d\u00b2 : Field K\u271d\nK : Type u_2\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u03b6 : K\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\nprod_monic : Monic (\u220f i in Nat.properDivisors n, cyclotomic' i K)\n\u22a2 0 + (\u220f i in Nat.properDivisors n, cyclotomic' i K) * cyclotomic' n K =\n      cyclotomic' n K * \u220f x in Nat.properDivisors n, cyclotomic' x K \u2227\n    degree 0 < degree (\u220f i in Nat.properDivisors n, cyclotomic' i K)"}, {"tactic": "simp only [degree_zero, zero_add]", "annotated_tactic": ["simp only [<a>degree_zero</a>, <a>zero_add</a>]", [{"full_name": "Polynomial.degree_zero", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [101, 9], "def_end_pos": [101, 20]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "K\u271d : Type u_1\ninst\u271d\u00b2 : Field K\u271d\nK : Type u_2\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u03b6 : K\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\nprod_monic : Monic (\u220f i in Nat.properDivisors n, cyclotomic' i K)\n\u22a2 0 + (\u220f i in Nat.properDivisors n, cyclotomic' i K) * cyclotomic' n K =\n      cyclotomic' n K * \u220f x in Nat.properDivisors n, cyclotomic' x K \u2227\n    degree 0 < degree (\u220f i in Nat.properDivisors n, cyclotomic' i K)", "state_after": "K\u271d : Type u_1\ninst\u271d\u00b2 : Field K\u271d\nK : Type u_2\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u03b6 : K\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\nprod_monic : Monic (\u220f i in Nat.properDivisors n, cyclotomic' i K)\n\u22a2 (\u220f i in Nat.properDivisors n, cyclotomic' i K) * cyclotomic' n K =\n      cyclotomic' n K * \u220f i in Nat.properDivisors n, cyclotomic' i K \u2227\n    \u22a5 < degree (\u220f i in Nat.properDivisors n, cyclotomic' i K)"}, {"tactic": "refine' \u27e8by rw [mul_comm], _\u27e9", "annotated_tactic": ["refine' \u27e8by rw [<a>mul_comm</a>], _\u27e9", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "K\u271d : Type u_1\ninst\u271d\u00b2 : Field K\u271d\nK : Type u_2\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u03b6 : K\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\nprod_monic : Monic (\u220f i in Nat.properDivisors n, cyclotomic' i K)\n\u22a2 (\u220f i in Nat.properDivisors n, cyclotomic' i K) * cyclotomic' n K =\n      cyclotomic' n K * \u220f i in Nat.properDivisors n, cyclotomic' i K \u2227\n    \u22a5 < degree (\u220f i in Nat.properDivisors n, cyclotomic' i K)", "state_after": "K\u271d : Type u_1\ninst\u271d\u00b2 : Field K\u271d\nK : Type u_2\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u03b6 : K\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\nprod_monic : Monic (\u220f i in Nat.properDivisors n, cyclotomic' i K)\n\u22a2 \u22a5 < degree (\u220f i in Nat.properDivisors n, cyclotomic' i K)"}, {"tactic": "rw [bot_lt_iff_ne_bot]", "annotated_tactic": ["rw [<a>bot_lt_iff_ne_bot</a>]", [{"full_name": "bot_lt_iff_ne_bot", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [371, 9], "def_end_pos": [371, 26]}]], "state_before": "K\u271d : Type u_1\ninst\u271d\u00b2 : Field K\u271d\nK : Type u_2\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u03b6 : K\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\nprod_monic : Monic (\u220f i in Nat.properDivisors n, cyclotomic' i K)\n\u22a2 \u22a5 < degree (\u220f i in Nat.properDivisors n, cyclotomic' i K)", "state_after": "K\u271d : Type u_1\ninst\u271d\u00b2 : Field K\u271d\nK : Type u_2\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u03b6 : K\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\nprod_monic : Monic (\u220f i in Nat.properDivisors n, cyclotomic' i K)\n\u22a2 degree (\u220f i in Nat.properDivisors n, cyclotomic' i K) \u2260 \u22a5"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "K\u271d : Type u_1\ninst\u271d\u00b2 : Field K\u271d\nK : Type u_2\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u03b6 : K\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\nprod_monic : Monic (\u220f i in Nat.properDivisors n, cyclotomic' i K)\n\u22a2 degree (\u220f i in Nat.properDivisors n, cyclotomic' i K) \u2260 \u22a5", "state_after": "K\u271d : Type u_1\ninst\u271d\u00b2 : Field K\u271d\nK : Type u_2\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u03b6 : K\nn : \u2115\nhpos : 0 < n\nh\u271d : IsPrimitiveRoot \u03b6 n\nprod_monic : Monic (\u220f i in Nat.properDivisors n, cyclotomic' i K)\nh : degree (\u220f i in Nat.properDivisors n, cyclotomic' i K) = \u22a5\n\u22a2 False"}, {"tactic": "exact Monic.ne_zero prod_monic (degree_eq_bot.1 h)", "annotated_tactic": ["exact <a>Monic.ne_zero</a> prod_monic (<a>degree_eq_bot</a>.1 h)", [{"full_name": "Polynomial.Monic.ne_zero", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [864, 9], "def_end_pos": [864, 22]}, {"full_name": "Polynomial.degree_eq_bot", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [116, 9], "def_end_pos": [116, 22]}]], "state_before": "K\u271d : Type u_1\ninst\u271d\u00b2 : Field K\u271d\nK : Type u_2\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u03b6 : K\nn : \u2115\nhpos : 0 < n\nh\u271d : IsPrimitiveRoot \u03b6 n\nprod_monic : Monic (\u220f i in Nat.properDivisors n, cyclotomic' i K)\nh : degree (\u220f i in Nat.properDivisors n, cyclotomic' i K) = \u22a5\n\u22a2 False", "state_after": "no goals"}, {"tactic": "apply monic_prod_of_monic", "annotated_tactic": ["apply <a>monic_prod_of_monic</a>", [{"full_name": "Polynomial.monic_prod_of_monic", "def_path": "Mathlib/Data/Polynomial/Monic.lean", "def_pos": [272, 9], "def_end_pos": [272, 28]}]], "state_before": "K\u271d : Type u_1\ninst\u271d\u00b2 : Field K\u271d\nK : Type u_2\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u03b6 : K\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\n\u22a2 Monic (\u220f i in Nat.properDivisors n, cyclotomic' i K)", "state_after": "case hs\nK\u271d : Type u_1\ninst\u271d\u00b2 : Field K\u271d\nK : Type u_2\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u03b6 : K\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\n\u22a2 \u2200 (i : \u2115), i \u2208 Nat.properDivisors n \u2192 Monic (cyclotomic' i K)"}, {"tactic": "intro i _", "annotated_tactic": ["intro i _", []], "state_before": "case hs\nK\u271d : Type u_1\ninst\u271d\u00b2 : Field K\u271d\nK : Type u_2\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u03b6 : K\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\n\u22a2 \u2200 (i : \u2115), i \u2208 Nat.properDivisors n \u2192 Monic (cyclotomic' i K)", "state_after": "case hs\nK\u271d : Type u_1\ninst\u271d\u00b2 : Field K\u271d\nK : Type u_2\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u03b6 : K\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\ni : \u2115\na\u271d : i \u2208 Nat.properDivisors n\n\u22a2 Monic (cyclotomic' i K)"}, {"tactic": "exact cyclotomic'.monic i K", "annotated_tactic": ["exact <a>cyclotomic'.monic</a> i K", [{"full_name": "Polynomial.cyclotomic'.monic", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 26]}]], "state_before": "case hs\nK\u271d : Type u_1\ninst\u271d\u00b2 : Field K\u271d\nK : Type u_2\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u03b6 : K\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\ni : \u2115\na\u271d : i \u2208 Nat.properDivisors n\n\u22a2 Monic (cyclotomic' i K)", "state_after": "no goals"}, {"tactic": "rw [mul_comm]", "annotated_tactic": ["rw [<a>mul_comm</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "K\u271d : Type u_1\ninst\u271d\u00b2 : Field K\u271d\nK : Type u_2\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u03b6 : K\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\nprod_monic : Monic (\u220f i in Nat.properDivisors n, cyclotomic' i K)\n\u22a2 (\u220f i in Nat.properDivisors n, cyclotomic' i K) * cyclotomic' n K =\n    cyclotomic' n K * \u220f i in Nat.properDivisors n, cyclotomic' i K", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/LiminfLimsup.lean", "full_name": "Filter.limsup_bot", "start": [714, 9], "end": [714, 76], "traced_tactics": [{"tactic": "simp [limsup]", "annotated_tactic": ["simp [<a>limsup</a>]", [{"full_name": "Filter.limsup", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [420, 5], "def_end_pos": [420, 11]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\ninst\u271d : CompleteLattice \u03b1\nf : \u03b2 \u2192 \u03b1\n\u22a2 limsup f \u22a5 = \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/ZetaValues.lean", "full_name": "hasSum_one_div_nat_pow_mul_cos", "start": [258, 1], "end": [292, 9], "traced_tactics": [{"tactic": "have ofReal_two : ((2 : \u211d) : \u2102) = 2 := by norm_cast", "annotated_tactic": ["have ofReal_two : ((2 : \u211d) : \u2102) = 2 := by norm_cast", []], "state_before": "k : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\n\u22a2 HasSum (fun n => 1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * x))\n    ((-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))", "state_after": "k : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 HasSum (fun n => 1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * x))\n    ((-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))"}, {"tactic": "convert ((hasSum_iff _ _).mp (this.div_const 2)).1 with n", "annotated_tactic": ["convert ((<a>hasSum_iff</a> _ _).<a>mp</a> (this.div_const 2)).1 with n", [{"full_name": "Complex.hasSum_iff", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [622, 9], "def_end_pos": [622, 19]}, {"full_name": "Iff.mp", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [90, 3], "def_end_pos": [90, 5]}]], "state_before": "k : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 HasSum (fun n => 1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * x))\n    ((-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))", "state_after": "case h.e'_5.h\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * x) = (1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x) / 2).re\n\ncase h.e'_6\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 (-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))) =\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x) / 2).re"}, {"tactic": "convert\n  hasSum_one_div_nat_pow_mul_fourier (by linarith [Nat.one_le_iff_ne_zero.mpr hk] : 2 \u2264 2 * k)\n    hx using 3", "annotated_tactic": ["convert\n      <a>hasSum_one_div_nat_pow_mul_fourier</a> (by linarith [Nat.one_le_iff_ne_zero.mpr hk] : 2 \u2264 2 * k)\n        hx using 3", [{"full_name": "hasSum_one_div_nat_pow_mul_fourier", "def_path": "Mathlib/NumberTheory/ZetaValues.lean", "def_pos": [241, 9], "def_end_pos": [241, 43]}]], "state_before": "k : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))", "state_after": "case h.e'_5.h.h.e'_6\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nx\u271d : \u2115\n\u22a2 \u2191(fourier \u2191x\u271d) \u2191x + \u2191(fourier (-\u2191x\u271d)) \u2191x = \u2191(fourier \u2191x\u271d) \u2191x + (-1) ^ (2 * k) * \u2191(fourier (-\u2191x\u271d)) \u2191x\n\ncase h.e'_6.h.e'_5.h.e'_5\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) = -(2 * \u2191\u03c0 * I) ^ (2 * k)"}, {"tactic": "linarith [Nat.one_le_iff_ne_zero.mpr hk]", "annotated_tactic": ["linarith [Nat.one_le_iff_ne_zero.mpr hk]", []], "state_before": "k : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 2 \u2264 2 * k", "state_after": "no goals"}, {"tactic": "rw [pow_mul (-1 : \u2102), neg_one_sq, one_pow, one_mul]", "annotated_tactic": ["rw [<a>pow_mul</a> (-1 : \u2102), <a>neg_one_sq</a>, <a>one_pow</a>, <a>one_mul</a>]", [{"full_name": "pow_mul", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [125, 9], "def_end_pos": [125, 16]}, {"full_name": "neg_one_sq", "def_path": "Mathlib/Algebra/GroupPower/Ring.lean", "def_pos": [223, 9], "def_end_pos": [223, 19]}, {"full_name": "one_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [90, 9], "def_end_pos": [90, 16]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "case h.e'_5.h.h.e'_6\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nx\u271d : \u2115\n\u22a2 \u2191(fourier \u2191x\u271d) \u2191x + \u2191(fourier (-\u2191x\u271d)) \u2191x = \u2191(fourier \u2191x\u271d) \u2191x + (-1) ^ (2 * k) * \u2191(fourier (-\u2191x\u271d)) \u2191x", "state_after": "no goals"}, {"tactic": "rw [pow_add, pow_one]", "annotated_tactic": ["rw [<a>pow_add</a>, <a>pow_one</a>]", [{"full_name": "pow_add", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 16]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}]], "state_before": "case h.e'_6.h.e'_5.h.e'_5\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) = -(2 * \u2191\u03c0 * I) ^ (2 * k)", "state_after": "case h.e'_6.h.e'_5.h.e'_5\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 (-1) ^ k * -1 * (2 * \u2191\u03c0) ^ (2 * k) = -(2 * \u2191\u03c0 * I) ^ (2 * k)"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case h.e'_6.h.e'_5.h.e'_5\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 (-1) ^ k * -1 * (2 * \u2191\u03c0) ^ (2 * k) = -((2 * \u2191\u03c0) ^ (2 * k) * (-1) ^ k)", "state_after": "no goals"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "k : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\n\u22a2 \u21912 = 2", "state_after": "no goals"}, {"tactic": "convert (ofReal_re _).symm", "annotated_tactic": ["convert (<a>ofReal_re</a> _).<a>symm</a>", [{"full_name": "Complex.ofReal_re", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [92, 9], "def_end_pos": [92, 18]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case h.e'_5.h\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * x) = (1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x) / 2).re", "state_after": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x) / 2 = \u2191(1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * x))"}, {"tactic": "rw [ofReal_mul]", "annotated_tactic": ["rw [<a>ofReal_mul</a>]", [{"full_name": "Complex.ofReal_mul", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [275, 9], "def_end_pos": [275, 19]}]], "state_before": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x) / 2 = \u2191(1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * x))", "state_after": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x) / 2 = \u2191(1 / \u2191n ^ (2 * k)) * \u2191(Real.cos (2 * \u03c0 * \u2191n * x))"}, {"tactic": "rw [\u2190 mul_div]", "annotated_tactic": ["rw [\u2190 <a>mul_div</a>]", [{"full_name": "mul_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [324, 9], "def_end_pos": [324, 16]}]], "state_before": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x) / 2 = \u2191(1 / \u2191n ^ (2 * k)) * \u2191(Real.cos (2 * \u03c0 * \u2191n * x))", "state_after": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) * ((\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x) / 2) = \u2191(1 / \u2191n ^ (2 * k)) * \u2191(Real.cos (2 * \u03c0 * \u2191n * x))"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) * ((\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x) / 2) = \u2191(1 / \u2191n ^ (2 * k)) * \u2191(Real.cos (2 * \u03c0 * \u2191n * x))", "state_after": "case h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) = \u2191(1 / \u2191n ^ (2 * k))\n\ncase h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x) / 2 = \u2191(Real.cos (2 * \u03c0 * \u2191n * x))"}, {"tactic": "rw [ofReal_div, ofReal_one, ofReal_pow]", "annotated_tactic": ["rw [<a>ofReal_div</a>, <a>ofReal_one</a>, <a>ofReal_pow</a>]", [{"full_name": "Complex.ofReal_div", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [863, 9], "def_end_pos": [863, 19]}, {"full_name": "Complex.ofReal_one", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [175, 9], "def_end_pos": [175, 19]}, {"full_name": "Complex.ofReal_pow", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [735, 9], "def_end_pos": [735, 19]}]], "state_before": "case h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) = \u2191(1 / \u2191n ^ (2 * k))", "state_after": "case h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) = 1 / \u2191\u2191n ^ (2 * k)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) = 1 / \u2191\u2191n ^ (2 * k)", "state_after": "no goals"}, {"tactic": "rw [ofReal_cos, ofReal_mul, fourier_coe_apply, fourier_coe_apply, cos, ofReal_one, div_one,\n  div_one, ofReal_mul, ofReal_mul, ofReal_two, Int.cast_neg, Int.cast_ofNat,\n  ofReal_nat_cast]", "annotated_tactic": ["rw [<a>ofReal_cos</a>, <a>ofReal_mul</a>, <a>fourier_coe_apply</a>, <a>fourier_coe_apply</a>, <a>cos</a>, <a>ofReal_one</a>, <a>div_one</a>,\n        <a>div_one</a>, <a>ofReal_mul</a>, <a>ofReal_mul</a>, ofReal_two, <a>Int.cast_neg</a>, <a>Int.cast_ofNat</a>,\n        <a>ofReal_nat_cast</a>]", [{"full_name": "Complex.ofReal_cos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [962, 9], "def_end_pos": [962, 19]}, {"full_name": "Complex.ofReal_mul", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [275, 9], "def_end_pos": [275, 19]}, {"full_name": "fourier_coe_apply", "def_path": "Mathlib/Analysis/Fourier/AddCircle.lean", "def_pos": [161, 9], "def_end_pos": [161, 26]}, {"full_name": "fourier_coe_apply", "def_path": "Mathlib/Analysis/Fourier/AddCircle.lean", "def_pos": [161, 9], "def_end_pos": [161, 26]}, {"full_name": "Complex.cos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [391, 5], "def_end_pos": [391, 8]}, {"full_name": "Complex.ofReal_one", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [175, 9], "def_end_pos": [175, 19]}, {"full_name": "div_one", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 16]}, {"full_name": "div_one", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 16]}, {"full_name": "Complex.ofReal_mul", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [275, 9], "def_end_pos": [275, 19]}, {"full_name": "Complex.ofReal_mul", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [275, 9], "def_end_pos": [275, 19]}, {"full_name": "Int.cast_neg", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 17]}, {"full_name": "Int.cast_ofNat", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [65, 9], "def_end_pos": [65, 19]}, {"full_name": "Complex.ofReal_nat_cast", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [787, 9], "def_end_pos": [787, 24]}]], "state_before": "case h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x) / 2 = \u2191(Real.cos (2 * \u03c0 * \u2191n * x))", "state_after": "case h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 (cexp (2 * \u2191\u03c0 * I * \u2191n * \u2191x) + cexp (2 * \u2191\u03c0 * I * -\u2191n * \u2191x)) / 2 =\n    (cexp (2 * \u2191\u03c0 * \u2191n * \u2191x * I) + cexp (-(2 * \u2191\u03c0 * \u2191n * \u2191x) * I)) / 2"}, {"tactic": "congr 3", "annotated_tactic": ["congr 3", []], "state_before": "case h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 (cexp (2 * \u2191\u03c0 * I * \u2191n * \u2191x) + cexp (2 * \u2191\u03c0 * I * -\u2191n * \u2191x)) / 2 =\n    (cexp (2 * \u2191\u03c0 * \u2191n * \u2191x * I) + cexp (-(2 * \u2191\u03c0 * \u2191n * \u2191x) * I)) / 2", "state_after": "case h.e'_3.h.e'_1.e_a.e_a.e_a.e_z\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 2 * \u2191\u03c0 * I * \u2191n * \u2191x = 2 * \u2191\u03c0 * \u2191n * \u2191x * I\n\ncase h.e'_3.h.e'_1.e_a.e_a.e_a.e_z\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 2 * \u2191\u03c0 * I * -\u2191n * \u2191x = -(2 * \u2191\u03c0 * \u2191n * \u2191x) * I"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case h.e'_3.h.e'_1.e_a.e_a.e_a.e_z\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 2 * \u2191\u03c0 * I * \u2191n * \u2191x = 2 * \u2191\u03c0 * \u2191n * \u2191x * I", "state_after": "no goals"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case h.e'_3.h.e'_1.e_a.e_a.e_a.e_z\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 2 * \u2191\u03c0 * I * -\u2191n * \u2191x = -(2 * \u2191\u03c0 * \u2191n * \u2191x) * I", "state_after": "no goals"}, {"tactic": "convert (ofReal_re _).symm", "annotated_tactic": ["convert (<a>ofReal_re</a> _).<a>symm</a>", [{"full_name": "Complex.ofReal_re", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [92, 9], "def_end_pos": [92, 18]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case h.e'_6\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 (-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))) =\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x) / 2).re", "state_after": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x) / 2 =\n    \u2191((-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n        Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))"}, {"tactic": "rw [ofReal_mul, ofReal_div, ofReal_div, ofReal_mul, ofReal_pow, ofReal_pow, ofReal_neg,\n  ofReal_nat_cast, ofReal_mul, ofReal_two, ofReal_one]", "annotated_tactic": ["rw [<a>ofReal_mul</a>, <a>ofReal_div</a>, <a>ofReal_div</a>, <a>ofReal_mul</a>, <a>ofReal_pow</a>, <a>ofReal_pow</a>, <a>ofReal_neg</a>,\n      <a>ofReal_nat_cast</a>, <a>ofReal_mul</a>, ofReal_two, <a>ofReal_one</a>]", [{"full_name": "Complex.ofReal_mul", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [275, 9], "def_end_pos": [275, 19]}, {"full_name": "Complex.ofReal_div", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [863, 9], "def_end_pos": [863, 19]}, {"full_name": "Complex.ofReal_div", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [863, 9], "def_end_pos": [863, 19]}, {"full_name": "Complex.ofReal_mul", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [275, 9], "def_end_pos": [275, 19]}, {"full_name": "Complex.ofReal_pow", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [735, 9], "def_end_pos": [735, 19]}, {"full_name": "Complex.ofReal_pow", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [735, 9], "def_end_pos": [735, 19]}, {"full_name": "Complex.ofReal_neg", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [254, 9], "def_end_pos": [254, 19]}, {"full_name": "Complex.ofReal_nat_cast", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [787, 9], "def_end_pos": [787, 24]}, {"full_name": "Complex.ofReal_mul", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [275, 9], "def_end_pos": [275, 19]}, {"full_name": "Complex.ofReal_one", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [175, 9], "def_end_pos": [175, 19]}]], "state_before": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x) / 2 =\n    \u2191((-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n        Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))", "state_after": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x) / 2 =\n    (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))"}, {"tactic": "rw [bernoulliFun]", "annotated_tactic": ["rw [<a>bernoulliFun</a>]", [{"full_name": "bernoulliFun", "def_path": "Mathlib/NumberTheory/ZetaValues.lean", "def_pos": [49, 5], "def_end_pos": [49, 17]}]], "state_before": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x) / 2 =\n    (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))", "state_after": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! *\n        \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k)))) /\n      2 =\n    (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * (\u2191(fourier \u2191n) \u2191x + \u2191(fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! *\n        \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k)))) /\n      2 =\n    (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Normed/Group/Hom.lean", "full_name": "NormedAddGroupHom.comp_assoc", "start": [698, 1], "end": [702, 6], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "V : Type u_1\nV\u2081 : Type u_2\nV\u2082 : Type u_3\nV\u2083 : Type u_4\ninst\u271d\u2074 : SeminormedAddCommGroup V\ninst\u271d\u00b3 : SeminormedAddCommGroup V\u2081\ninst\u271d\u00b2 : SeminormedAddCommGroup V\u2082\ninst\u271d\u00b9 : SeminormedAddCommGroup V\u2083\nf\u271d g\u271d : NormedAddGroupHom V\u2081 V\u2082\nV\u2084 : Type u_5\ninst\u271d : SeminormedAddCommGroup V\u2084\nh : NormedAddGroupHom V\u2083 V\u2084\ng : NormedAddGroupHom V\u2082 V\u2083\nf : NormedAddGroupHom V\u2081 V\u2082\n\u22a2 NormedAddGroupHom.comp (NormedAddGroupHom.comp h g) f = NormedAddGroupHom.comp h (NormedAddGroupHom.comp g f)", "state_after": "case H\nV : Type u_1\nV\u2081 : Type u_2\nV\u2082 : Type u_3\nV\u2083 : Type u_4\ninst\u271d\u2074 : SeminormedAddCommGroup V\ninst\u271d\u00b3 : SeminormedAddCommGroup V\u2081\ninst\u271d\u00b2 : SeminormedAddCommGroup V\u2082\ninst\u271d\u00b9 : SeminormedAddCommGroup V\u2083\nf\u271d g\u271d : NormedAddGroupHom V\u2081 V\u2082\nV\u2084 : Type u_5\ninst\u271d : SeminormedAddCommGroup V\u2084\nh : NormedAddGroupHom V\u2083 V\u2084\ng : NormedAddGroupHom V\u2082 V\u2083\nf : NormedAddGroupHom V\u2081 V\u2082\nx\u271d : V\u2081\n\u22a2 \u2191(NormedAddGroupHom.comp (NormedAddGroupHom.comp h g) f) x\u271d =\n    \u2191(NormedAddGroupHom.comp h (NormedAddGroupHom.comp g f)) x\u271d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case H\nV : Type u_1\nV\u2081 : Type u_2\nV\u2082 : Type u_3\nV\u2083 : Type u_4\ninst\u271d\u2074 : SeminormedAddCommGroup V\ninst\u271d\u00b3 : SeminormedAddCommGroup V\u2081\ninst\u271d\u00b2 : SeminormedAddCommGroup V\u2082\ninst\u271d\u00b9 : SeminormedAddCommGroup V\u2083\nf\u271d g\u271d : NormedAddGroupHom V\u2081 V\u2082\nV\u2084 : Type u_5\ninst\u271d : SeminormedAddCommGroup V\u2084\nh : NormedAddGroupHom V\u2083 V\u2084\ng : NormedAddGroupHom V\u2082 V\u2083\nf : NormedAddGroupHom V\u2081 V\u2082\nx\u271d : V\u2081\n\u22a2 \u2191(NormedAddGroupHom.comp (NormedAddGroupHom.comp h g) f) x\u271d =\n    \u2191(NormedAddGroupHom.comp h (NormedAddGroupHom.comp g f)) x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/Basic.lean", "full_name": "Polynomial.erase_ne", "start": [1104, 1], "end": [1105, 24], "traced_tactics": [{"tactic": "simp [coeff_erase, h]", "annotated_tactic": ["simp [<a>coeff_erase</a>, h]", [{"full_name": "Polynomial.coeff_erase", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [1081, 9], "def_end_pos": [1081, 20]}]], "state_before": "R : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q p : R[X]\nn i : \u2115\nh : i \u2260 n\n\u22a2 coeff (erase n p) i = coeff p i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.ext", "start": [736, 8], "end": [742, 63], "traced_tactics": [{"tactic": "have h0 : some a = some a' := h 0", "annotated_tactic": ["have h0 : <a>some</a> a = <a>some</a> a' := h 0", [{"full_name": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}, {"full_name": "Option.some", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2143, 5], "def_end_pos": [2143, 9]}]], "state_before": "\u03b1 : Type u_1\na : \u03b1\nl\u2081 : List \u03b1\na' : \u03b1\nl\u2082 : List \u03b1\nh : \u2200 (n : Nat), get? (a :: l\u2081) n = get? (a' :: l\u2082) n\n\u22a2 a :: l\u2081 = a' :: l\u2082", "state_after": "\u03b1 : Type u_1\na : \u03b1\nl\u2081 : List \u03b1\na' : \u03b1\nl\u2082 : List \u03b1\nh : \u2200 (n : Nat), get? (a :: l\u2081) n = get? (a' :: l\u2082) n\nh0 : some a = some a'\n\u22a2 a :: l\u2081 = a' :: l\u2082"}, {"tactic": "injection h0 with aa", "annotated_tactic": ["injection h0 with aa", []], "state_before": "\u03b1 : Type u_1\na : \u03b1\nl\u2081 : List \u03b1\na' : \u03b1\nl\u2082 : List \u03b1\nh : \u2200 (n : Nat), get? (a :: l\u2081) n = get? (a' :: l\u2082) n\nh0 : some a = some a'\n\u22a2 a :: l\u2081 = a' :: l\u2082", "state_after": "\u03b1 : Type u_1\na : \u03b1\nl\u2081 : List \u03b1\na' : \u03b1\nl\u2082 : List \u03b1\nh : \u2200 (n : Nat), get? (a :: l\u2081) n = get? (a' :: l\u2082) n\naa : a = a'\n\u22a2 a :: l\u2081 = a' :: l\u2082"}, {"tactic": "simp only [aa, ext fun n => h (n+1)]", "annotated_tactic": ["simp only [aa, ext fun n => h (n+1)]", []], "state_before": "\u03b1 : Type u_1\na : \u03b1\nl\u2081 : List \u03b1\na' : \u03b1\nl\u2082 : List \u03b1\nh : \u2200 (n : Nat), get? (a :: l\u2081) n = get? (a' :: l\u2082) n\naa : a = a'\n\u22a2 a :: l\u2081 = a' :: l\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Basic.lean", "full_name": "Set.Icc_subset_Ioi_iff", "start": [573, 1], "end": [574, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Order/Basic.lean", "full_name": "Int.abs_lt_one_iff", "start": [120, 1], "end": [126, 60], "traced_tactics": [{"tactic": "let \u27e8hn, hp\u27e9 := abs_lt.mp a0", "annotated_tactic": ["let \u27e8hn, hp\u27e9 := abs_lt.mp a0", []], "state_before": "a : \u2124\na0 : |a| < 1\n\u22a2 a = 0", "state_after": "a : \u2124\na0 : |a| < 1\nhn : -1 < a\nhp : a < 1\n\u22a2 a = 0"}, {"tactic": "rw [\u2190zero_add 1, lt_add_one_iff] at hp", "annotated_tactic": ["rw [\u2190<a>zero_add</a> 1, <a>lt_add_one_iff</a>] at hp", [{"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "Int.lt_add_one_iff", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [851, 9], "def_end_pos": [851, 23]}]], "state_before": "a : \u2124\na0 : |a| < 1\nhn : -1 < a\nhp : a < 1\n\u22a2 a = 0", "state_after": "a : \u2124\na0 : |a| < 1\nhn : -1 < a\nhp\u271d : a < 1\nhp : a \u2264 0\n\u22a2 a = 0"}, {"tactic": "exact hp.antisymm hn", "annotated_tactic": ["exact hp.antisymm hn", []], "state_before": "a : \u2124\na0 : |a| < 1\nhn : -1 < a\nhp\u271d : a < 1\nhp : a \u2264 0\n\u22a2 a = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Generator.lean", "full_name": "CategoryTheory.isSeparator_coprod_of_isSeparator_left", "start": [559, 1], "end": [561, 64], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b9 : HasZeroMorphisms C\nG H : C\ninst\u271d : HasBinaryCoproduct G H\nhG : IsSeparator G\n\u22a2 {G} \u2286 {G, H}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "full_name": "Algebra.toSubsemiring_eq_top", "start": [816, 1], "end": [817, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Real/NNReal.lean", "full_name": "Set.OrdConnected.image_real_toNNReal", "start": [1043, 1], "end": [1050, 79], "traced_tactics": [{"tactic": "refine' \u27e8ball_image_iff.2 fun x hx => ball_image_iff.2 fun y hy z hz => _\u27e9", "annotated_tactic": ["refine' \u27e8<a>ball_image_iff</a>.2 fun x hx => <a>ball_image_iff</a>.2 fun y hy z hz => _\u27e9", [{"full_name": "Set.ball_image_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [249, 9], "def_end_pos": [249, 23]}, {"full_name": "Set.ball_image_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [249, 9], "def_end_pos": [249, 23]}]], "state_before": "s : Set \u211d\nt : Set \u211d\u22650\nh : OrdConnected s\n\u22a2 OrdConnected (toNNReal '' s)", "state_after": "s : Set \u211d\nt : Set \u211d\u22650\nh : OrdConnected s\nx : \u211d\nhx : x \u2208 s\ny : \u211d\nhy : y \u2208 s\nz : \u211d\u22650\nhz : z \u2208 Icc (toNNReal x) (toNNReal y)\n\u22a2 z \u2208 toNNReal '' s"}, {"tactic": "cases' le_total y 0 with hy\u2080 hy\u2080", "annotated_tactic": ["cases' <a>le_total</a> y 0 with hy\u2080 hy\u2080", [{"full_name": "le_total", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [295, 9], "def_end_pos": [295, 17]}]], "state_before": "s : Set \u211d\nt : Set \u211d\u22650\nh : OrdConnected s\nx : \u211d\nhx : x \u2208 s\ny : \u211d\nhy : y \u2208 s\nz : \u211d\u22650\nhz : z \u2208 Icc (toNNReal x) (toNNReal y)\n\u22a2 z \u2208 toNNReal '' s", "state_after": "case inl\ns : Set \u211d\nt : Set \u211d\u22650\nh : OrdConnected s\nx : \u211d\nhx : x \u2208 s\ny : \u211d\nhy : y \u2208 s\nz : \u211d\u22650\nhz : z \u2208 Icc (toNNReal x) (toNNReal y)\nhy\u2080 : y \u2264 0\n\u22a2 z \u2208 toNNReal '' s\n\ncase inr\ns : Set \u211d\nt : Set \u211d\u22650\nh : OrdConnected s\nx : \u211d\nhx : x \u2208 s\ny : \u211d\nhy : y \u2208 s\nz : \u211d\u22650\nhz : z \u2208 Icc (toNNReal x) (toNNReal y)\nhy\u2080 : 0 \u2264 y\n\u22a2 z \u2208 toNNReal '' s"}, {"tactic": "rw [mem_Icc, Real.toNNReal_of_nonpos hy\u2080, nonpos_iff_eq_zero] at hz", "annotated_tactic": ["rw [<a>mem_Icc</a>, <a>Real.toNNReal_of_nonpos</a> hy\u2080, <a>nonpos_iff_eq_zero</a>] at hz", [{"full_name": "Set.mem_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}, {"full_name": "Real.toNNReal_of_nonpos", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [637, 9], "def_end_pos": [637, 27]}, {"full_name": "nonpos_iff_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [237, 3], "def_end_pos": [237, 14]}]], "state_before": "case inl\ns : Set \u211d\nt : Set \u211d\u22650\nh : OrdConnected s\nx : \u211d\nhx : x \u2208 s\ny : \u211d\nhy : y \u2208 s\nz : \u211d\u22650\nhz : z \u2208 Icc (toNNReal x) (toNNReal y)\nhy\u2080 : y \u2264 0\n\u22a2 z \u2208 toNNReal '' s", "state_after": "case inl\ns : Set \u211d\nt : Set \u211d\u22650\nh : OrdConnected s\nx : \u211d\nhx : x \u2208 s\ny : \u211d\nhy : y \u2208 s\nz : \u211d\u22650\nhz : toNNReal x \u2264 z \u2227 z = 0\nhy\u2080 : y \u2264 0\n\u22a2 z \u2208 toNNReal '' s"}, {"tactic": "exact \u27e8y, hy, (toNNReal_of_nonpos hy\u2080).trans hz.2.symm\u27e9", "annotated_tactic": ["exact \u27e8y, hy, (<a>toNNReal_of_nonpos</a> hy\u2080).<a>trans</a> hz.2.<a>symm</a>\u27e9", [{"full_name": "Real.toNNReal_of_nonpos", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [637, 9], "def_end_pos": [637, 27]}, {"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case inl\ns : Set \u211d\nt : Set \u211d\u22650\nh : OrdConnected s\nx : \u211d\nhx : x \u2208 s\ny : \u211d\nhy : y \u2208 s\nz : \u211d\u22650\nhz : toNNReal x \u2264 z \u2227 z = 0\nhy\u2080 : y \u2264 0\n\u22a2 z \u2208 toNNReal '' s", "state_after": "no goals"}, {"tactic": "lift y to \u211d\u22650 using hy\u2080", "annotated_tactic": ["lift y to \u211d\u22650 using hy\u2080", []], "state_before": "case inr\ns : Set \u211d\nt : Set \u211d\u22650\nh : OrdConnected s\nx : \u211d\nhx : x \u2208 s\ny : \u211d\nhy : y \u2208 s\nz : \u211d\u22650\nhz : z \u2208 Icc (toNNReal x) (toNNReal y)\nhy\u2080 : 0 \u2264 y\n\u22a2 z \u2208 toNNReal '' s", "state_after": "case inr.intro\ns : Set \u211d\nt : Set \u211d\u22650\nh : OrdConnected s\nx : \u211d\nhx : x \u2208 s\nz y : \u211d\u22650\nhy : \u2191y \u2208 s\nhz : z \u2208 Icc (toNNReal x) (toNNReal \u2191y)\n\u22a2 z \u2208 toNNReal '' s"}, {"tactic": "rw [toNNReal_coe] at hz", "annotated_tactic": ["rw [<a>toNNReal_coe</a>] at hz", [{"full_name": "Real.toNNReal_coe", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [391, 9], "def_end_pos": [391, 33]}]], "state_before": "case inr.intro\ns : Set \u211d\nt : Set \u211d\u22650\nh : OrdConnected s\nx : \u211d\nhx : x \u2208 s\nz y : \u211d\u22650\nhy : \u2191y \u2208 s\nhz : z \u2208 Icc (toNNReal x) (toNNReal \u2191y)\n\u22a2 z \u2208 toNNReal '' s", "state_after": "case inr.intro\ns : Set \u211d\nt : Set \u211d\u22650\nh : OrdConnected s\nx : \u211d\nhx : x \u2208 s\nz y : \u211d\u22650\nhy : \u2191y \u2208 s\nhz : z \u2208 Icc (toNNReal x) y\n\u22a2 z \u2208 toNNReal '' s"}, {"tactic": "exact \u27e8z, h.out hx hy \u27e8toNNReal_le_iff_le_coe.1 hz.1, hz.2\u27e9, toNNReal_coe\u27e9", "annotated_tactic": ["exact \u27e8z, h.out hx hy \u27e8<a>toNNReal_le_iff_le_coe</a>.1 hz.1, hz.2\u27e9, <a>toNNReal_coe</a>\u27e9", [{"full_name": "Real.toNNReal_le_iff_le_coe", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [685, 9], "def_end_pos": [685, 31]}, {"full_name": "Real.toNNReal_coe", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [391, 9], "def_end_pos": [391, 33]}]], "state_before": "case inr.intro\ns : Set \u211d\nt : Set \u211d\u22650\nh : OrdConnected s\nx : \u211d\nhx : x \u2208 s\nz y : \u211d\u22650\nhy : \u2191y \u2208 s\nhz : z \u2208 Icc (toNNReal x) y\n\u22a2 z \u2208 toNNReal '' s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.mem_union_left", "start": [749, 1], "end": [750, 9], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Associated.lean", "full_name": "not_irreducible_one", "start": [196, 1], "end": [196, 91], "traced_tactics": [{"tactic": "simp [irreducible_iff]", "annotated_tactic": ["simp [<a>irreducible_iff</a>]", [{"full_name": "irreducible_iff", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [190, 9], "def_end_pos": [190, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : Monoid \u03b1\n\u22a2 \u00acIrreducible 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/InnerProductSpace/LinearPMap.lean", "full_name": "LinearPMap.adjoint_apply_eq", "start": [195, 1], "end": [197, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Init/CCLemmas.lean", "full_name": "not_eq_of_eq_true", "start": [72, 1], "end": [73, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "full_name": "Multiset.pow_count", "start": [151, 1], "end": [152, 33], "traced_tactics": [{"tactic": "rw [filter_eq, prod_replicate]", "annotated_tactic": ["rw [<a>filter_eq</a>, <a>prod_replicate</a>]", [{"full_name": "Multiset.filter_eq", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2583, 9], "def_end_pos": [2583, 18]}, {"full_name": "Multiset.prod_replicate", "def_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "def_pos": [129, 9], "def_end_pos": [129, 23]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ns t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\n\u22a2 a ^ count a s = prod (filter (Eq a) s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "Set.Countable.ae_not_mem", "start": [3137, 1], "end": [3139, 64], "traced_tactics": [{"tactic": "simpa only [ae_iff, Classical.not_not] using h.measure_zero \u03bc", "annotated_tactic": ["simpa only [<a>ae_iff</a>, <a>Classical.not_not</a>] using h.measure_zero \u03bc", [{"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [388, 9], "def_end_pos": [388, 15]}, {"full_name": "Classical.not_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [683, 24], "def_end_pos": [683, 31]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\u271d\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d\ns\u271d s' t : Set \u03b1\u271d\ninst\u271d\u00b9 : NoAtoms \u03bc\u271d\n\u03b1 : Type u_8\nm : MeasurableSpace \u03b1\ns : Set \u03b1\nh : Set.Countable s\n\u03bc : Measure \u03b1\ninst\u271d : NoAtoms \u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean", "full_name": "ne\u2082\u2083_of_not_collinear", "start": [509, 1], "end": [512, 29], "traced_tactics": [{"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np\u2081 p\u2082 p\u2083 : P\nh : \u00acCollinear k {p\u2081, p\u2082, p\u2083}\n\u22a2 p\u2082 \u2260 p\u2083", "state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np\u2081 p\u2082 : P\nh : \u00acCollinear k {p\u2081, p\u2082, p\u2082}\n\u22a2 False"}, {"tactic": "simp [collinear_pair] at h", "annotated_tactic": ["simp [<a>collinear_pair</a>] at h", [{"full_name": "collinear_pair", "def_path": "Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean", "def_pos": [433, 9], "def_end_pos": [433, 23]}]], "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DivisionRing k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np\u2081 p\u2082 : P\nh : \u00acCollinear k {p\u2081, p\u2082, p\u2082}\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Homology/ImageToKernel.lean", "full_name": "imageToKernel'_kernelSubobjectIso", "start": [429, 1], "end": [433, 24], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw\u271d : f \u226b g = 0\ninst\u271d\u00b9 : HasKernels V\ninst\u271d : HasImages V\nw : f \u226b g = 0\n\u22a2 imageToKernel' f g w \u226b (kernelSubobjectIso g).inv = (imageSubobjectIso f).inv \u226b imageToKernel f g w", "state_after": "case h\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw\u271d : f \u226b g = 0\ninst\u271d\u00b9 : HasKernels V\ninst\u271d : HasImages V\nw : f \u226b g = 0\n\u22a2 (imageToKernel' f g w \u226b (kernelSubobjectIso g).inv) \u226b Subobject.arrow (kernelSubobject g) =\n    ((imageSubobjectIso f).inv \u226b imageToKernel f g w) \u226b Subobject.arrow (kernelSubobject g)"}, {"tactic": "simp [imageToKernel']", "annotated_tactic": ["simp [<a>imageToKernel'</a>]", [{"full_name": "imageToKernel'", "def_path": "Mathlib/Algebra/Homology/ImageToKernel.lean", "def_pos": [414, 5], "def_end_pos": [414, 19]}]], "state_before": "case h\n\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasZeroMorphisms V\nA B C : V\nf : A \u27f6 B\ng : B \u27f6 C\nw\u271d : f \u226b g = 0\ninst\u271d\u00b9 : HasKernels V\ninst\u271d : HasImages V\nw : f \u226b g = 0\n\u22a2 (imageToKernel' f g w \u226b (kernelSubobjectIso g).inv) \u226b Subobject.arrow (kernelSubobject g) =\n    ((imageSubobjectIso f).inv \u226b imageToKernel f g w) \u226b Subobject.arrow (kernelSubobject g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/InnerProductSpace/Adjoint.lean", "full_name": "ContinuousLinearMap.apply_norm_sq_eq_inner_adjoint_right", "start": [159, 1], "end": [162, 45], "traced_tactics": [{"tactic": "have h : \u27eax, (A\u2020 * A) x\u27eb = \u27eaA x, A x\u27eb := by rw [\u2190 adjoint_inner_right]; rfl", "annotated_tactic": ["have h : \u27eax, (A\u2020 * A) x\u27eb = \u27eaA x, A x\u27eb := by rw [\u2190 <a>adjoint_inner_right</a>]; rfl", [{"full_name": "ContinuousLinearMap.adjoint_inner_right", "def_path": "Mathlib/Analysis/InnerProductSpace/Adjoint.lean", "def_pos": [129, 9], "def_end_pos": [129, 28]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] E\nx : E\n\u22a2 \u2016\u2191A x\u2016 ^ 2 = \u2191re (inner x (\u2191(\u2191adjoint A * A) x))", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] E\nx : E\nh : inner x (\u2191(\u2191adjoint A * A) x) = inner (\u2191A x) (\u2191A x)\n\u22a2 \u2016\u2191A x\u2016 ^ 2 = \u2191re (inner x (\u2191(\u2191adjoint A * A) x))"}, {"tactic": "rw [h, \u2190 inner_self_eq_norm_sq (\ud835\udd5c := \ud835\udd5c) _]", "annotated_tactic": ["rw [h, \u2190 <a>inner_self_eq_norm_sq</a> (\ud835\udd5c := \ud835\udd5c) _]", [{"full_name": "inner_self_eq_norm_sq", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [1008, 9], "def_end_pos": [1008, 30]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] E\nx : E\nh : inner x (\u2191(\u2191adjoint A * A) x) = inner (\u2191A x) (\u2191A x)\n\u22a2 \u2016\u2191A x\u2016 ^ 2 = \u2191re (inner x (\u2191(\u2191adjoint A * A) x))", "state_after": "no goals"}, {"tactic": "rw [\u2190 adjoint_inner_right]", "annotated_tactic": ["rw [\u2190 <a>adjoint_inner_right</a>]", [{"full_name": "ContinuousLinearMap.adjoint_inner_right", "def_path": "Mathlib/Analysis/InnerProductSpace/Adjoint.lean", "def_pos": [129, 9], "def_end_pos": [129, 28]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] E\nx : E\n\u22a2 inner x (\u2191(\u2191adjoint A * A) x) = inner (\u2191A x) (\u2191A x)", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] E\nx : E\n\u22a2 inner x (\u2191(\u2191adjoint A * A) x) = inner x (\u2191(\u2191adjoint A) (\u2191A x))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] E\nx : E\n\u22a2 inner x (\u2191(\u2191adjoint A * A) x) = inner x (\u2191(\u2191adjoint A) (\u2191A x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Bases.lean", "full_name": "Filter.HasBasis.frequently_iff", "start": [368, 1], "end": [370, 66], "traced_tactics": [{"tactic": "simp only [Filter.Frequently, hl.eventually_iff]", "annotated_tactic": ["simp only [<a>Filter.Frequently</a>, hl.eventually_iff]", [{"full_name": "Filter.Frequently", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1259, 15], "def_end_pos": [1259, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nq : \u03b1 \u2192 Prop\n\u22a2 (\u2203\u1da0 (x : \u03b1) in l, q x) \u2194 \u2200 (i : \u03b9), p i \u2192 \u2203 x, x \u2208 s i \u2227 q x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nq : \u03b1 \u2192 Prop\n\u22a2 (\u00ac\u2203 i, p i \u2227 \u2200 \u2983x : \u03b1\u2984, x \u2208 s i \u2192 \u00acq x) \u2194 \u2200 (i : \u03b9), p i \u2192 \u2203 x, x \u2208 s i \u2227 q x"}, {"tactic": "push_neg", "annotated_tactic": ["push_neg", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nq : \u03b1 \u2192 Prop\n\u22a2 (\u00ac\u2203 i, p i \u2227 \u2200 \u2983x : \u03b1\u2984, x \u2208 s i \u2192 \u00acq x) \u2194 \u2200 (i : \u03b9), p i \u2192 \u2203 x, x \u2208 s i \u2227 q x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nq : \u03b1 \u2192 Prop\n\u22a2 (\u2200 (i : \u03b9), p i \u2192 Exists fun \u2983x\u2984 => x \u2208 s i \u2227 q x) \u2194 \u2200 (i : \u03b9), p i \u2192 \u2203 x, x \u2208 s i \u2227 q x"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nq : \u03b1 \u2192 Prop\n\u22a2 (\u2200 (i : \u03b9), p i \u2192 Exists fun \u2983x\u2984 => x \u2208 s i \u2227 q x) \u2194 \u2200 (i : \u03b9), p i \u2192 \u2203 x, x \u2208 s i \u2227 q x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/StarSubalgebra.lean", "full_name": "elementalStarAlgebra.isClosed", "start": [227, 11], "end": [228, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/DFinsupp/NeLocus.lean", "full_name": "DFinsupp.zipWith_neLocus_eq_right", "start": [102, 1], "end": [107, 49], "traced_tactics": [{"tactic": "ext a", "annotated_tactic": ["ext a", []], "state_before": "\u03b1 : Type u_1\nN : \u03b1 \u2192 Type u_2\ninst\u271d\u2075 : DecidableEq \u03b1\nM : \u03b1 \u2192 Type u_3\nP : \u03b1 \u2192 Type u_4\ninst\u271d\u2074 : (a : \u03b1) \u2192 Zero (N a)\ninst\u271d\u00b3 : (a : \u03b1) \u2192 Zero (M a)\ninst\u271d\u00b2 : (a : \u03b1) \u2192 Zero (P a)\ninst\u271d\u00b9 : (a : \u03b1) \u2192 DecidableEq (M a)\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (P a)\nF : (a : \u03b1) \u2192 M a \u2192 N a \u2192 P a\nF0 : \u2200 (a : \u03b1), F a 0 0 = 0\nf\u2081 f\u2082 : \u03a0\u2080 (a : \u03b1), M a\ng : \u03a0\u2080 (a : \u03b1), N a\nhF : \u2200 (a : \u03b1) (g : N a), Function.Injective fun f => F a f g\n\u22a2 neLocus (zipWith F F0 f\u2081 g) (zipWith F F0 f\u2082 g) = neLocus f\u2081 f\u2082", "state_after": "case a\n\u03b1 : Type u_1\nN : \u03b1 \u2192 Type u_2\ninst\u271d\u2075 : DecidableEq \u03b1\nM : \u03b1 \u2192 Type u_3\nP : \u03b1 \u2192 Type u_4\ninst\u271d\u2074 : (a : \u03b1) \u2192 Zero (N a)\ninst\u271d\u00b3 : (a : \u03b1) \u2192 Zero (M a)\ninst\u271d\u00b2 : (a : \u03b1) \u2192 Zero (P a)\ninst\u271d\u00b9 : (a : \u03b1) \u2192 DecidableEq (M a)\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (P a)\nF : (a : \u03b1) \u2192 M a \u2192 N a \u2192 P a\nF0 : \u2200 (a : \u03b1), F a 0 0 = 0\nf\u2081 f\u2082 : \u03a0\u2080 (a : \u03b1), M a\ng : \u03a0\u2080 (a : \u03b1), N a\nhF : \u2200 (a : \u03b1) (g : N a), Function.Injective fun f => F a f g\na : \u03b1\n\u22a2 a \u2208 neLocus (zipWith F F0 f\u2081 g) (zipWith F F0 f\u2082 g) \u2194 a \u2208 neLocus f\u2081 f\u2082"}, {"tactic": "simpa only [mem_neLocus] using (hF a _).ne_iff", "annotated_tactic": ["simpa only [<a>mem_neLocus</a>] using (hF a _).<a>ne_iff</a>", [{"full_name": "DFinsupp.mem_neLocus", "def_path": "Mathlib/Data/DFinsupp/NeLocus.lean", "def_pos": [41, 9], "def_end_pos": [41, 20]}, {"full_name": "Function.Injective.ne_iff", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [111, 9], "def_end_pos": [111, 25]}]], "state_before": "case a\n\u03b1 : Type u_1\nN : \u03b1 \u2192 Type u_2\ninst\u271d\u2075 : DecidableEq \u03b1\nM : \u03b1 \u2192 Type u_3\nP : \u03b1 \u2192 Type u_4\ninst\u271d\u2074 : (a : \u03b1) \u2192 Zero (N a)\ninst\u271d\u00b3 : (a : \u03b1) \u2192 Zero (M a)\ninst\u271d\u00b2 : (a : \u03b1) \u2192 Zero (P a)\ninst\u271d\u00b9 : (a : \u03b1) \u2192 DecidableEq (M a)\ninst\u271d : (a : \u03b1) \u2192 DecidableEq (P a)\nF : (a : \u03b1) \u2192 M a \u2192 N a \u2192 P a\nF0 : \u2200 (a : \u03b1), F a 0 0 = 0\nf\u2081 f\u2082 : \u03a0\u2080 (a : \u03b1), M a\ng : \u03a0\u2080 (a : \u03b1), N a\nhF : \u2200 (a : \u03b1) (g : N a), Function.Injective fun f => F a f g\na : \u03b1\n\u22a2 a \u2208 neLocus (zipWith F F0 f\u2081 g) (zipWith F F0 f\u2082 g) \u2194 a \u2208 neLocus f\u2081 f\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "full_name": "Polynomial.X_pow_sub_one_eq_prod", "start": [133, 1], "end": [143, 40], "traced_tactics": [{"tactic": "classical\nrw [nthRootsFinset, \u2190 Multiset.toFinset_eq (IsPrimitiveRoot.nthRoots_nodup h)]\nsimp only [Finset.prod_mk, RingHom.map_one]\nrw [nthRoots]\nhave hmonic : (X ^ n - C (1 : R)).Monic := monic_X_pow_sub_C (1 : R) (ne_of_lt hpos).symm\nsymm\napply prod_multiset_X_sub_C_of_monic_of_roots_card_eq hmonic\nrw [@natDegree_X_pow_sub_C R _ _ n 1, \u2190 nthRoots]\nexact IsPrimitiveRoot.card_nthRoots h", "annotated_tactic": ["classical\n  rw [<a>nthRootsFinset</a>, \u2190 <a>Multiset.toFinset_eq</a> (<a>IsPrimitiveRoot.nthRoots_nodup</a> h)]\n  simp only [<a>Finset.prod_mk</a>, <a>RingHom.map_one</a>]\n  rw [<a>nthRoots</a>]\n  have hmonic : (<a>X</a> ^ n - <a>C</a> (1 : R)).<a>Monic</a> := <a>monic_X_pow_sub_C</a> (1 : R) (<a>ne_of_lt</a> hpos).<a>symm</a>\n  symm\n  apply <a>prod_multiset_X_sub_C_of_monic_of_roots_card_eq</a> hmonic\n  rw [@<a>natDegree_X_pow_sub_C</a> R _ _ n 1, \u2190 <a>nthRoots</a>]\n  exact <a>IsPrimitiveRoot.card_nthRoots</a> h", [{"full_name": "Polynomial.nthRootsFinset", "def_path": "Mathlib/Data/Polynomial/RingDivision.lean", "def_pos": [835, 5], "def_end_pos": [835, 19]}, {"full_name": "Multiset.toFinset_eq", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3190, 9], "def_end_pos": [3190, 20]}, {"full_name": "IsPrimitiveRoot.nthRoots_nodup", "def_path": "Mathlib/RingTheory/RootsOfUnity/Basic.lean", "def_pos": [843, 9], "def_end_pos": [843, 23]}, {"full_name": "Finset.prod_mk", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [67, 9], "def_end_pos": [67, 16]}, {"full_name": "RingHom.map_one", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [559, 19], "def_end_pos": [559, 26]}, {"full_name": "Polynomial.nthRoots", "def_path": "Mathlib/Data/Polynomial/RingDivision.lean", "def_pos": [796, 5], "def_end_pos": [796, 13]}, {"full_name": "Polynomial.X", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [566, 5], "def_end_pos": [566, 6]}, {"full_name": "Polynomial.C", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [498, 5], "def_end_pos": [498, 6]}, {"full_name": "Polynomial.Monic", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [75, 5], "def_end_pos": [75, 10]}, {"full_name": "Polynomial.monic_X_pow_sub_C", "def_path": "Mathlib/Data/Polynomial/Monic.lean", "def_pos": [384, 9], "def_end_pos": [384, 26]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "Ne.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [575, 9], "def_end_pos": [575, 16]}, {"full_name": "Polynomial.prod_multiset_X_sub_C_of_monic_of_roots_card_eq", "def_path": "Mathlib/Data/Polynomial/RingDivision.lean", "def_pos": [1249, 9], "def_end_pos": [1249, 56]}, {"full_name": "Polynomial.natDegree_X_pow_sub_C", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [1584, 9], "def_end_pos": [1584, 30]}, {"full_name": "Polynomial.nthRoots", "def_path": "Mathlib/Data/Polynomial/RingDivision.lean", "def_pos": [796, 5], "def_end_pos": [796, 13]}, {"full_name": "IsPrimitiveRoot.card_nthRoots", "def_path": "Mathlib/RingTheory/RootsOfUnity/Basic.lean", "def_pos": [824, 16], "def_end_pos": [824, 29]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\n\u22a2 X ^ n - 1 = \u220f \u03b6 in nthRootsFinset n R, (X - \u2191C \u03b6)", "state_after": "no goals"}, {"tactic": "rw [nthRootsFinset, \u2190 Multiset.toFinset_eq (IsPrimitiveRoot.nthRoots_nodup h)]", "annotated_tactic": ["rw [<a>nthRootsFinset</a>, \u2190 <a>Multiset.toFinset_eq</a> (<a>IsPrimitiveRoot.nthRoots_nodup</a> h)]", [{"full_name": "Polynomial.nthRootsFinset", "def_path": "Mathlib/Data/Polynomial/RingDivision.lean", "def_pos": [835, 5], "def_end_pos": [835, 19]}, {"full_name": "Multiset.toFinset_eq", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3190, 9], "def_end_pos": [3190, 20]}, {"full_name": "IsPrimitiveRoot.nthRoots_nodup", "def_path": "Mathlib/RingTheory/RootsOfUnity/Basic.lean", "def_pos": [843, 9], "def_end_pos": [843, 23]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\n\u22a2 X ^ n - 1 = \u220f \u03b6 in nthRootsFinset n R, (X - \u2191C \u03b6)", "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\n\u22a2 X ^ n - 1 = \u220f \u03b6 in { val := nthRoots n 1, nodup := (_ : Multiset.Nodup (nthRoots n 1)) }, (X - \u2191C \u03b6)"}, {"tactic": "simp only [Finset.prod_mk, RingHom.map_one]", "annotated_tactic": ["simp only [<a>Finset.prod_mk</a>, <a>RingHom.map_one</a>]", [{"full_name": "Finset.prod_mk", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [67, 9], "def_end_pos": [67, 16]}, {"full_name": "RingHom.map_one", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [559, 19], "def_end_pos": [559, 26]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\n\u22a2 X ^ n - 1 = \u220f \u03b6 in { val := nthRoots n 1, nodup := (_ : Multiset.Nodup (nthRoots n 1)) }, (X - \u2191C \u03b6)", "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\n\u22a2 X ^ n - 1 = Multiset.prod (Multiset.map (fun x => X - \u2191C x) (nthRoots n 1))"}, {"tactic": "rw [nthRoots]", "annotated_tactic": ["rw [<a>nthRoots</a>]", [{"full_name": "Polynomial.nthRoots", "def_path": "Mathlib/Data/Polynomial/RingDivision.lean", "def_pos": [796, 5], "def_end_pos": [796, 13]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\n\u22a2 X ^ n - 1 = Multiset.prod (Multiset.map (fun x => X - \u2191C x) (nthRoots n 1))", "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\n\u22a2 X ^ n - 1 = Multiset.prod (Multiset.map (fun x => X - \u2191C x) (roots (X ^ n - \u2191C 1)))"}, {"tactic": "have hmonic : (X ^ n - C (1 : R)).Monic := monic_X_pow_sub_C (1 : R) (ne_of_lt hpos).symm", "annotated_tactic": ["have hmonic : (<a>X</a> ^ n - <a>C</a> (1 : R)).<a>Monic</a> := <a>monic_X_pow_sub_C</a> (1 : R) (<a>ne_of_lt</a> hpos).<a>symm</a>", [{"full_name": "Polynomial.X", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [566, 5], "def_end_pos": [566, 6]}, {"full_name": "Polynomial.C", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [498, 5], "def_end_pos": [498, 6]}, {"full_name": "Polynomial.Monic", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [75, 5], "def_end_pos": [75, 10]}, {"full_name": "Polynomial.monic_X_pow_sub_C", "def_path": "Mathlib/Data/Polynomial/Monic.lean", "def_pos": [384, 9], "def_end_pos": [384, 26]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "Ne.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [575, 9], "def_end_pos": [575, 16]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\n\u22a2 X ^ n - 1 = Multiset.prod (Multiset.map (fun x => X - \u2191C x) (roots (X ^ n - \u2191C 1)))", "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\nhmonic : Monic (X ^ n - \u2191C 1)\n\u22a2 X ^ n - 1 = Multiset.prod (Multiset.map (fun x => X - \u2191C x) (roots (X ^ n - \u2191C 1)))"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\nhmonic : Monic (X ^ n - \u2191C 1)\n\u22a2 X ^ n - 1 = Multiset.prod (Multiset.map (fun x => X - \u2191C x) (roots (X ^ n - \u2191C 1)))", "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\nhmonic : Monic (X ^ n - \u2191C 1)\n\u22a2 Multiset.prod (Multiset.map (fun x => X - \u2191C x) (roots (X ^ n - \u2191C 1))) = X ^ n - 1"}, {"tactic": "apply prod_multiset_X_sub_C_of_monic_of_roots_card_eq hmonic", "annotated_tactic": ["apply <a>prod_multiset_X_sub_C_of_monic_of_roots_card_eq</a> hmonic", [{"full_name": "Polynomial.prod_multiset_X_sub_C_of_monic_of_roots_card_eq", "def_path": "Mathlib/Data/Polynomial/RingDivision.lean", "def_pos": [1249, 9], "def_end_pos": [1249, 56]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\nhmonic : Monic (X ^ n - \u2191C 1)\n\u22a2 Multiset.prod (Multiset.map (fun x => X - \u2191C x) (roots (X ^ n - \u2191C 1))) = X ^ n - 1", "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\nhmonic : Monic (X ^ n - \u2191C 1)\n\u22a2 \u2191Multiset.card (roots (X ^ n - \u2191C 1)) = natDegree (X ^ n - \u2191C 1)"}, {"tactic": "rw [@natDegree_X_pow_sub_C R _ _ n 1, \u2190 nthRoots]", "annotated_tactic": ["rw [@<a>natDegree_X_pow_sub_C</a> R _ _ n 1, \u2190 <a>nthRoots</a>]", [{"full_name": "Polynomial.natDegree_X_pow_sub_C", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [1584, 9], "def_end_pos": [1584, 30]}, {"full_name": "Polynomial.nthRoots", "def_path": "Mathlib/Data/Polynomial/RingDivision.lean", "def_pos": [796, 5], "def_end_pos": [796, 13]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\nhmonic : Monic (X ^ n - \u2191C 1)\n\u22a2 \u2191Multiset.card (roots (X ^ n - \u2191C 1)) = natDegree (X ^ n - \u2191C 1)", "state_after": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\nhmonic : Monic (X ^ n - \u2191C 1)\n\u22a2 \u2191Multiset.card (nthRoots n 1) = n"}, {"tactic": "exact IsPrimitiveRoot.card_nthRoots h", "annotated_tactic": ["exact <a>IsPrimitiveRoot.card_nthRoots</a> h", [{"full_name": "IsPrimitiveRoot.card_nthRoots", "def_path": "Mathlib/RingTheory/RootsOfUnity/Basic.lean", "def_pos": [824, 16], "def_end_pos": [824, 29]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\n\u03b6 : R\nn : \u2115\nhpos : 0 < n\nh : IsPrimitiveRoot \u03b6 n\nhmonic : Monic (X ^ n - \u2191C 1)\n\u22a2 \u2191Multiset.card (nthRoots n 1) = n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/NAry.lean", "full_name": "Finset.mem_image\u2082", "start": [44, 1], "end": [45, 27], "traced_tactics": [{"tactic": "simp [image\u2082, and_assoc]", "annotated_tactic": ["simp [<a>image\u2082</a>, <a>and_assoc</a>]", [{"full_name": "Finset.image\u2082", "def_path": "Mathlib/Data/Finset/NAry.lean", "def_pos": [39, 5], "def_end_pos": [39, 11]}, {"full_name": "and_assoc", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [177, 9], "def_end_pos": [177, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\ninst\u271d\u2077 : DecidableEq \u03b1'\ninst\u271d\u2076 : DecidableEq \u03b2'\ninst\u271d\u2075 : DecidableEq \u03b3\ninst\u271d\u2074 : DecidableEq \u03b3'\ninst\u271d\u00b3 : DecidableEq \u03b4\ninst\u271d\u00b2 : DecidableEq \u03b4'\ninst\u271d\u00b9 : DecidableEq \u03b5\ninst\u271d : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na a' : \u03b1\nb b' : \u03b2\nc : \u03b3\n\u22a2 c \u2208 image\u2082 f s t \u2194 \u2203 a b, a \u2208 s \u2227 b \u2208 t \u2227 f a b = c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Basic.lean", "full_name": "Equiv.Perm.cycleOf_apply_apply_self", "start": [1037, 1], "end": [1038, 53], "traced_tactics": [{"tactic": "convert cycleOf_apply_apply_pow_self f x 1 using 1", "annotated_tactic": ["convert <a>cycleOf_apply_apply_pow_self</a> f x 1 using 1", [{"full_name": "Equiv.Perm.cycleOf_apply_apply_pow_self", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Basic.lean", "def_pos": [1031, 9], "def_end_pos": [1031, 37]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g : Perm \u03b1\nx\u271d y : \u03b1\nf : Perm \u03b1\nx : \u03b1\n\u22a2 \u2191(cycleOf f x) (\u2191f x) = \u2191f (\u2191f x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "full_name": "Matrix.toLin'_apply", "start": [349, 1], "end": [350, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.blsub_id", "start": [1913, 1], "end": [1914, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "full_name": "measurable_quotient_mk'", "start": [536, 1], "end": [537, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean", "full_name": "EuclideanGeometry.left_ne_right_of_oangle_eq_pi_div_two", "start": [114, 1], "end": [116, 92], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "full_name": "MeasureTheory.condexp_undef", "start": [159, 1], "end": [165, 41], "traced_tactics": [{"tactic": "by_cases hm : m \u2264 m0", "annotated_tactic": ["by_cases hm : m \u2264 m0", []], "state_before": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhf : \u00acIntegrable f\n\u22a2 \u03bc[f|m] = 0", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhf : \u00acIntegrable f\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] = 0\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhf : \u00acIntegrable f\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] = 0"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhf : \u00acIntegrable f\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] = 0\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhf : \u00acIntegrable f\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] = 0", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhf : \u00acIntegrable f\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] = 0\n\ncase pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhf : \u00acIntegrable f\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] = 0"}, {"tactic": "by_cases h\u03bcm : SigmaFinite (\u03bc.trim hm)", "annotated_tactic": ["by_cases h\u03bcm : <a>SigmaFinite</a> (\u03bc.trim hm)", [{"full_name": "MeasureTheory.SigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3289, 7], "def_end_pos": [3289, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhf : \u00acIntegrable f\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] = 0", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhf : \u00acIntegrable f\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] = 0\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhf : \u00acIntegrable f\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] = 0"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhf : \u00acIntegrable f\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] = 0\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhf : \u00acIntegrable f\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] = 0", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhf : \u00acIntegrable f\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] = 0\n\ncase pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhf : \u00acIntegrable f\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] = 0"}, {"tactic": "haveI : SigmaFinite (\u03bc.trim hm) := h\u03bcm", "annotated_tactic": ["haveI : <a>SigmaFinite</a> (\u03bc.trim hm) := h\u03bcm", [{"full_name": "MeasureTheory.SigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3289, 7], "def_end_pos": [3289, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhf : \u00acIntegrable f\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] = 0", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhf : \u00acIntegrable f\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] = 0"}, {"tactic": "rw [condexp_of_sigmaFinite, if_neg hf]", "annotated_tactic": ["rw [<a>condexp_of_sigmaFinite</a>, <a>if_neg</a> hf]", [{"full_name": "MeasureTheory.condexp_of_sigmaFinite", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [113, 9], "def_end_pos": [113, 31]}, {"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhf : \u00acIntegrable f\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] = 0", "state_after": "no goals"}, {"tactic": "rw [condexp_of_not_le hm]", "annotated_tactic": ["rw [<a>condexp_of_not_le</a> hm]", [{"full_name": "MeasureTheory.condexp_of_not_le", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [106, 9], "def_end_pos": [106, 26]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhf : \u00acIntegrable f\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] = 0", "state_after": "no goals"}, {"tactic": "rw [condexp_of_not_sigmaFinite hm h\u03bcm]", "annotated_tactic": ["rw [<a>condexp_of_not_sigmaFinite</a> hm h\u03bcm]", [{"full_name": "MeasureTheory.condexp_of_not_sigmaFinite", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [109, 9], "def_end_pos": [109, 35]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9 : NormedSpace \u211d F'\ninst\u271d : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhf : \u00acIntegrable f\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Polynomial/Chebyshev.lean", "full_name": "Polynomial.Chebyshev.T_mul", "start": [269, 1], "end": [276, 42], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 \u2200 (n : \u2115), T R (0 * n) = comp (T R 0) (T R n)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 \u2200 (n : \u2115), T R (1 * n) = comp (T R 1) (T R n)", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm : \u2115\n\u22a2 \u2200 (n : \u2115), T R ((m + 2) * n) = comp (T R (m + 2)) (T R n)", "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm n : \u2115\n\u22a2 T R ((m + 2) * n) = comp (T R (m + 2)) (T R n)"}, {"tactic": "have : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) := by\n  convert mul_T R n (m * n) using 1 <;> ring_nf", "annotated_tactic": ["have : 2 * <a>T</a> R n * <a>T</a> R ((m + 1) * n) = <a>T</a> R ((m + 2) * n) + <a>T</a> R (m * n) := by\n      convert <a>mul_T</a> R n (m * n) using 1 <;> ring_nf", [{"full_name": "Polynomial.Chebyshev.T", "def_path": "Mathlib/RingTheory/Polynomial/Chebyshev.lean", "def_pos": [67, 19], "def_end_pos": [67, 20]}, {"full_name": "Polynomial.Chebyshev.T", "def_path": "Mathlib/RingTheory/Polynomial/Chebyshev.lean", "def_pos": [67, 19], "def_end_pos": [67, 20]}, {"full_name": "Polynomial.Chebyshev.T", "def_path": "Mathlib/RingTheory/Polynomial/Chebyshev.lean", "def_pos": [67, 19], "def_end_pos": [67, 20]}, {"full_name": "Polynomial.Chebyshev.T", "def_path": "Mathlib/RingTheory/Polynomial/Chebyshev.lean", "def_pos": [67, 19], "def_end_pos": [67, 20]}, {"full_name": "Polynomial.Chebyshev.mul_T", "def_path": "Mathlib/RingTheory/Polynomial/Chebyshev.lean", "def_pos": [239, 9], "def_end_pos": [239, 14]}]], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm n : \u2115\n\u22a2 T R ((m + 2) * n) = comp (T R (m + 2)) (T R n)", "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm n : \u2115\nthis : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n)\n\u22a2 T R ((m + 2) * n) = comp (T R (m + 2)) (T R n)"}, {"tactic": "simp [this, T_mul m, \u2190 T_mul (m + 1)]", "annotated_tactic": ["simp [this, T_mul m, \u2190 T_mul (m + 1)]", []], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm n : \u2115\nthis : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n)\n\u22a2 T R ((m + 2) * n) = comp (T R (m + 2)) (T R n)", "state_after": "no goals"}, {"tactic": "convert mul_T R n (m * n) using 1 <;> ring_nf", "annotated_tactic": ["convert <a>mul_T</a> R n (m * n) using 1 <;> ring_nf", [{"full_name": "Polynomial.Chebyshev.mul_T", "def_path": "Mathlib/RingTheory/Polynomial/Chebyshev.lean", "def_pos": [239, 9], "def_end_pos": [239, 14]}]], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm n : \u2115\n\u22a2 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Clopen.lean", "full_name": "isClopen_compl_iff", "start": [66, 1], "end": [67, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Powerset.lean", "full_name": "Finset.not_mem_of_mem_powerset_of_not_mem", "start": [92, 1], "end": [95, 26], "traced_tactics": [{"tactic": "apply mt _ h", "annotated_tactic": ["apply <a>mt</a> _ h", [{"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t\u271d s t : Finset \u03b1\na : \u03b1\nht : t \u2208 powerset s\nh : \u00aca \u2208 s\n\u22a2 \u00aca \u2208 t", "state_after": "\u03b1 : Type u_1\ns\u271d t\u271d s t : Finset \u03b1\na : \u03b1\nht : t \u2208 powerset s\nh : \u00aca \u2208 s\n\u22a2 a \u2208 t \u2192 a \u2208 s"}, {"tactic": "apply mem_powerset.1 ht", "annotated_tactic": ["apply <a>mem_powerset</a>.1 ht", [{"full_name": "Finset.mem_powerset", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [34, 9], "def_end_pos": [34, 21]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t\u271d s t : Finset \u03b1\na : \u03b1\nht : t \u2208 powerset s\nh : \u00aca \u2208 s\n\u22a2 a \u2208 t \u2192 a \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.sup'_le", "start": [795, 1], "end": [797, 64], "traced_tactics": [{"tactic": "rw [\u2190 WithBot.coe_le_coe, coe_sup']", "annotated_tactic": ["rw [\u2190 <a>WithBot.coe_le_coe</a>, <a>coe_sup'</a>]", [{"full_name": "WithBot.coe_le_coe", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [206, 9], "def_end_pos": [206, 19]}, {"full_name": "Finset.coe_sup'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [772, 9], "def_end_pos": [772, 17]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : SemilatticeSup \u03b1\ns : Finset \u03b2\nH : Finset.Nonempty s\nf : \u03b2 \u2192 \u03b1\na : \u03b1\nhs : \u2200 (b : \u03b2), b \u2208 s \u2192 f b \u2264 a\n\u22a2 sup' s H f \u2264 a", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Type u_5\n\u03ba : Type u_6\ninst\u271d : SemilatticeSup \u03b1\ns : Finset \u03b2\nH : Finset.Nonempty s\nf : \u03b2 \u2192 \u03b1\na : \u03b1\nhs : \u2200 (b : \u03b2), b \u2208 s \u2192 f b \u2264 a\n\u22a2 sup s (WithBot.some \u2218 f) \u2264 \u2191a"}, {"tactic": "exact Finset.sup_le fun b h => WithBot.coe_le_coe.2 <| hs b h", "annotated_tactic": ["exact <a>Finset.sup_le</a> fun b h => <a>WithBot.coe_le_coe</a>.2 <| hs b h", [{"full_name": "Finset.sup_le", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [105, 11], 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"start": [939, 1], "end": [940, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Monotone/Monovary.lean", "full_name": "antivaryOn_toDual_right", "start": [280, 1], "end": [281, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean", "full_name": "Asymptotics.SuperpolynomialDecay.mul_param_zpow", "start": [264, 1], "end": [266, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.image_prod_mk_subset_prod_right", "start": [370, 1], "end": [372, 17], "traced_tactics": [{"tactic": "rintro _ \u27e8b, hb, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8b, hb, rfl\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\nha : a \u2208 s\n\u22a2 Prod.mk a '' t \u2286 s \u00d7\u02e2 t", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb\u271d : \u03b2\nha : a \u2208 s\nb : \u03b2\nhb : b \u2208 t\n\u22a2 (a, b) \u2208 s \u00d7\u02e2 t"}, {"tactic": "exact \u27e8ha, hb\u27e9", "annotated_tactic": ["exact \u27e8ha, hb\u27e9", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb\u271d : \u03b2\nha : a \u2208 s\nb : \u03b2\nhb : b \u2208 t\n\u22a2 (a, b) \u2208 s \u00d7\u02e2 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.mk_out", "start": [153, 1], "end": [154, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Vector/Basic.lean", "full_name": "Vector.ne_cons_iff", "start": [69, 1], "end": [70, 94], "traced_tactics": [{"tactic": "rw [Ne.def, eq_cons_iff a v v', not_and_or]", "annotated_tactic": ["rw [<a>Ne.def</a>, <a>eq_cons_iff</a> a v v', <a>not_and_or</a>]", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "Vector.eq_cons_iff", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [63, 9], "def_end_pos": [63, 20]}, {"full_name": "not_and_or", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [473, 9], "def_end_pos": [473, 19]}]], "state_before": "n : \u2115\n\u03b1 : Type u_1\na : \u03b1\nv : Vector \u03b1 (Nat.succ n)\nv' : Vector \u03b1 n\n\u22a2 v \u2260 a ::\u1d65 v' \u2194 head v \u2260 a \u2228 tail v \u2260 v'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Lift.lean", "full_name": "Filter.lift_principal2", "start": [190, 1], "end": [192, 77], "traced_tactics": [{"tactic": "simp only [hs, le_principal_iff]", "annotated_tactic": ["simp only [hs, <a>le_principal_iff</a>]", [{"full_name": "Filter.le_principal_iff", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [655, 9], "def_end_pos": [655, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf\u271d f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\nf : Filter \u03b1\ns : Set \u03b1\nhs : s \u2208 f\n\u22a2 f \u2264 \ud835\udcdf s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Ray.lean", "full_name": "Module.Ray.someRayVector_ray", "start": [362, 1], "end": [363, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Hom/Monoid.lean", "full_name": "strictMono_iff_map_pos", "start": [256, 1], "end": [261, 53], "traced_tactics": [{"tactic": "refine \u27e8fun h a => ?_, fun h a b hl => ?_\u27e9", "annotated_tactic": ["refine \u27e8fun h a => ?_, fun h a b hl => ?_\u27e9", []], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : AddMonoidHomClass F \u03b1 \u03b2\nf : F\ninst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\n\u22a2 StrictMono \u2191f \u2194 \u2200 (a : \u03b1), 0 < a \u2192 0 < \u2191f a", "state_after": "case refine_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : AddMonoidHomClass F \u03b1 \u03b2\nf : F\ninst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\nh : StrictMono \u2191f\na : \u03b1\n\u22a2 0 < a \u2192 0 < \u2191f a\n\ncase refine_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : AddMonoidHomClass F \u03b1 \u03b2\nf : F\ninst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\nh : \u2200 (a : \u03b1), 0 < a \u2192 0 < \u2191f a\na b : \u03b1\nhl : a < b\n\u22a2 \u2191f a < \u2191f b"}, {"tactic": "rw [\u2190 map_zero f]", "annotated_tactic": ["rw [\u2190 <a>map_zero</a> f]", [{"full_name": "map_zero", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [201, 3], "def_end_pos": [201, 14]}]], "state_before": "case refine_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : AddMonoidHomClass F \u03b1 \u03b2\nf : F\ninst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\nh : StrictMono \u2191f\na : \u03b1\n\u22a2 0 < a \u2192 0 < \u2191f a", "state_after": "case refine_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : AddMonoidHomClass F \u03b1 \u03b2\nf : F\ninst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\nh : StrictMono \u2191f\na : \u03b1\n\u22a2 0 < a \u2192 \u2191f 0 < \u2191f a"}, {"tactic": "apply h", "annotated_tactic": ["apply h", []], "state_before": "case refine_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : AddMonoidHomClass F \u03b1 \u03b2\nf : F\ninst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\nh : StrictMono \u2191f\na : \u03b1\n\u22a2 0 < a \u2192 \u2191f 0 < \u2191f a", "state_after": "no goals"}, {"tactic": "rw [\u2190 sub_add_cancel b a, map_add f]", "annotated_tactic": ["rw [\u2190 <a>sub_add_cancel</a> b a, <a>map_add</a> f]", [{"full_name": "sub_add_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [728, 30], "def_end_pos": [728, 44]}, {"full_name": "map_add", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [298, 3], "def_end_pos": [298, 14]}]], "state_before": "case refine_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : AddMonoidHomClass F \u03b1 \u03b2\nf : F\ninst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\nh : \u2200 (a : \u03b1), 0 < a \u2192 0 < \u2191f a\na b : \u03b1\nhl : a < b\n\u22a2 \u2191f a < \u2191f b", "state_after": "case refine_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : AddMonoidHomClass F \u03b1 \u03b2\nf : F\ninst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\nh : \u2200 (a : \u03b1), 0 < a \u2192 0 < \u2191f a\na b : \u03b1\nhl : a < b\n\u22a2 \u2191f a < \u2191f (b - a) + \u2191f a"}, {"tactic": "exact lt_add_of_pos_left _ (h _ <| sub_pos.2 hl)", "annotated_tactic": ["exact <a>lt_add_of_pos_left</a> _ (h _ <| <a>sub_pos</a>.2 hl)", [{"full_name": "lt_add_of_pos_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [513, 15], "def_end_pos": [513, 33]}, {"full_name": "sub_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [883, 30], "def_end_pos": [883, 37]}]], "state_before": "case refine_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u00b3 : OrderedAddCommGroup \u03b1\ninst\u271d\u00b2 : OrderedAddCommMonoid \u03b2\ninst\u271d\u00b9 : AddMonoidHomClass F \u03b1 \u03b2\nf : F\ninst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\nh : \u2200 (a : \u03b1), 0 < a \u2192 0 < \u2191f a\na b : \u03b1\nhl : a < b\n\u22a2 \u2191f a < \u2191f (b - a) + \u2191f a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean", "full_name": "exists_dual_vector'", "start": [140, 1], "end": [146, 36], "traced_tactics": [{"tactic": "by_cases hx : x = 0", "annotated_tactic": ["by_cases hx : x = 0", []], "state_before": "\ud835\udd5c : Type v\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : Nontrivial E\nx : E\n\u22a2 \u2203 g, \u2016g\u2016 = 1 \u2227 \u2191g x = \u2191\u2016x\u2016", "state_after": "case pos\n\ud835\udd5c : Type v\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : Nontrivial E\nx : E\nhx : x = 0\n\u22a2 \u2203 g, \u2016g\u2016 = 1 \u2227 \u2191g x = \u2191\u2016x\u2016\n\ncase neg\n\ud835\udd5c : Type v\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : Nontrivial E\nx : E\nhx : \u00acx = 0\n\u22a2 \u2203 g, \u2016g\u2016 = 1 \u2227 \u2191g x = \u2191\u2016x\u2016"}, {"tactic": "obtain \u27e8y, hy\u27e9 := exists_ne (0 : E)", "annotated_tactic": ["obtain \u27e8y, hy\u27e9 := <a>exists_ne</a> (0 : E)", [{"full_name": "exists_ne", "def_path": "Mathlib/Logic/Nontrivial/Defs.lean", "def_pos": [51, 9], "def_end_pos": [51, 18]}]], "state_before": "case pos\n\ud835\udd5c : Type v\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : Nontrivial E\nx : E\nhx : x = 0\n\u22a2 \u2203 g, \u2016g\u2016 = 1 \u2227 \u2191g x = \u2191\u2016x\u2016", "state_after": "case pos.intro\n\ud835\udd5c : Type v\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : Nontrivial E\nx : E\nhx : x = 0\ny : E\nhy : y \u2260 0\n\u22a2 \u2203 g, \u2016g\u2016 = 1 \u2227 \u2191g x = \u2191\u2016x\u2016"}, {"tactic": "obtain \u27e8g, hg\u27e9 : \u2203 g : E \u2192L[\ud835\udd5c] \ud835\udd5c, \u2016g\u2016 = 1 \u2227 g y = \u2016y\u2016 := exists_dual_vector \ud835\udd5c y hy", "annotated_tactic": ["obtain \u27e8g, hg\u27e9 : \u2203 g : E \u2192L[\ud835\udd5c] \ud835\udd5c, \u2016g\u2016 = 1 \u2227 g y = \u2016y\u2016 := <a>exists_dual_vector</a> \ud835\udd5c y hy", [{"full_name": "exists_dual_vector", "def_path": "Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean", "def_pos": [126, 9], "def_end_pos": [126, 27]}]], "state_before": "case pos.intro\n\ud835\udd5c : Type v\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : Nontrivial E\nx : E\nhx : x = 0\ny : E\nhy : y \u2260 0\n\u22a2 \u2203 g, \u2016g\u2016 = 1 \u2227 \u2191g x = \u2191\u2016x\u2016", "state_after": "case pos.intro.intro\n\ud835\udd5c : Type v\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : Nontrivial E\nx : E\nhx : x = 0\ny : E\nhy : y \u2260 0\ng : E \u2192L[\ud835\udd5c] \ud835\udd5c\nhg : \u2016g\u2016 = 1 \u2227 \u2191g y = \u2191\u2016y\u2016\n\u22a2 \u2203 g, \u2016g\u2016 = 1 \u2227 \u2191g x = \u2191\u2016x\u2016"}, {"tactic": "refine' \u27e8g, hg.left, _\u27e9", "annotated_tactic": ["refine' \u27e8g, hg.left, _\u27e9", []], "state_before": "case pos.intro.intro\n\ud835\udd5c : Type v\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : Nontrivial E\nx : E\nhx : x = 0\ny : E\nhy : y \u2260 0\ng : E \u2192L[\ud835\udd5c] \ud835\udd5c\nhg : \u2016g\u2016 = 1 \u2227 \u2191g y = \u2191\u2016y\u2016\n\u22a2 \u2203 g, \u2016g\u2016 = 1 \u2227 \u2191g x = \u2191\u2016x\u2016", "state_after": "case pos.intro.intro\n\ud835\udd5c : Type v\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : Nontrivial E\nx : E\nhx : x = 0\ny : E\nhy : y \u2260 0\ng : E \u2192L[\ud835\udd5c] \ud835\udd5c\nhg : \u2016g\u2016 = 1 \u2227 \u2191g y = \u2191\u2016y\u2016\n\u22a2 \u2191g x = \u2191\u2016x\u2016"}, {"tactic": "simp [hx]", "annotated_tactic": ["simp [hx]", []], "state_before": "case pos.intro.intro\n\ud835\udd5c : Type v\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : Nontrivial E\nx : E\nhx : x = 0\ny : E\nhy : y \u2260 0\ng : E \u2192L[\ud835\udd5c] \ud835\udd5c\nhg : \u2016g\u2016 = 1 \u2227 \u2191g y = \u2191\u2016y\u2016\n\u22a2 \u2191g x = \u2191\u2016x\u2016", "state_after": "no goals"}, {"tactic": "exact exists_dual_vector \ud835\udd5c x hx", "annotated_tactic": ["exact <a>exists_dual_vector</a> \ud835\udd5c x hx", [{"full_name": "exists_dual_vector", "def_path": "Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean", "def_pos": [126, 9], "def_end_pos": [126, 27]}]], "state_before": "case neg\n\ud835\udd5c : Type v\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : Nontrivial E\nx : E\nhx : \u00acx = 0\n\u22a2 \u2203 g, \u2016g\u2016 = 1 \u2227 \u2191g x = \u2191\u2016x\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Hom/Monoid.lean", "full_name": "OrderMonoidHom.toMonoidHom_eq_coe", "start": [543, 1], "end": [544, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Fin/Tuple/Basic.lean", "full_name": "Fin.forall_fin_succ_pi", "start": [212, 1], "end": [213, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Cover.lean", "full_name": "apply_wcovby_apply_iff", "start": [136, 1], "end": [137, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Group/Defs.lean", "full_name": "div_lt_self_iff", "start": [418, 1], "end": [419, 24], "traced_tactics": [{"tactic": "simp [div_eq_mul_inv]", "annotated_tactic": ["simp [<a>div_eq_mul_inv</a>]", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}]], "state_before": "\u03b1 : Type u\ninst\u271d\u00b3 : Group \u03b1\ninst\u271d\u00b2 : LT \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na\u271d b\u271d c d a b : \u03b1\n\u22a2 a / b < a \u2194 1 < b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/SymmDiff.lean", "full_name": "sup_sdiff_symmDiff", "start": [393, 1], "end": [394, 63], "traced_tactics": [{"tactic": "rw [symmDiff_eq_sup_sdiff_inf]", "annotated_tactic": ["rw [<a>symmDiff_eq_sup_sdiff_inf</a>]", [{"full_name": "symmDiff_eq_sup_sdiff_inf", "def_path": "Mathlib/Order/SymmDiff.lean", "def_pos": [158, 9], "def_end_pos": [158, 34]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\na b c d : \u03b1\n\u22a2 (a \u2294 b) \\ (a \u2293 b) = a \u2206 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Basic.lean", "full_name": "Subtype.mk_le_mk", "start": [1150, 1], "end": [1152, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Star/Subalgebra.lean", "full_name": "StarSubalgebra.range_le", "start": [149, 1], "end": [150, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Finrank.lean", "full_name": "FiniteDimensional.lt_rank_of_lt_finrank", "start": [88, 1], "end": [93, 20], "traced_tactics": [{"tactic": "rwa [\u2190 Cardinal.toNat_lt_iff_lt_of_lt_aleph0, toNat_cast]", "annotated_tactic": ["rwa [\u2190 <a>Cardinal.toNat_lt_iff_lt_of_lt_aleph0</a>, <a>toNat_cast</a>]", [{"full_name": "Cardinal.toNat_lt_iff_lt_of_lt_aleph0", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1733, 9], "def_end_pos": [1733, 37]}, {"full_name": "Cardinal.toNat_cast", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1749, 9], "def_end_pos": [1749, 19]}]], "state_before": "K : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : n < finrank K V\n\u22a2 \u2191n < Module.rank K V", "state_after": "case hc\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : n < finrank K V\n\u22a2 \u2191n < \u2135\u2080\n\ncase hd\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : n < finrank K V\n\u22a2 Module.rank K V < \u2135\u2080"}, {"tactic": "exact nat_lt_aleph0 n", "annotated_tactic": ["exact <a>nat_lt_aleph0</a> n", [{"full_name": "Cardinal.nat_lt_aleph0", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1414, 9], "def_end_pos": [1414, 22]}]], "state_before": "case hc\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : n < finrank K V\n\u22a2 \u2191n < \u2135\u2080", "state_after": "no goals"}, {"tactic": "contrapose! h", "annotated_tactic": ["contrapose! h", []], "state_before": "case hd\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : n < finrank K V\n\u22a2 Module.rank K V < \u2135\u2080", "state_after": "case hd\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : \u2135\u2080 \u2264 Module.rank K V\n\u22a2 finrank K V \u2264 n"}, {"tactic": "rw [finrank, Cardinal.toNat_apply_of_aleph0_le h]", "annotated_tactic": ["rw [<a>finrank</a>, <a>Cardinal.toNat_apply_of_aleph0_le</a> h]", [{"full_name": "FiniteDimensional.finrank", "def_path": "Mathlib/LinearAlgebra/Finrank.lean", "def_pos": [58, 19], "def_end_pos": [58, 26]}, {"full_name": "Cardinal.toNat_apply_of_aleph0_le", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1710, 9], "def_end_pos": [1710, 33]}]], "state_before": "case hd\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : \u2135\u2080 \u2264 Module.rank K V\n\u22a2 finrank K V \u2264 n", "state_after": "case hd\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : \u2135\u2080 \u2264 Module.rank K V\n\u22a2 0 \u2264 n"}, {"tactic": "exact n.zero_le", "annotated_tactic": ["exact n.zero_le", []], "state_before": "case hd\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : \u2135\u2080 \u2264 Module.rank K V\n\u22a2 0 \u2264 n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "full_name": "ciSup_mono", "start": [809, 1], "end": [813, 53], "traced_tactics": [{"tactic": "cases isEmpty_or_nonempty \u03b9", "annotated_tactic": ["cases <a>isEmpty_or_nonempty</a> \u03b9", [{"full_name": "isEmpty_or_nonempty", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [207, 9], "def_end_pos": [207, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLattice \u03b1\ns t : Set \u03b1\na b : \u03b1\nf g : \u03b9 \u2192 \u03b1\nB : BddAbove (range g)\nH : \u2200 (x : \u03b9), f x \u2264 g x\n\u22a2 iSup f \u2264 iSup g", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLattice \u03b1\ns t : Set \u03b1\na b : \u03b1\nf g : \u03b9 \u2192 \u03b1\nB : BddAbove (range g)\nH : \u2200 (x : \u03b9), f x \u2264 g x\nh\u271d : IsEmpty \u03b9\n\u22a2 iSup f \u2264 iSup g\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLattice \u03b1\ns t : Set \u03b1\na b : \u03b1\nf g : \u03b9 \u2192 \u03b1\nB : BddAbove (range g)\nH : \u2200 (x : \u03b9), f x \u2264 g x\nh\u271d : Nonempty \u03b9\n\u22a2 iSup f \u2264 iSup g"}, {"tactic": "rw [iSup_of_empty', iSup_of_empty']", "annotated_tactic": ["rw [<a>iSup_of_empty'</a>, <a>iSup_of_empty'</a>]", [{"full_name": "iSup_of_empty'", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1559, 9], "def_end_pos": [1559, 23]}, {"full_name": "iSup_of_empty'", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1559, 9], "def_end_pos": [1559, 23]}]], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLattice \u03b1\ns t : Set \u03b1\na b : \u03b1\nf g : \u03b9 \u2192 \u03b1\nB : BddAbove (range g)\nH : \u2200 (x : \u03b9), f x \u2264 g x\nh\u271d : IsEmpty \u03b9\n\u22a2 iSup f \u2264 iSup g", "state_after": "no goals"}, {"tactic": "exact ciSup_le fun x => le_ciSup_of_le B x (H x)", "annotated_tactic": ["exact <a>ciSup_le</a> fun x => <a>le_ciSup_of_le</a> B x (H x)", [{"full_name": "ciSup_le", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [795, 9], "def_end_pos": [795, 17]}, {"full_name": "le_ciSup_of_le", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [804, 9], "def_end_pos": [804, 23]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\ninst\u271d : ConditionallyCompleteLattice \u03b1\ns t : Set \u03b1\na b : \u03b1\nf g : \u03b9 \u2192 \u03b1\nB : BddAbove (range g)\nH : \u2200 (x : \u03b9), f x \u2264 g x\nh\u271d : Nonempty \u03b9\n\u22a2 iSup f \u2264 iSup g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Order/Hom/Basic.lean", "full_name": "ContinuousOrderHom.coe_id", "start": [147, 1], "end": [148, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "IntervalIntegrable.aestronglyMeasurable", "start": [225, 11], "end": [227, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Cover.lean", "full_name": "AntisymmRel.trans_wcovby", "start": [80, 1], "end": [81, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Constructions.lean", "full_name": "Continuous.prod_mk", "start": [402, 1], "end": [404, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "full_name": "le_nhds_of_cauchy_adhp_aux", "start": [139, 1], "end": [149, 79], "traced_tactics": [{"tactic": "intro s hs", "annotated_tactic": ["intro s hs", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\nf : Filter \u03b1\nx : \u03b1\nadhs : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 f \u2227 t \u00d7\u02e2 t \u2286 s \u2227 \u2203 y, (x, y) \u2208 s \u2227 y \u2208 t\n\u22a2 f \u2264 \ud835\udcdd x", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\nf : Filter \u03b1\nx : \u03b1\nadhs : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 f \u2227 t \u00d7\u02e2 t \u2286 s \u2227 \u2203 y, (x, y) \u2208 s \u2227 y \u2208 t\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd x\n\u22a2 s \u2208 f"}, {"tactic": "rcases comp_mem_uniformity_sets (mem_nhds_uniformity_iff_right.1 hs) with \u27e8U, U_mem, hU\u27e9", "annotated_tactic": ["rcases <a>comp_mem_uniformity_sets</a> (<a>mem_nhds_uniformity_iff_right</a>.1 hs) with \u27e8U, U_mem, hU\u27e9", [{"full_name": "comp_mem_uniformity_sets", "def_path": "Mathlib/Topology/UniformSpace/Basic.lean", "def_pos": [461, 9], "def_end_pos": [461, 33]}, {"full_name": "mem_nhds_uniformity_iff_right", "def_path": "Mathlib/Topology/UniformSpace/Basic.lean", "def_pos": [695, 9], "def_end_pos": [695, 38]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\nf : Filter \u03b1\nx : \u03b1\nadhs : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 f \u2227 t \u00d7\u02e2 t \u2286 s \u2227 \u2203 y, (x, y) \u2208 s \u2227 y \u2208 t\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd x\n\u22a2 s \u2208 f", "state_after": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\nf : Filter \u03b1\nx : \u03b1\nadhs : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 f \u2227 t \u00d7\u02e2 t \u2286 s \u2227 \u2203 y, (x, y) \u2208 s \u2227 y \u2208 t\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd x\nU : Set (\u03b1 \u00d7 \u03b1)\nU_mem : U \u2208 \ud835\udce4 \u03b1\nhU : U \u25cb U \u2286 {p | p.1 = x \u2192 p.2 \u2208 s}\n\u22a2 s \u2208 f"}, {"tactic": "rcases adhs U U_mem with \u27e8t, t_mem, ht, y, hxy, hy\u27e9", "annotated_tactic": ["rcases adhs U U_mem with \u27e8t, t_mem, ht, y, hxy, hy\u27e9", []], "state_before": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\nf : Filter \u03b1\nx : \u03b1\nadhs : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 f \u2227 t \u00d7\u02e2 t \u2286 s \u2227 \u2203 y, (x, y) \u2208 s \u2227 y \u2208 t\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd x\nU : Set (\u03b1 \u00d7 \u03b1)\nU_mem : U \u2208 \ud835\udce4 \u03b1\nhU : U \u25cb U \u2286 {p | p.1 = x \u2192 p.2 \u2208 s}\n\u22a2 s \u2208 f", "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\nf : Filter \u03b1\nx : \u03b1\nadhs : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 f \u2227 t \u00d7\u02e2 t \u2286 s \u2227 \u2203 y, (x, y) \u2208 s \u2227 y \u2208 t\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd x\nU : Set (\u03b1 \u00d7 \u03b1)\nU_mem : U \u2208 \ud835\udce4 \u03b1\nhU : U \u25cb U \u2286 {p | p.1 = x \u2192 p.2 \u2208 s}\nt : Set \u03b1\nt_mem : t \u2208 f\nht : t \u00d7\u02e2 t \u2286 U\ny : \u03b1\nhxy : (x, y) \u2208 U\nhy : y \u2208 t\n\u22a2 s \u2208 f"}, {"tactic": "apply mem_of_superset t_mem", "annotated_tactic": ["apply <a>mem_of_superset</a> t_mem", [{"full_name": "Filter.mem_of_superset", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [152, 9], "def_end_pos": [152, 24]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\nf : Filter \u03b1\nx : \u03b1\nadhs : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 f \u2227 t \u00d7\u02e2 t \u2286 s \u2227 \u2203 y, (x, y) \u2208 s \u2227 y \u2208 t\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd x\nU : Set (\u03b1 \u00d7 \u03b1)\nU_mem : U \u2208 \ud835\udce4 \u03b1\nhU : U \u25cb U \u2286 {p | p.1 = x \u2192 p.2 \u2208 s}\nt : Set \u03b1\nt_mem : t \u2208 f\nht : t \u00d7\u02e2 t \u2286 U\ny : \u03b1\nhxy : (x, y) \u2208 U\nhy : y \u2208 t\n\u22a2 s \u2208 f", "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\nf : Filter \u03b1\nx : \u03b1\nadhs : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 f \u2227 t \u00d7\u02e2 t \u2286 s \u2227 \u2203 y, (x, y) \u2208 s \u2227 y \u2208 t\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd x\nU : Set (\u03b1 \u00d7 \u03b1)\nU_mem : U \u2208 \ud835\udce4 \u03b1\nhU : U \u25cb U \u2286 {p | p.1 = x \u2192 p.2 \u2208 s}\nt : Set \u03b1\nt_mem : t \u2208 f\nht : t \u00d7\u02e2 t \u2286 U\ny : \u03b1\nhxy : (x, y) \u2208 U\nhy : y \u2208 t\n\u22a2 t \u2286 s"}, {"tactic": "exact fun z hz => hU (prod_mk_mem_compRel hxy (ht <| mk_mem_prod hy hz)) rfl", "annotated_tactic": ["exact fun z hz => hU (<a>prod_mk_mem_compRel</a> hxy (ht <| <a>mk_mem_prod</a> hy hz)) <a>rfl</a>", [{"full_name": "prod_mk_mem_compRel", "def_path": "Mathlib/Topology/UniformSpace/Basic.lean", "def_pos": [175, 9], "def_end_pos": [175, 28]}, {"full_name": "Set.mk_mem_prod", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [66, 9], "def_end_pos": [66, 20]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\nf : Filter \u03b1\nx : \u03b1\nadhs : \u2200 (s : Set (\u03b1 \u00d7 \u03b1)), s \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 t, t \u2208 f \u2227 t \u00d7\u02e2 t \u2286 s \u2227 \u2203 y, (x, y) \u2208 s \u2227 y \u2208 t\ns : Set \u03b1\nhs : s \u2208 \ud835\udcdd x\nU : Set (\u03b1 \u00d7 \u03b1)\nU_mem : U \u2208 \ud835\udce4 \u03b1\nhU : U \u25cb U \u2286 {p | p.1 = x \u2192 p.2 \u2208 s}\nt : Set \u03b1\nt_mem : t \u2208 f\nht : t \u00d7\u02e2 t \u2286 U\ny : \u03b1\nhxy : (x, y) \u2208 U\nhy : y \u2208 t\n\u22a2 t \u2286 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Control/Fold.lean", "full_name": "Traversable.foldrm_toList", "start": [412, 1], "end": [416, 71], "traced_tactics": [{"tactic": "change _ = foldrM.ofFreeMonoid f (FreeMonoid.ofList <| toList xs) x", "annotated_tactic": ["change _ = <a>foldrM.ofFreeMonoid</a> f (<a>FreeMonoid.ofList</a> <| <a>toList</a> xs) x", [{"full_name": "Monoid.foldrM.ofFreeMonoid", "def_path": "Mathlib/Control/Fold.lean", "def_pos": [185, 5], "def_end_pos": [185, 24]}, {"full_name": "FreeMonoid.ofList", "def_path": "Mathlib/Algebra/FreeMonoid/Basic.lean", "def_pos": [45, 5], "def_end_pos": [45, 11]}, {"full_name": "Traversable.toList", "def_path": "Mathlib/Control/Fold.lean", "def_pos": [230, 5], "def_end_pos": [230, 11]}]], "state_before": "\u03b1 \u03b2 \u03b3 : Type u\nt : Type u \u2192 Type u\ninst\u271d\u00b3 : Traversable t\ninst\u271d\u00b2 : LawfulTraversable t\nm : Type u \u2192 Type u\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nf : \u03b1 \u2192 \u03b2 \u2192 m \u03b2\nx : \u03b2\nxs : t \u03b1\n\u22a2 foldrm f x xs = List.foldrM f x (toList xs)", "state_after": "\u03b1 \u03b2 \u03b3 : Type u\nt : Type u \u2192 Type u\ninst\u271d\u00b3 : Traversable t\ninst\u271d\u00b2 : LawfulTraversable t\nm : Type u \u2192 Type u\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nf : \u03b1 \u2192 \u03b2 \u2192 m \u03b2\nx : \u03b2\nxs : t \u03b1\n\u22a2 foldrm f x xs = \u2191(foldrM.ofFreeMonoid f) (\u2191FreeMonoid.ofList (toList xs)) x"}, {"tactic": "simp only [foldrm, toList_spec, foldMap_hom_free (foldrM.ofFreeMonoid f),\n  foldrm.ofFreeMonoid_comp_of, foldrM.get, FreeMonoid.ofList_toList]", "annotated_tactic": ["simp only [<a>foldrm</a>, <a>toList_spec</a>, <a>foldMap_hom_free</a> (<a>foldrM.ofFreeMonoid</a> f),\n    <a>foldrm.ofFreeMonoid_comp_of</a>, <a>foldrM.get</a>, <a>FreeMonoid.ofList_toList</a>]", [{"full_name": "Traversable.foldrm", "def_path": "Mathlib/Control/Fold.lean", "def_pos": [244, 5], "def_end_pos": [244, 11]}, {"full_name": "Traversable.toList_spec", "def_path": "Mathlib/Control/Fold.lean", "def_pos": [332, 9], "def_end_pos": [332, 20]}, {"full_name": "Traversable.foldMap_hom_free", "def_path": "Mathlib/Control/Fold.lean", "def_pos": [288, 9], "def_end_pos": [288, 25]}, {"full_name": "Monoid.foldrM.ofFreeMonoid", "def_path": "Mathlib/Control/Fold.lean", "def_pos": [185, 5], "def_end_pos": [185, 24]}, {"full_name": "Traversable.foldrm.ofFreeMonoid_comp_of", "def_path": "Mathlib/Control/Fold.lean", "def_pos": [326, 9], "def_end_pos": [326, 36]}, {"full_name": "Monoid.foldrM.get", "def_path": "Mathlib/Control/Fold.lean", "def_pos": [180, 5], "def_end_pos": [180, 15]}, {"full_name": "FreeMonoid.ofList_toList", "def_path": "Mathlib/Algebra/FreeMonoid/Basic.lean", "def_pos": [65, 9], "def_end_pos": [65, 22]}]], "state_before": "\u03b1 \u03b2 \u03b3 : Type u\nt : Type u \u2192 Type u\ninst\u271d\u00b3 : Traversable t\ninst\u271d\u00b2 : LawfulTraversable t\nm : Type u \u2192 Type u\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nf : \u03b1 \u2192 \u03b2 \u2192 m \u03b2\nx : \u03b2\nxs : t \u03b1\n\u22a2 foldrm f x xs = \u2191(foldrM.ofFreeMonoid f) (\u2191FreeMonoid.ofList (toList xs)) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/Implicit.lean", "full_name": "HasStrictFDerivAt.eq_implicitFunctionOfComplemented", "start": [331, 1], "end": [335, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Matrix/Basic.lean", "full_name": "Matrix.of_add_of", "start": [323, 1], "end": [324, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/ContinuousFunction/Algebra.lean", "full_name": "ContinuousMap.one_comp", "start": [102, 1], "end": [103, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/ZeroObjects.lean", "full_name": "CategoryTheory.Limits.IsZero.eq_of_tgt", "start": [86, 1], "end": [87, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/EpsilonNFA.lean", "full_name": "\u03b5NFA.start_one", "start": [246, 1], "end": [247, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean", "full_name": "CategoryTheory.Limits.WalkingPair.equivBool_symm_apply_false", "start": [108, 1], "end": [109, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/Log.lean", "full_name": "Nat.clog_anti_left", "start": [339, 1], "end": [343, 51], "traced_tactics": [{"tactic": "rw [\u2190 le_pow_iff_clog_le (lt_of_lt_of_le hc hb)]", "annotated_tactic": ["rw [\u2190 <a>le_pow_iff_clog_le</a> (<a>lt_of_lt_of_le</a> hc hb)]", [{"full_name": "Nat.le_pow_iff_clog_le", "def_path": "Mathlib/Data/Nat/Log.lean", "def_pos": [292, 9], "def_end_pos": [292, 27]}, {"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}]], "state_before": "b c n : \u2115\nhc : 1 < c\nhb : c \u2264 b\n\u22a2 clog b n \u2264 clog c n", "state_after": "b c n : \u2115\nhc : 1 < c\nhb : c \u2264 b\n\u22a2 n \u2264 b ^ clog c n"}, {"tactic": "calc\n  n \u2264 c ^ clog c n := le_pow_clog hc _\n  _ \u2264 b ^ clog c n := pow_le_pow_of_le_left hb _", "annotated_tactic": ["calc\n    n \u2264 c ^ <a>clog</a> c n := <a>le_pow_clog</a> hc _\n    _ \u2264 b ^ <a>clog</a> c n := <a>pow_le_pow_of_le_left</a> hb _", [{"full_name": "Nat.clog", "def_path": "Mathlib/Data/Nat/Log.lean", "def_pos": [238, 5], "def_end_pos": [238, 9]}, {"full_name": "Nat.le_pow_clog", "def_path": "Mathlib/Data/Nat/Log.lean", "def_pos": [325, 9], "def_end_pos": [325, 20]}, {"full_name": "Nat.clog", "def_path": "Mathlib/Data/Nat/Log.lean", "def_pos": [238, 5], "def_end_pos": [238, 9]}, {"full_name": "Nat.pow_le_pow_of_le_left", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [478, 9], "def_end_pos": [478, 30]}]], "state_before": "b c n : \u2115\nhc : 1 < c\nhb : c \u2264 b\n\u22a2 n \u2264 b ^ clog c n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Matrix/Basic.lean", "full_name": "Matrix.conjTranspose_list_prod", "start": [2359, 1], "end": [2361, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "full_name": "Cardinal.card_le_iff", "start": [1381, 1], "end": [1382, 27], "traced_tactics": [{"tactic": "rw [lt_ord, lt_succ_iff]", "annotated_tactic": ["rw [<a>lt_ord</a>, <a>lt_succ_iff</a>]", [{"full_name": "Cardinal.lt_ord", "def_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "def_pos": [1356, 9], "def_end_pos": [1356, 15]}, {"full_name": "Order.lt_succ_iff", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [338, 9], "def_end_pos": [338, 20]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\nc : Cardinal.{u_3}\n\u22a2 card o \u2264 c \u2194 o < ord (succ c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/Sqrt.lean", "full_name": "Nat.sqrt_add_eq'", "start": [148, 1], "end": [149, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Game/Nim.lean", "full_name": "SetTheory.PGame.moveRight_nim'", "start": [115, 1], "end": [116, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/FreeGroup/Basic.lean", "full_name": "FreeGroup.lift.range_le", "start": [761, 1], "end": [766, 61], "traced_tactics": [{"tactic": "rintro _ \u27e8\u27e8L\u27e9, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8\u27e8L\u27e9, rfl\u27e9", []], "state_before": "\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\ns : Subgroup \u03b2\nH : Set.range f \u2286 \u2191s\n\u22a2 MonoidHom.range (\u2191lift f) \u2264 s", "state_after": "case intro.mk\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\ns : Subgroup \u03b2\nH : Set.range f \u2286 \u2191s\nw\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\n\u22a2 \u2191(\u2191lift f) (Quot.mk Red.Step L) \u2208 s"}, {"tactic": "exact\n  List.recOn L s.one_mem fun \u27e8x, b\u27e9 tl ih =>\n    Bool.recOn b (by simp at ih \u22a2; exact s.mul_mem (s.inv_mem <| H \u27e8x, rfl\u27e9) ih)\n      (by simp at ih \u22a2; exact s.mul_mem (H \u27e8x, rfl\u27e9) ih)", "annotated_tactic": ["exact\n      <a>List.recOn</a> L s.one_mem fun \u27e8x, b\u27e9 tl ih =>\n        <a>Bool.recOn</a> b (by simp at ih \u22a2; exact s.mul_mem (s.inv_mem <| H \u27e8x, <a>rfl</a>\u27e9) ih)\n          (by simp at ih \u22a2; exact s.mul_mem (H \u27e8x, <a>rfl</a>\u27e9) ih)", [{"full_name": "List.recOn", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}, {"full_name": "Bool.recOn", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [545, 11], "def_end_pos": [545, 15]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro.mk\n\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx y : FreeGroup \u03b1\ns : Subgroup \u03b2\nH : Set.range f \u2286 \u2191s\nw\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\n\u22a2 \u2191(\u2191lift f) (Quot.mk Red.Step L) \u2208 s", "state_after": "no goals"}, {"tactic": "simp at ih \u22a2", "annotated_tactic": ["simp at ih \u22a2", []], "state_before": "\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx\u271d\u00b9 y : FreeGroup \u03b1\ns : Subgroup \u03b2\nH : Set.range f \u2286 \u2191s\nw\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1 \u00d7 Bool\ntl : List (\u03b1 \u00d7 Bool)\nih : \u2191(\u2191lift f) (Quot.mk Red.Step tl) \u2208 s\nx : \u03b1\nb : Bool\n\u22a2 \u2191(\u2191lift f) (Quot.mk Red.Step ((x, false) :: tl)) \u2208 s", "state_after": "\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx\u271d\u00b9 y : FreeGroup \u03b1\ns : Subgroup \u03b2\nH : Set.range f \u2286 \u2191s\nw\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1 \u00d7 Bool\ntl : List (\u03b1 \u00d7 Bool)\nih : List.prod (List.map (fun x => bif x.2 then f x.1 else (f x.1)\u207b\u00b9) tl) \u2208 s\nx : \u03b1\nb : Bool\n\u22a2 (f x)\u207b\u00b9 * List.prod (List.map (fun x => bif x.2 then f x.1 else (f x.1)\u207b\u00b9) tl) \u2208 s"}, {"tactic": "exact s.mul_mem (s.inv_mem <| H \u27e8x, rfl\u27e9) ih", "annotated_tactic": ["exact s.mul_mem (s.inv_mem <| H \u27e8x, <a>rfl</a>\u27e9) ih", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx\u271d\u00b9 y : FreeGroup \u03b1\ns : Subgroup \u03b2\nH : Set.range f \u2286 \u2191s\nw\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1 \u00d7 Bool\ntl : List (\u03b1 \u00d7 Bool)\nih : List.prod (List.map (fun x => bif x.2 then f x.1 else (f x.1)\u207b\u00b9) tl) \u2208 s\nx : \u03b1\nb : Bool\n\u22a2 (f x)\u207b\u00b9 * List.prod (List.map (fun x => bif x.2 then f x.1 else (f x.1)\u207b\u00b9) tl) \u2208 s", "state_after": "no goals"}, {"tactic": "simp at ih \u22a2", "annotated_tactic": ["simp at ih \u22a2", []], "state_before": "\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx\u271d\u00b9 y : FreeGroup \u03b1\ns : Subgroup \u03b2\nH : Set.range f \u2286 \u2191s\nw\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1 \u00d7 Bool\ntl : List (\u03b1 \u00d7 Bool)\nih : \u2191(\u2191lift f) (Quot.mk Red.Step tl) \u2208 s\nx : \u03b1\nb : Bool\n\u22a2 \u2191(\u2191lift f) (Quot.mk Red.Step ((x, true) :: tl)) \u2208 s", "state_after": "\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx\u271d\u00b9 y : FreeGroup \u03b1\ns : Subgroup \u03b2\nH : Set.range f \u2286 \u2191s\nw\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1 \u00d7 Bool\ntl : List (\u03b1 \u00d7 Bool)\nih : List.prod (List.map (fun x => bif x.2 then f x.1 else (f x.1)\u207b\u00b9) tl) \u2208 s\nx : \u03b1\nb : Bool\n\u22a2 f x * List.prod (List.map (fun x => bif x.2 then f x.1 else (f x.1)\u207b\u00b9) tl) \u2208 s"}, {"tactic": "exact s.mul_mem (H \u27e8x, rfl\u27e9) ih", "annotated_tactic": ["exact s.mul_mem (H \u27e8x, <a>rfl</a>\u27e9) ih", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03b1 : Type u\nL\u271d L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\n\u03b2 : Type v\ninst\u271d : Group \u03b2\nf : \u03b1 \u2192 \u03b2\nx\u271d\u00b9 y : FreeGroup \u03b1\ns : Subgroup \u03b2\nH : Set.range f \u2286 \u2191s\nw\u271d : FreeGroup \u03b1\nL : List (\u03b1 \u00d7 Bool)\nx\u271d : \u03b1 \u00d7 Bool\ntl : List (\u03b1 \u00d7 Bool)\nih : List.prod (List.map (fun x => bif x.2 then f x.1 else (f x.1)\u207b\u00b9) tl) \u2208 s\nx : \u03b1\nb : Bool\n\u22a2 f x * List.prod (List.map (fun x => bif x.2 then f x.1 else (f x.1)\u207b\u00b9) tl) \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Polynomial/Tower.lean", "full_name": "Polynomial.aeval_algebraMap_apply", "start": [54, 1], "end": [56, 70], "traced_tactics": [{"tactic": "rw [aeval_def, aeval_def, hom_eval\u2082, \u2190 IsScalarTower.algebraMap_eq]", "annotated_tactic": ["rw [<a>aeval_def</a>, <a>aeval_def</a>, <a>hom_eval\u2082</a>, \u2190 <a>IsScalarTower.algebraMap_eq</a>]", [{"full_name": "Polynomial.aeval_def", "def_path": "Mathlib/Data/Polynomial/AlgebraMap.lean", "def_pos": [187, 9], "def_end_pos": [187, 18]}, {"full_name": "Polynomial.aeval_def", "def_path": "Mathlib/Data/Polynomial/AlgebraMap.lean", "def_pos": [187, 9], "def_end_pos": [187, 18]}, {"full_name": "Polynomial.hom_eval\u2082", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [1029, 9], "def_end_pos": [1029, 18]}, {"full_name": "IsScalarTower.algebraMap_eq", "def_path": "Mathlib/Algebra/Algebra/Tower.lean", "def_pos": [122, 9], "def_end_pos": [122, 22]}]], "state_before": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring A\ninst\u271d\u2074 : Semiring B\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : Algebra R B\ninst\u271d : IsScalarTower R A B\nx : A\np : R[X]\n\u22a2 \u2191(aeval (\u2191(algebraMap A B) x)) p = \u2191(algebraMap A B) (\u2191(aeval x) p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.Tape.write_nth", "start": [667, 1], "end": [671, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Real/Hyperreal.lean", "full_name": "Hyperreal.Infinite.ne_zero", "start": [440, 1], "end": [441, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/NormedSpace/ProdLp.lean", "full_name": "WithLp.prod_antilipschitzWith_equiv_aux", "start": [418, 1], "end": [439, 78], "traced_tactics": [{"tactic": "intro x y", "annotated_tactic": ["intro x y", []], "state_before": "p : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\n\u22a2 AntilipschitzWith (2 ^ ENNReal.toReal (1 / p)) \u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2))", "state_after": "p : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\n\u22a2 edist x y \u2264 \u2191(2 ^ ENNReal.toReal (1 / p)) * edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y)"}, {"tactic": "rcases p.dichotomy with (rfl | h)", "annotated_tactic": ["rcases p.dichotomy with (rfl | h)", []], "state_before": "p : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\n\u22a2 edist x y \u2264 \u2191(2 ^ ENNReal.toReal (1 / p)) * edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y)", "state_after": "case inl\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nhp : Fact (1 \u2264 \u22a4)\nx y : WithLp \u22a4 (\u03b1 \u00d7 \u03b2)\n\u22a2 edist x y \u2264 \u2191(2 ^ ENNReal.toReal (1 / \u22a4)) * edist (\u2191(WithLp.equiv \u22a4 (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv \u22a4 (\u03b1 \u00d7 \u03b2)) y)\n\ncase inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\n\u22a2 edist x y \u2264 \u2191(2 ^ ENNReal.toReal (1 / p)) * edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y)"}, {"tactic": "simp [edist]", "annotated_tactic": ["simp [<a>edist</a>]", [{"full_name": "EDist.edist", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [48, 3], "def_end_pos": [48, 8]}]], "state_before": "case inl\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nhp : Fact (1 \u2264 \u22a4)\nx y : WithLp \u22a4 (\u03b1 \u00d7 \u03b2)\n\u22a2 edist x y \u2264 \u2191(2 ^ ENNReal.toReal (1 / \u22a4)) * edist (\u2191(WithLp.equiv \u22a4 (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv \u22a4 (\u03b1 \u00d7 \u03b2)) y)", "state_after": "no goals"}, {"tactic": "have pos : 0 < p.toReal := by positivity", "annotated_tactic": ["have pos : 0 < p.toReal := by positivity", []], "state_before": "case inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\n\u22a2 edist x y \u2264 \u2191(2 ^ ENNReal.toReal (1 / p)) * edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y)", "state_after": "case inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\n\u22a2 edist x y \u2264 \u2191(2 ^ ENNReal.toReal (1 / p)) * edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y)"}, {"tactic": "have nonneg : 0 \u2264 1 / p.toReal := by positivity", "annotated_tactic": ["have nonneg : 0 \u2264 1 / p.toReal := by positivity", []], "state_before": "case inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\n\u22a2 edist x y \u2264 \u2191(2 ^ ENNReal.toReal (1 / p)) * edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y)", "state_after": "case inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\n\u22a2 edist x y \u2264 \u2191(2 ^ ENNReal.toReal (1 / p)) * edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y)"}, {"tactic": "have cancel : p.toReal * (1 / p.toReal) = 1 := mul_div_cancel' 1 (ne_of_gt pos)", "annotated_tactic": ["have cancel : p.toReal * (1 / p.toReal) = 1 := <a>mul_div_cancel'</a> 1 (<a>ne_of_gt</a> pos)", [{"full_name": "mul_div_cancel'", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [173, 9], "def_end_pos": [173, 24]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}]], "state_before": "case inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\n\u22a2 edist x y \u2264 \u2191(2 ^ ENNReal.toReal (1 / p)) * edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y)", "state_after": "case inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 edist x y \u2264 \u2191(2 ^ ENNReal.toReal (1 / p)) * edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y)"}, {"tactic": "rw [prod_edist_eq_add pos, ENNReal.toReal_div 1 p]", "annotated_tactic": ["rw [<a>prod_edist_eq_add</a> pos, <a>ENNReal.toReal_div</a> 1 p]", [{"full_name": "WithLp.prod_edist_eq_add", "def_path": "Mathlib/Analysis/NormedSpace/ProdLp.lean", "def_pos": [169, 9], "def_end_pos": [169, 26]}, {"full_name": "ENNReal.toReal_div", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2385, 9], "def_end_pos": [2385, 19]}]], "state_before": "case inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 edist x y \u2264 \u2191(2 ^ ENNReal.toReal (1 / p)) * edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y)", "state_after": "case inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 (edist x.1 y.1 ^ ENNReal.toReal p + edist x.2 y.2 ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) \u2264\n    \u2191(2 ^ (ENNReal.toReal 1 / ENNReal.toReal p)) * edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y)"}, {"tactic": "simp only [edist, \u2190 one_div, ENNReal.one_toReal]", "annotated_tactic": ["simp only [<a>edist</a>, \u2190 <a>one_div</a>, <a>ENNReal.one_toReal</a>]", [{"full_name": "EDist.edist", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [48, 3], "def_end_pos": [48, 8]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "ENNReal.one_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [230, 17], "def_end_pos": [230, 27]}]], "state_before": "case inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 (edist x.1 y.1 ^ ENNReal.toReal p + edist x.2 y.2 ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) \u2264\n    \u2191(2 ^ (ENNReal.toReal 1 / ENNReal.toReal p)) * edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y)", "state_after": "case inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 (edist x.1 y.1 ^ ENNReal.toReal p + edist x.2 y.2 ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) \u2264\n    \u2191(2 ^ (1 / ENNReal.toReal p)) *\n      (edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x).1 (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y).1 \u2294\n        edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x).2 (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y).2)"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "p : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\n\u22a2 0 < ENNReal.toReal p", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "p : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\n\u22a2 0 \u2264 1 / ENNReal.toReal p", "state_after": "no goals"}, {"tactic": "refine ENNReal.rpow_le_rpow (add_le_add ?_ ?_) nonneg", "annotated_tactic": ["refine <a>ENNReal.rpow_le_rpow</a> (<a>add_le_add</a> ?_ ?_) nonneg", [{"full_name": "ENNReal.rpow_le_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [642, 9], "def_end_pos": [642, 21]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}]], "state_before": "p : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 (edist x.1 y.1 ^ ENNReal.toReal p + edist x.2 y.2 ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) \u2264\n    (edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y) ^ ENNReal.toReal p +\n        edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y) ^ ENNReal.toReal p) ^\n      (1 / ENNReal.toReal p)", "state_after": "case refine_1\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 edist x.1 y.1 ^ ENNReal.toReal p \u2264\n    edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y) ^ ENNReal.toReal p\n\ncase refine_2\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 edist x.2 y.2 ^ ENNReal.toReal p \u2264\n    edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y) ^ ENNReal.toReal p"}, {"tactic": "refine ENNReal.rpow_le_rpow ?_ (le_of_lt pos)", "annotated_tactic": ["refine <a>ENNReal.rpow_le_rpow</a> ?_ (<a>le_of_lt</a> pos)", [{"full_name": "ENNReal.rpow_le_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [642, 9], "def_end_pos": [642, 21]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case refine_1\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 edist x.1 y.1 ^ ENNReal.toReal p \u2264\n    edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y) ^ ENNReal.toReal p", "state_after": "case refine_1\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 edist x.1 y.1 \u2264 edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y)"}, {"tactic": "simp [edist]", "annotated_tactic": ["simp [<a>edist</a>]", [{"full_name": "EDist.edist", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [48, 3], "def_end_pos": [48, 8]}]], "state_before": "case refine_1\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 edist x.1 y.1 \u2264 edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y)", "state_after": "no goals"}, {"tactic": "refine ENNReal.rpow_le_rpow ?_ (le_of_lt pos)", "annotated_tactic": ["refine <a>ENNReal.rpow_le_rpow</a> ?_ (<a>le_of_lt</a> pos)", [{"full_name": "ENNReal.rpow_le_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [642, 9], "def_end_pos": [642, 21]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case refine_2\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 edist x.2 y.2 ^ ENNReal.toReal p \u2264\n    edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y) ^ ENNReal.toReal p", "state_after": "case refine_2\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 edist x.2 y.2 \u2264 edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y)"}, {"tactic": "simp [edist]", "annotated_tactic": ["simp [<a>edist</a>]", [{"full_name": "EDist.edist", "def_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "def_pos": [48, 3], "def_end_pos": [48, 8]}]], "state_before": "case refine_2\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 edist x.2 y.2 \u2264 edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y)", "state_after": "no goals"}, {"tactic": "simp only [\u2190 two_mul, ENNReal.mul_rpow_of_nonneg _ _ nonneg, \u2190 ENNReal.rpow_mul, cancel,\n  ENNReal.rpow_one, \u2190 ENNReal.coe_rpow_of_nonneg _ nonneg, coe_ofNat]", "annotated_tactic": ["simp only [\u2190 <a>two_mul</a>, <a>ENNReal.mul_rpow_of_nonneg</a> _ _ nonneg, \u2190 <a>ENNReal.rpow_mul</a>, cancel,\n          <a>ENNReal.rpow_one</a>, \u2190 <a>ENNReal.coe_rpow_of_nonneg</a> _ nonneg, <a>coe_ofNat</a>]", [{"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}, {"full_name": "ENNReal.mul_rpow_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [594, 9], "def_end_pos": [594, 27]}, {"full_name": "ENNReal.rpow_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [532, 9], "def_end_pos": [532, 17]}, {"full_name": "ENNReal.rpow_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [450, 9], "def_end_pos": [450, 17]}, {"full_name": "ENNReal.coe_rpow_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [436, 9], "def_end_pos": [436, 27]}, {"full_name": "ENNReal.coe_ofNat", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [398, 9], "def_end_pos": [398, 18]}]], "state_before": "p : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp : Fact (1 \u2264 p)\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : WithLp p (\u03b1 \u00d7 \u03b2)\nh : 1 \u2264 ENNReal.toReal p\npos : 0 < ENNReal.toReal p\nnonneg : 0 \u2264 1 / ENNReal.toReal p\ncancel : ENNReal.toReal p * (1 / ENNReal.toReal p) = 1\n\u22a2 (edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y) ^ ENNReal.toReal p +\n        edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y) ^ ENNReal.toReal p) ^\n      (1 / ENNReal.toReal p) =\n    \u2191(2 ^ (1 / ENNReal.toReal p)) * edist (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) x) (\u2191(WithLp.equiv p (\u03b1 \u00d7 \u03b2)) y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/SimpleGraph/Subgraph.lean", "full_name": "SimpleGraph.Subgraph.neighborSet_sup", "start": [487, 1], "end": [488, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/BooleanAlgebra.lean", "full_name": "sdiff_sdiff_eq_self", "start": [381, 1], "end": [382, 49], "traced_tactics": [{"tactic": "rw [sdiff_sdiff_right_self, inf_of_le_right h]", "annotated_tactic": ["rw [<a>sdiff_sdiff_right_self</a>, <a>inf_of_le_right</a> h]", [{"full_name": "sdiff_sdiff_right_self", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [377, 9], "def_end_pos": [377, 31]}, {"full_name": "inf_of_le_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [458, 11], "def_end_pos": [458, 26]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nh : y \u2264 x\n\u22a2 x \\ (x \\ y) = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/InnerProductSpace/LinearPMap.lean", "full_name": "LinearPMap.IsFormalAdjoint.le_adjoint", "start": [206, 1], "end": [212, 76], "traced_tactics": [{"tactic": "rw [h.symm, hxy]", "annotated_tactic": ["rw [h.symm, hxy]", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c F\nT : E \u2192\u2097.[\ud835\udd5c] F\nS : F \u2192\u2097.[\ud835\udd5c] E\nhT : Dense \u2191T.domain\ninst\u271d : CompleteSpace E\nh : IsFormalAdjoint T S\nx\u271d\u00b2 : { x // x \u2208 S.domain }\nx\u271d\u00b9 : { x // x \u2208 T\u2020.domain }\nhxy : \u2191x\u271d\u00b2 = \u2191x\u271d\u00b9\nx\u271d : { x // x \u2208 T.domain }\n\u22a2 inner (\u2191S x\u271d\u00b2) \u2191x\u271d = inner (\u2191x\u271d\u00b9) (\u2191T x\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Simple.lean", "full_name": "CategoryTheory.isIso_of_epi_of_nonzero", "start": [164, 1], "end": [168, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.coe_add", "start": [794, 1], "end": [795, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/EuclideanDomain/Basic.lean", "full_name": "EuclideanDomain.gcd_val", "start": [145, 1], "end": [147, 70], "traced_tactics": [{"tactic": "rw [gcd]", "annotated_tactic": ["rw [<a>gcd</a>]", [{"full_name": "EuclideanDomain.gcd", "def_path": "Mathlib/Algebra/EuclideanDomain/Defs.lean", "def_pos": [200, 5], "def_end_pos": [200, 8]}]], "state_before": "R : Type u\ninst\u271d\u00b9 : EuclideanDomain R\ninst\u271d : DecidableEq R\na b : R\n\u22a2 gcd a b = gcd (b % a) a", "state_after": "R : Type u\ninst\u271d\u00b9 : EuclideanDomain R\ninst\u271d : DecidableEq R\na b : R\n\u22a2 (if a0 : a = 0 then b\n    else\n      let_fun x := (_ : EuclideanDomain.r (b % a) a);\n      gcd (b % a) a) =\n    gcd (b % a) a"}, {"tactic": "split_ifs with h <;> [simp only [h, mod_zero, gcd_zero_right]; rfl]", "annotated_tactic": ["split_ifs with h <;> [simp only [h, <a>mod_zero</a>, <a>gcd_zero_right</a>]; rfl]", [{"full_name": "EuclideanDomain.mod_zero", "def_path": "Mathlib/Algebra/EuclideanDomain/Defs.lean", "def_pos": [157, 9], "def_end_pos": [157, 17]}, {"full_name": "EuclideanDomain.gcd_zero_right", "def_path": "Mathlib/Algebra/EuclideanDomain/Basic.lean", "def_pos": [140, 9], "def_end_pos": [140, 23]}]], "state_before": "R : Type u\ninst\u271d\u00b9 : EuclideanDomain R\ninst\u271d : DecidableEq R\na b : R\n\u22a2 (if a0 : a = 0 then b\n    else\n      let_fun x := (_ : EuclideanDomain.r (b % a) a);\n      gcd (b % a) a) =\n    gcd (b % a) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Real/ENNReal.lean", "full_name": "ENNReal.lt_ofReal_iff_toReal_lt", "start": [2216, 1], "end": [2219, 75], "traced_tactics": [{"tactic": "lift a to \u211d\u22650 using ha", "annotated_tactic": ["lift a to \u211d\u22650 using ha", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\na : \u211d\u22650\u221e\nb : \u211d\nha : a \u2260 \u22a4\n\u22a2 a < ENNReal.ofReal b \u2194 ENNReal.toReal a < b", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nb : \u211d\na : \u211d\u22650\n\u22a2 \u2191a < ENNReal.ofReal b \u2194 ENNReal.toReal \u2191a < b"}, {"tactic": "simpa [ENNReal.ofReal, ENNReal.toReal] using Real.lt_toNNReal_iff_coe_lt", "annotated_tactic": ["simpa [<a>ENNReal.ofReal</a>, <a>ENNReal.toReal</a>] using <a>Real.lt_toNNReal_iff_coe_lt</a>", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "Real.lt_toNNReal_iff_coe_lt", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [702, 9], "def_end_pos": [702, 31]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nb : \u211d\na : \u211d\u22650\n\u22a2 \u2191a < ENNReal.ofReal b \u2194 ENNReal.toReal \u2191a < b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Option/Basic.lean", "full_name": "Option.Mem.leftUnique", "start": [73, 1], "end": [74, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Quot.lean", "full_name": "Quot.liftOn\u2082_mk", "start": [139, 1], "end": [142, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Homology/HomologicalComplex.lean", "full_name": "CochainComplex.mkHom_f_1", "start": [1166, 1], "end": [1167, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Field/Basic.lean", "full_name": "div_neg_of_pos_of_neg", "start": [706, 1], "end": [707, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Finiteness.lean", "full_name": "RingHom.Finite.of_surjective", "start": [693, 1], "end": [695, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.ne_empty_of_mem", "start": [562, 1], "end": [563, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/FractionalIdeal.lean", "full_name": "FractionalIdeal.map_comp", "start": [747, 1], "end": [748, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean", "full_name": "CircleDeg1Lift.monotone", "start": [148, 11], "end": [148, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean", "full_name": "GaussianInt.normSq_le_normSq_of_re_le_of_im_le", "start": [207, 1], "end": [213, 101], "traced_tactics": [{"tactic": "rw [normSq_apply, normSq_apply, \u2190 _root_.abs_mul_self, _root_.abs_mul, \u2190\n    _root_.abs_mul_self y.re, _root_.abs_mul y.re, \u2190 _root_.abs_mul_self x.im,\n    _root_.abs_mul x.im, \u2190 _root_.abs_mul_self y.im, _root_.abs_mul y.im]", "annotated_tactic": ["rw [<a>normSq_apply</a>, <a>normSq_apply</a>, \u2190 <a>_root_.abs_mul_self</a>, <a>_root_.abs_mul</a>, \u2190\n      <a>_root_.abs_mul_self</a> y.re, <a>_root_.abs_mul</a> y.re, \u2190 <a>_root_.abs_mul_self</a> x.im,\n      <a>_root_.abs_mul</a> x.im, \u2190 <a>_root_.abs_mul_self</a> y.im, <a>_root_.abs_mul</a> y.im]", [{"full_name": "Complex.normSq_apply", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [604, 9], "def_end_pos": [604, 21]}, {"full_name": "Complex.normSq_apply", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [604, 9], "def_end_pos": [604, 21]}, {"full_name": "abs_mul_self", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [54, 9], "def_end_pos": [54, 21]}, {"full_name": "abs_mul", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [33, 9], "def_end_pos": [33, 16]}, {"full_name": "abs_mul_self", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [54, 9], "def_end_pos": [54, 21]}, {"full_name": "abs_mul", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [33, 9], "def_end_pos": [33, 16]}, {"full_name": "abs_mul_self", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [54, 9], "def_end_pos": [54, 21]}, {"full_name": "abs_mul", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [33, 9], "def_end_pos": [33, 16]}, {"full_name": "abs_mul_self", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [54, 9], "def_end_pos": [54, 21]}, {"full_name": "abs_mul", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [33, 9], "def_end_pos": [33, 16]}]], "state_before": "x y : \u2102\nhre : |x.re| \u2264 |y.re|\nhim : |x.im| \u2264 |y.im|\n\u22a2 \u2191normSq x \u2264 \u2191normSq y", "state_after": "x y : \u2102\nhre : |x.re| \u2264 |y.re|\nhim : |x.im| \u2264 |y.im|\n\u22a2 |x.re| * |x.re| + |x.im| * |x.im| \u2264 |y.re| * |y.re| + |y.im| * |y.im|"}, {"tactic": "exact\n    add_le_add (mul_self_le_mul_self (abs_nonneg _) hre) (mul_self_le_mul_self (abs_nonneg _) him)", "annotated_tactic": ["exact\n      <a>add_le_add</a> (<a>mul_self_le_mul_self</a> (<a>abs_nonneg</a> _) hre) (<a>mul_self_le_mul_self</a> (<a>abs_nonneg</a> _) him)", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "mul_self_le_mul_self", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [419, 9], "def_end_pos": [419, 29]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}, {"full_name": "mul_self_le_mul_self", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [419, 9], "def_end_pos": [419, 29]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}]], "state_before": "x y : \u2102\nhre : |x.re| \u2264 |y.re|\nhim : |x.im| \u2264 |y.im|\n\u22a2 |x.re| * |x.re| + |x.im| * |x.im| \u2264 |y.re| * |y.re| + |y.im| * |y.im|", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/ConstMulAction.lean", "full_name": "isOpenMap_quotient_mk'_mul", "start": [491, 1], "end": [495, 53], "traced_tactics": [{"tactic": "rw [isOpen_coinduced, MulAction.quotient_preimage_image_eq_union_mul U]", "annotated_tactic": ["rw [<a>isOpen_coinduced</a>, <a>MulAction.quotient_preimage_image_eq_union_mul</a> U]", [{"full_name": "isOpen_coinduced", "def_path": "Mathlib/Topology/Order.lean", "def_pos": [417, 9], "def_end_pos": [417, 25]}, {"full_name": "MulAction.quotient_preimage_image_eq_union_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [296, 9], "def_end_pos": [296, 45]}]], "state_before": "M : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u0393 : Type u_4\ninst\u271d\u00b3 : Group \u0393\nT : Type u_5\ninst\u271d\u00b2 : TopologicalSpace T\ninst\u271d\u00b9 : MulAction \u0393 T\ninst\u271d : ContinuousConstSMul \u0393 T\nU : Set T\nhU : IsOpen U\n\u22a2 IsOpen (Quotient.mk' '' U)", "state_after": "M : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u0393 : Type u_4\ninst\u271d\u00b3 : Group \u0393\nT : Type u_5\ninst\u271d\u00b2 : TopologicalSpace T\ninst\u271d\u00b9 : MulAction \u0393 T\ninst\u271d : ContinuousConstSMul \u0393 T\nU : Set T\nhU : IsOpen U\n\u22a2 IsOpen (\u22c3 g, (fun x x_1 => x \u2022 x_1) g '' U)"}, {"tactic": "exact isOpen_iUnion fun \u03b3 => isOpenMap_smul \u03b3 U hU", "annotated_tactic": ["exact <a>isOpen_iUnion</a> fun \u03b3 => <a>isOpenMap_smul</a> \u03b3 U hU", [{"full_name": "isOpen_iUnion", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [147, 9], "def_end_pos": [147, 22]}, {"full_name": "isOpenMap_smul", "def_path": "Mathlib/Topology/Algebra/ConstMulAction.lean", "def_pos": [248, 9], "def_end_pos": [248, 23]}]], "state_before": "M : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u0393 : Type u_4\ninst\u271d\u00b3 : Group \u0393\nT : Type u_5\ninst\u271d\u00b2 : TopologicalSpace T\ninst\u271d\u00b9 : MulAction \u0393 T\ninst\u271d : ContinuousConstSMul \u0393 T\nU : Set T\nhU : IsOpen U\n\u22a2 IsOpen (\u22c3 g, (fun x x_1 => x \u2022 x_1) g '' U)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.disjiUnion_cons", "start": [3536, 1], "end": [3543, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/DFinsupp/Basic.lean", "full_name": "AddSubmonoid.iSup_eq_mrange_dfinsupp_sumAddHom", "start": [1972, 1], "end": [1980, 85], "traced_tactics": [{"tactic": "apply le_antisymm", "annotated_tactic": ["apply <a>le_antisymm</a>", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\n\u22a2 iSup S = AddMonoidHom.mrange (sumAddHom fun i => AddSubmonoid.subtype (S i))", "state_after": "case a\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\n\u22a2 iSup S \u2264 AddMonoidHom.mrange (sumAddHom fun i => AddSubmonoid.subtype (S i))\n\ncase a\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\n\u22a2 AddMonoidHom.mrange (sumAddHom fun i => AddSubmonoid.subtype (S i)) \u2264 iSup S"}, {"tactic": "apply iSup_le _", "annotated_tactic": ["apply <a>iSup_le</a> _", [{"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}]], "state_before": "case a\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\n\u22a2 iSup S \u2264 AddMonoidHom.mrange (sumAddHom fun i => AddSubmonoid.subtype (S i))", "state_after": "\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\n\u22a2 \u2200 (i : \u03b9), S i \u2264 AddMonoidHom.mrange (sumAddHom fun i => AddSubmonoid.subtype (S i))"}, {"tactic": "intro i y hy", "annotated_tactic": ["intro i y hy", []], "state_before": "\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\n\u22a2 \u2200 (i : \u03b9), S i \u2264 AddMonoidHom.mrange (sumAddHom fun i => AddSubmonoid.subtype (S i))", "state_after": "\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\ni : \u03b9\ny : \u03b3\nhy : y \u2208 S i\n\u22a2 y \u2208 AddMonoidHom.mrange (sumAddHom fun i => AddSubmonoid.subtype (S i))"}, {"tactic": "exact \u27e8DFinsupp.single i \u27e8y, hy\u27e9, DFinsupp.sumAddHom_single _ _ _\u27e9", "annotated_tactic": ["exact \u27e8<a>DFinsupp.single</a> i \u27e8y, hy\u27e9, <a>DFinsupp.sumAddHom_single</a> _ _ _\u27e9", [{"full_name": "DFinsupp.single", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [639, 5], "def_end_pos": [639, 11]}, {"full_name": "DFinsupp.sumAddHom_single", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [1936, 9], "def_end_pos": [1936, 25]}]], "state_before": "\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\ni : \u03b9\ny : \u03b3\nhy : y \u2208 S i\n\u22a2 y \u2208 AddMonoidHom.mrange (sumAddHom fun i => AddSubmonoid.subtype (S i))", "state_after": "no goals"}, {"tactic": "rintro x \u27e8v, rfl\u27e9", "annotated_tactic": ["rintro x \u27e8v, rfl\u27e9", []], "state_before": "case a\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\n\u22a2 AddMonoidHom.mrange (sumAddHom fun i => AddSubmonoid.subtype (S i)) \u2264 iSup S", "state_after": "case a.intro\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\nv : \u03a0\u2080 (i : \u03b9), { x // x \u2208 S i }\n\u22a2 \u2191(sumAddHom fun i => AddSubmonoid.subtype (S i)) v \u2208 iSup S"}, {"tactic": "exact dfinsupp_sumAddHom_mem _ v _ fun i _ => (le_iSup S i : S i \u2264 _) (v i).prop", "annotated_tactic": ["exact <a>dfinsupp_sumAddHom_mem</a> _ v _ fun i _ => (<a>le_iSup</a> S i : S i \u2264 _) (v i).<a>prop</a>", [{"full_name": "dfinsupp_sumAddHom_mem", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [1961, 9], "def_end_pos": [1961, 38]}, {"full_name": "le_iSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [810, 9], "def_end_pos": [810, 16]}, {"full_name": "Subtype.prop", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [38, 9], "def_end_pos": [38, 13]}]], "state_before": "case a.intro\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\ninst\u271d : AddCommMonoid \u03b3\nS : \u03b9 \u2192 AddSubmonoid \u03b3\nv : \u03a0\u2080 (i : \u03b9), { x // x \u2208 S i }\n\u22a2 \u2191(sumAddHom fun i => AddSubmonoid.subtype (S i)) v \u2208 iSup S", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Instances/Real.lean", "full_name": "Real.isBounded_iff_bddBelow_bddAbove", "start": [182, 1], "end": [187, 56], "traced_tactics": [{"tactic": "obtain \u27e8r, hr\u27e9 : \u2203 r : \u211d, s \u2286 Icc (-r) r := by\n  simpa [Real.closedBall_eq_Icc] using bdd.subset_closedBall 0", "annotated_tactic": ["obtain \u27e8r, hr\u27e9 : \u2203 r : \u211d, s \u2286 <a>Icc</a> (-r) r := by\n      simpa [<a>Real.closedBall_eq_Icc</a>] using bdd.subset_closedBall 0", [{"full_name": "Set.Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Real.closedBall_eq_Icc", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1413, 9], "def_end_pos": [1413, 31]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Set \u211d\nbdd : IsBounded s\n\u22a2 BddBelow s \u2227 BddAbove s", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Set \u211d\nbdd : IsBounded s\nr : \u211d\nhr : s \u2286 Icc (-r) r\n\u22a2 BddBelow s \u2227 BddAbove s"}, {"tactic": "exact \u27e8bddBelow_Icc.mono hr, bddAbove_Icc.mono hr\u27e9", "annotated_tactic": ["exact \u27e8bddBelow_Icc.mono hr, bddAbove_Icc.mono hr\u27e9", []], "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Set \u211d\nbdd : IsBounded s\nr : \u211d\nhr : s \u2286 Icc (-r) r\n\u22a2 BddBelow s \u2227 BddAbove s", "state_after": "no goals"}, {"tactic": "simpa [Real.closedBall_eq_Icc] using bdd.subset_closedBall 0", "annotated_tactic": ["simpa [<a>Real.closedBall_eq_Icc</a>] using bdd.subset_closedBall 0", [{"full_name": "Real.closedBall_eq_Icc", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1413, 9], "def_end_pos": [1413, 31]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Set \u211d\nbdd : IsBounded s\n\u22a2 \u2203 r, s \u2286 Icc (-r) r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Tropical/Basic.lean", "full_name": "Tropical.add_eq_right", "start": [331, 1], "end": [332, 38], "traced_tactics": [{"tactic": "simpa using h", "annotated_tactic": ["simpa using h", []], "state_before": "R : Type u\ninst\u271d : LinearOrder R\nx y : Tropical R\nh : y \u2264 x\n\u22a2 untrop (x + y) = untrop y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean", "full_name": "AddChar.to_mulShift_inj_of_isPrimitive", "start": [252, 1], "end": [259, 37], "traced_tactics": [{"tactic": "intro a b h", "annotated_tactic": ["intro a b h", []], "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\nR' : Type v\ninst\u271d : CommRing R'\n\u03c8 : AddChar R R'\nh\u03c8 : IsPrimitive \u03c8\n\u22a2 Function.Injective (mulShift \u03c8)", "state_after": "R : Type u\ninst\u271d\u00b9 : CommRing R\nR' : Type v\ninst\u271d : CommRing R'\n\u03c8 : AddChar R R'\nh\u03c8 : IsPrimitive \u03c8\na b : R\nh : mulShift \u03c8 a = mulShift \u03c8 b\n\u22a2 a = b"}, {"tactic": "apply_fun fun x => x * mulShift \u03c8 (-b) at h", "annotated_tactic": ["apply_fun fun x => x * <a>mulShift</a> \u03c8 (-b) at h", [{"full_name": "AddChar.mulShift", "def_path": "Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean", "def_pos": [204, 5], "def_end_pos": [204, 13]}]], "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\nR' : Type v\ninst\u271d : CommRing R'\n\u03c8 : AddChar R R'\nh\u03c8 : IsPrimitive \u03c8\na b : R\nh : mulShift \u03c8 a = mulShift \u03c8 b\n\u22a2 a = b", "state_after": "R : Type u\ninst\u271d\u00b9 : CommRing R\nR' : Type v\ninst\u271d : CommRing R'\n\u03c8 : AddChar R R'\nh\u03c8 : IsPrimitive \u03c8\na b : R\nh : mulShift \u03c8 a * mulShift \u03c8 (-b) = mulShift \u03c8 b * mulShift \u03c8 (-b)\n\u22a2 a = b"}, {"tactic": "simp only [mulShift_mul, mulShift_zero, add_right_neg] at h", "annotated_tactic": ["simp only [<a>mulShift_mul</a>, <a>mulShift_zero</a>, <a>add_right_neg</a>] at h", [{"full_name": "AddChar.mulShift_mul", "def_path": "Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean", "def_pos": [231, 9], "def_end_pos": [231, 21]}, {"full_name": "AddChar.mulShift_zero", "def_path": "Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean", "def_pos": [239, 9], "def_end_pos": [239, 22]}, {"full_name": "add_right_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1134, 3], "def_end_pos": [1134, 14]}]], "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\nR' : Type v\ninst\u271d : CommRing R'\n\u03c8 : AddChar R R'\nh\u03c8 : IsPrimitive \u03c8\na b : R\nh : mulShift \u03c8 a * mulShift \u03c8 (-b) = mulShift \u03c8 b * mulShift \u03c8 (-b)\n\u22a2 a = b", "state_after": "R : Type u\ninst\u271d\u00b9 : CommRing R\nR' : Type v\ninst\u271d : CommRing R'\n\u03c8 : AddChar R R'\nh\u03c8 : IsPrimitive \u03c8\na b : R\nh : mulShift \u03c8 (a + -b) = 1\n\u22a2 a = b"}, {"tactic": "have h\u2082 := h\u03c8 (a + -b)", "annotated_tactic": ["have h\u2082 := h\u03c8 (a + -b)", []], "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\nR' : Type v\ninst\u271d : CommRing R'\n\u03c8 : AddChar R R'\nh\u03c8 : IsPrimitive \u03c8\na b : R\nh : mulShift \u03c8 (a + -b) = 1\n\u22a2 a = b", "state_after": "R : Type u\ninst\u271d\u00b9 : CommRing R\nR' : Type v\ninst\u271d : CommRing R'\n\u03c8 : AddChar R R'\nh\u03c8 : IsPrimitive \u03c8\na b : R\nh : mulShift \u03c8 (a + -b) = 1\nh\u2082 : a + -b \u2260 0 \u2192 IsNontrivial (mulShift \u03c8 (a + -b))\n\u22a2 a = b"}, {"tactic": "rw [h, isNontrivial_iff_ne_trivial, \u2190 sub_eq_add_neg, sub_ne_zero] at h\u2082", "annotated_tactic": ["rw [h, <a>isNontrivial_iff_ne_trivial</a>, \u2190 <a>sub_eq_add_neg</a>, <a>sub_ne_zero</a>] at h\u2082", [{"full_name": "AddChar.isNontrivial_iff_ne_trivial", "def_path": "Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean", "def_pos": [196, 9], "def_end_pos": [196, 36]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "sub_ne_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [813, 3], "def_end_pos": [813, 14]}]], "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\nR' : Type v\ninst\u271d : CommRing R'\n\u03c8 : AddChar R R'\nh\u03c8 : IsPrimitive \u03c8\na b : R\nh : mulShift \u03c8 (a + -b) = 1\nh\u2082 : a + -b \u2260 0 \u2192 IsNontrivial (mulShift \u03c8 (a + -b))\n\u22a2 a = b", "state_after": "R : Type u\ninst\u271d\u00b9 : CommRing R\nR' : Type v\ninst\u271d : CommRing R'\n\u03c8 : AddChar R R'\nh\u03c8 : IsPrimitive \u03c8\na b : R\nh : mulShift \u03c8 (a + -b) = 1\nh\u2082 : a \u2260 b \u2192 1 \u2260 1\n\u22a2 a = b"}, {"tactic": "exact not_not.mp fun h => h\u2082 h rfl", "annotated_tactic": ["exact not_not.mp fun h => h\u2082 h <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\nR' : Type v\ninst\u271d : CommRing R'\n\u03c8 : AddChar R R'\nh\u03c8 : IsPrimitive \u03c8\na b : R\nh : mulShift \u03c8 (a + -b) = 1\nh\u2082 : a \u2260 b \u2192 1 \u2260 1\n\u22a2 a = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Preadditive/Opposite.lean", "full_name": "CategoryTheory.op_zsmul", "start": [56, 1], "end": [57, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "full_name": "Ordinal.nat_cast_succ", "start": [1095, 1], "end": [1096, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "full_name": "AffineEquiv.span_eq_top_iff", "start": [1610, 1], "end": [1616, 77], "traced_tactics": [{"tactic": "refine' \u27e8(e : P\u2081 \u2192\u1d43[k] P\u2082).span_eq_top_of_surjective e.surjective, _\u27e9", "annotated_tactic": ["refine' \u27e8(e : P\u2081 \u2192\u1d43[k] P\u2082).<a>span_eq_top_of_surjective</a> e.surjective, _\u27e9", [{"full_name": "AffineMap.span_eq_top_of_surjective", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [1601, 9], "def_end_pos": [1601, 34]}]], "state_before": "k : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\ne : P\u2081 \u2243\u1d43[k] P\u2082\n\u22a2 affineSpan k s = \u22a4 \u2194 affineSpan k (\u2191e '' s) = \u22a4", "state_after": "k : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\ne : P\u2081 \u2243\u1d43[k] P\u2082\n\u22a2 affineSpan k (\u2191e '' s) = \u22a4 \u2192 affineSpan k s = \u22a4"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "k : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\ne : P\u2081 \u2243\u1d43[k] P\u2082\n\u22a2 affineSpan k (\u2191e '' s) = \u22a4 \u2192 affineSpan k s = \u22a4", "state_after": "k : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\ne : P\u2081 \u2243\u1d43[k] P\u2082\nh : affineSpan k (\u2191e '' s) = \u22a4\n\u22a2 affineSpan k s = \u22a4"}, {"tactic": "have : s = e.symm '' (e '' s) := by rw [\u2190 image_comp]; simp", "annotated_tactic": ["have : s = e.symm '' (e '' s) := by rw [\u2190 <a>image_comp</a>]; simp", [{"full_name": "Set.image_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [293, 9], "def_end_pos": [293, 19]}]], "state_before": "k : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\ne : P\u2081 \u2243\u1d43[k] P\u2082\nh : affineSpan k (\u2191e '' s) = \u22a4\n\u22a2 affineSpan k s = \u22a4", "state_after": "k : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\ne : P\u2081 \u2243\u1d43[k] P\u2082\nh : affineSpan k (\u2191e '' s) = \u22a4\nthis : s = \u2191(symm e) '' (\u2191e '' s)\n\u22a2 affineSpan k s = \u22a4"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "k : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\ne : P\u2081 \u2243\u1d43[k] P\u2082\nh : affineSpan k (\u2191e '' s) = \u22a4\nthis : s = \u2191(symm e) '' (\u2191e '' s)\n\u22a2 affineSpan k s = \u22a4", "state_after": "k : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\ne : P\u2081 \u2243\u1d43[k] P\u2082\nh : affineSpan k (\u2191e '' s) = \u22a4\nthis : s = \u2191(symm e) '' (\u2191e '' s)\n\u22a2 affineSpan k (\u2191(symm e) '' (\u2191e '' s)) = \u22a4"}, {"tactic": "exact (e.symm : P\u2082 \u2192\u1d43[k] P\u2081).span_eq_top_of_surjective e.symm.surjective h", "annotated_tactic": ["exact (e.symm : P\u2082 \u2192\u1d43[k] P\u2081).<a>span_eq_top_of_surjective</a> e.symm.surjective h", [{"full_name": "AffineMap.span_eq_top_of_surjective", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [1601, 9], "def_end_pos": [1601, 34]}]], "state_before": "k : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\ne : P\u2081 \u2243\u1d43[k] P\u2082\nh : affineSpan k (\u2191e '' s) = \u22a4\nthis : s = \u2191(symm e) '' (\u2191e '' s)\n\u22a2 affineSpan k (\u2191(symm e) '' (\u2191e '' s)) = \u22a4", "state_after": "no goals"}, {"tactic": "rw [\u2190 image_comp]", "annotated_tactic": ["rw [\u2190 <a>image_comp</a>]", [{"full_name": "Set.image_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [293, 9], "def_end_pos": [293, 19]}]], "state_before": "k : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\ne : P\u2081 \u2243\u1d43[k] P\u2082\nh : affineSpan k (\u2191e '' s) = \u22a4\n\u22a2 s = \u2191(symm e) '' (\u2191e '' s)", "state_after": "k : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\ne : P\u2081 \u2243\u1d43[k] P\u2082\nh : affineSpan k (\u2191e '' s) = \u22a4\n\u22a2 s = \u2191(symm e) \u2218 \u2191e '' s"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "k : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\ne : P\u2081 \u2243\u1d43[k] P\u2082\nh : affineSpan k (\u2191e '' s) = \u22a4\n\u22a2 s = \u2191(symm e) \u2218 \u2191e '' s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/IsROrC/Basic.lean", "full_name": "IsROrC.conj_re", "start": [345, 1], "end": [346, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "full_name": "Set.div_eq_empty", "start": [634, 1], "end": [635, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Counit.lean", "full_name": "MvPolynomial.counit_surjective", "start": [82, 1], "end": [83, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "full_name": "MeasureTheory.Mem\u2112p.snorm_indicator_norm_ge_pos_le", "start": [297, 1], "end": [306, 28], "traced_tactics": [{"tactic": "obtain \u27e8M, hM\u27e9 := hf.snorm_indicator_norm_ge_le \u03bc hmeas h\u03b5", "annotated_tactic": ["obtain \u27e8M, hM\u27e9 := hf.snorm_indicator_norm_ge_le \u03bc hmeas h\u03b5", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 M, 0 < M \u2227 snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 M, 0 < M \u2227 snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "refine'\n  \u27e8max M 1, lt_of_lt_of_le zero_lt_one (le_max_right _ _), le_trans (snorm_mono fun x => _) hM\u27e9", "annotated_tactic": ["refine'\n    \u27e8<a>max</a> M 1, <a>lt_of_lt_of_le</a> <a>zero_lt_one</a> (<a>le_max_right</a> _ _), <a>le_trans</a> (<a>snorm_mono</a> fun x => _) hM\u27e9", [{"full_name": "Max.max", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1090, 3], "def_end_pos": [1090, 6]}, {"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}, {"full_name": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "MeasureTheory.snorm_mono", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [434, 9], "def_end_pos": [434, 19]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 M, 0 < M \u2227 snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\nx : \u03b1\n\u22a2 \u2016Set.indicator {x | max M 1 \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016 \u2264 \u2016Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016"}, {"tactic": "rw [norm_indicator_eq_indicator_norm, norm_indicator_eq_indicator_norm]", "annotated_tactic": ["rw [<a>norm_indicator_eq_indicator_norm</a>, <a>norm_indicator_eq_indicator_norm</a>]", [{"full_name": "norm_indicator_eq_indicator_norm", "def_path": "Mathlib/Analysis/NormedSpace/IndicatorFunction.lean", "def_pos": [25, 9], "def_end_pos": [25, 41]}, {"full_name": "norm_indicator_eq_indicator_norm", "def_path": "Mathlib/Analysis/NormedSpace/IndicatorFunction.lean", "def_pos": [25, 9], "def_end_pos": [25, 41]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\nx : \u03b1\n\u22a2 \u2016Set.indicator {x | max M 1 \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016 \u2264 \u2016Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f x\u2016", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\nx : \u03b1\n\u22a2 Set.indicator {x | max M 1 \u2264 \u2191\u2016f x\u2016\u208a} (fun a => \u2016f a\u2016) x \u2264 Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} (fun a => \u2016f a\u2016) x"}, {"tactic": "refine' Set.indicator_le_indicator_of_subset (fun x hx => _) (fun x => norm_nonneg (f x)) x", "annotated_tactic": ["refine' <a>Set.indicator_le_indicator_of_subset</a> (fun x hx => _) (fun x => <a>norm_nonneg</a> (f x)) x", [{"full_name": "Set.indicator_le_indicator_of_subset", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [849, 3], "def_end_pos": [849, 14]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [500, 30], "def_end_pos": [500, 41]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\nx : \u03b1\n\u22a2 Set.indicator {x | max M 1 \u2264 \u2191\u2016f x\u2016\u208a} (fun a => \u2016f a\u2016) x \u2264 Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} (fun a => \u2016f a\u2016) x", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\nx\u271d x : \u03b1\nhx : x \u2208 {x | max M 1 \u2264 \u2191\u2016f x\u2016\u208a}\n\u22a2 x \u2208 {x | M \u2264 \u2191\u2016f x\u2016\u208a}"}, {"tactic": "rw [Set.mem_setOf_eq] at hx", "annotated_tactic": ["rw [<a>Set.mem_setOf_eq</a>] at hx", [{"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\nx\u271d x : \u03b1\nhx : x \u2208 {x | max M 1 \u2264 \u2191\u2016f x\u2016\u208a}\n\u22a2 x \u2208 {x | M \u2264 \u2191\u2016f x\u2016\u208a}", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\nx\u271d x : \u03b1\nhx : max M 1 \u2264 \u2191\u2016f x\u2016\u208a\n\u22a2 x \u2208 {x | M \u2264 \u2191\u2016f x\u2016\u208a}"}, {"tactic": "exact (max_le_iff.1 hx).1", "annotated_tactic": ["exact (<a>max_le_iff</a>.1 hx).1", [{"full_name": "max_le_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [48, 9], "def_end_pos": [48, 19]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhf : Mem\u2112p f p\nhmeas : StronglyMeasurable f\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nM : \u211d\nhM : snorm (Set.indicator {x | M \u2264 \u2191\u2016f x\u2016\u208a} f) p \u03bc \u2264 ENNReal.ofReal \u03b5\nx\u271d x : \u03b1\nhx : max M 1 \u2264 \u2191\u2016f x\u2016\u208a\n\u22a2 x \u2208 {x | M \u2264 \u2191\u2016f x\u2016\u208a}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Real/CauSeqCompletion.lean", "full_name": "CauSeq.Completion.mk_mul", "start": [99, 1], "end": [100, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Ordinal/Notation.lean", "full_name": "ONote.NFBelow.repr_lt", "start": [266, 1], "end": [274, 58], "traced_tactics": [{"tactic": "induction' h with _ e n a eb b h\u2081 h\u2082 h\u2083 _ IH", "annotated_tactic": ["induction' h with _ e n a eb b h\u2081 h\u2082 h\u2083 _ IH", []], "state_before": "o : ONote\nb : Ordinal.{0}\nh : NFBelow o b\n\u22a2 repr o < \u03c9 ^ b", "state_after": "case zero\no : ONote\nb b\u271d : Ordinal.{0}\n\u22a2 repr 0 < \u03c9 ^ b\u271d\n\ncase oadd'\no : ONote\nb\u271d : Ordinal.{0}\ne : ONote\nn : \u2115+\na : ONote\neb b : Ordinal.{0}\nh\u2081 : NFBelow e eb\nh\u2082 : NFBelow a (repr e)\nh\u2083 : repr e < b\na_ih\u271d : repr e < \u03c9 ^ eb\nIH : repr a < \u03c9 ^ repr e\n\u22a2 repr (ONote.oadd e n a) < \u03c9 ^ b"}, {"tactic": "exact opow_pos _ omega_pos", "annotated_tactic": ["exact <a>opow_pos</a> _ <a>omega_pos</a>", [{"full_name": "Ordinal.opow_pos", "def_path": "Mathlib/SetTheory/Ordinal/Exponential.lean", "def_pos": [93, 9], "def_end_pos": [93, 17]}, {"full_name": "Ordinal.omega_pos", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [2423, 9], "def_end_pos": [2423, 18]}]], "state_before": "case zero\no : ONote\nb b\u271d : Ordinal.{0}\n\u22a2 repr 0 < \u03c9 ^ b\u271d", "state_after": "no goals"}, {"tactic": "rw [repr]", "annotated_tactic": ["rw [<a>repr</a>]", [{"full_name": "ONote.repr", "def_path": "Mathlib/SetTheory/Ordinal/Notation.lean", "def_pos": [71, 19], "def_end_pos": [71, 23]}]], "state_before": "case oadd'\no : ONote\nb\u271d : Ordinal.{0}\ne : ONote\nn : \u2115+\na : ONote\neb b : Ordinal.{0}\nh\u2081 : NFBelow e eb\nh\u2082 : NFBelow a (repr e)\nh\u2083 : repr e < b\na_ih\u271d : repr e < \u03c9 ^ eb\nIH : repr a < \u03c9 ^ repr e\n\u22a2 repr (ONote.oadd e n a) < \u03c9 ^ b", "state_after": "case oadd'\no : ONote\nb\u271d : Ordinal.{0}\ne : ONote\nn : \u2115+\na : ONote\neb b : Ordinal.{0}\nh\u2081 : NFBelow e eb\nh\u2082 : NFBelow a (repr e)\nh\u2083 : repr e < b\na_ih\u271d : repr e < \u03c9 ^ eb\nIH : repr a < \u03c9 ^ repr e\n\u22a2 \u03c9 ^ repr e * \u2191\u2191n + repr a < \u03c9 ^ b"}, {"tactic": "apply ((add_lt_add_iff_left _).2 IH).trans_le", "annotated_tactic": ["apply ((<a>add_lt_add_iff_left</a> _).2 IH).<a>trans_le</a>", [{"full_name": "add_lt_add_iff_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [104, 3], "def_end_pos": [104, 14]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}]], "state_before": "case oadd'\no : ONote\nb\u271d : Ordinal.{0}\ne : ONote\nn : \u2115+\na : ONote\neb b : Ordinal.{0}\nh\u2081 : NFBelow e eb\nh\u2082 : NFBelow a (repr e)\nh\u2083 : repr e < b\na_ih\u271d : repr e < \u03c9 ^ eb\nIH : repr a < \u03c9 ^ repr e\n\u22a2 \u03c9 ^ repr e * \u2191\u2191n + repr a < \u03c9 ^ b", "state_after": "case oadd'\no : ONote\nb\u271d : Ordinal.{0}\ne : ONote\nn : \u2115+\na : ONote\neb b : Ordinal.{0}\nh\u2081 : NFBelow e eb\nh\u2082 : NFBelow a (repr e)\nh\u2083 : repr e < b\na_ih\u271d : repr e < \u03c9 ^ eb\nIH : repr a < \u03c9 ^ repr e\n\u22a2 \u03c9 ^ repr e * \u2191\u2191n + \u03c9 ^ repr e \u2264 \u03c9 ^ b"}, {"tactic": "rw [\u2190 mul_succ]", "annotated_tactic": ["rw [\u2190 <a>mul_succ</a>]", [{"full_name": "Ordinal.mul_succ", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [718, 9], "def_end_pos": [718, 17]}]], "state_before": "case oadd'\no : ONote\nb\u271d : Ordinal.{0}\ne : ONote\nn : \u2115+\na : ONote\neb b : Ordinal.{0}\nh\u2081 : NFBelow e eb\nh\u2082 : NFBelow a (repr e)\nh\u2083 : repr e < b\na_ih\u271d : repr e < \u03c9 ^ eb\nIH : repr a < \u03c9 ^ repr e\n\u22a2 \u03c9 ^ repr e * \u2191\u2191n + \u03c9 ^ repr e \u2264 \u03c9 ^ b", "state_after": "case oadd'\no : ONote\nb\u271d : Ordinal.{0}\ne : ONote\nn : \u2115+\na : ONote\neb b : Ordinal.{0}\nh\u2081 : NFBelow e eb\nh\u2082 : NFBelow a (repr e)\nh\u2083 : repr e < b\na_ih\u271d : repr e < \u03c9 ^ eb\nIH : repr a < \u03c9 ^ repr e\n\u22a2 \u03c9 ^ repr e * succ \u2191\u2191n \u2264 \u03c9 ^ b"}, {"tactic": "apply (mul_le_mul_left' (succ_le_of_lt (nat_lt_omega _)) _).trans", "annotated_tactic": ["apply (<a>mul_le_mul_left'</a> (<a>succ_le_of_lt</a> (<a>nat_lt_omega</a> _)) _).<a>trans</a>", [{"full_name": "mul_le_mul_left'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [50, 9], "def_end_pos": [50, 25]}, {"full_name": "Order.succ_le_of_lt", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [221, 9], "def_end_pos": [221, 22]}, {"full_name": "Ordinal.nat_lt_omega", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [2419, 9], "def_end_pos": [2419, 21]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case oadd'\no : ONote\nb\u271d : Ordinal.{0}\ne : ONote\nn : \u2115+\na : ONote\neb b : Ordinal.{0}\nh\u2081 : NFBelow e eb\nh\u2082 : NFBelow a (repr e)\nh\u2083 : repr e < b\na_ih\u271d : repr e < \u03c9 ^ eb\nIH : repr a < \u03c9 ^ repr e\n\u22a2 \u03c9 ^ repr e * succ \u2191\u2191n \u2264 \u03c9 ^ b", "state_after": "case oadd'\no : ONote\nb\u271d : Ordinal.{0}\ne : ONote\nn : \u2115+\na : ONote\neb b : Ordinal.{0}\nh\u2081 : NFBelow e eb\nh\u2082 : NFBelow a (repr e)\nh\u2083 : repr e < b\na_ih\u271d : repr e < \u03c9 ^ eb\nIH : repr a < \u03c9 ^ repr e\n\u22a2 \u03c9 ^ repr e * \u03c9 \u2264 \u03c9 ^ b"}, {"tactic": "rw [\u2190 opow_succ]", "annotated_tactic": ["rw [\u2190 <a>opow_succ</a>]", [{"full_name": "Ordinal.opow_succ", "def_path": "Mathlib/SetTheory/Ordinal/Exponential.lean", "def_pos": [57, 9], "def_end_pos": [57, 18]}]], "state_before": "case oadd'\no : ONote\nb\u271d : Ordinal.{0}\ne : ONote\nn : \u2115+\na : ONote\neb b : Ordinal.{0}\nh\u2081 : NFBelow e eb\nh\u2082 : NFBelow a (repr e)\nh\u2083 : repr e < b\na_ih\u271d : repr e < \u03c9 ^ eb\nIH : repr a < \u03c9 ^ repr e\n\u22a2 \u03c9 ^ repr e * \u03c9 \u2264 \u03c9 ^ b", "state_after": "case oadd'\no : ONote\nb\u271d : Ordinal.{0}\ne : ONote\nn : \u2115+\na : ONote\neb b : Ordinal.{0}\nh\u2081 : NFBelow e eb\nh\u2082 : NFBelow a (repr e)\nh\u2083 : repr e < b\na_ih\u271d : repr e < \u03c9 ^ eb\nIH : repr a < \u03c9 ^ repr e\n\u22a2 \u03c9 ^ succ (repr e) \u2264 \u03c9 ^ b"}, {"tactic": "exact opow_le_opow_right omega_pos (succ_le_of_lt h\u2083)", "annotated_tactic": ["exact <a>opow_le_opow_right</a> <a>omega_pos</a> (<a>succ_le_of_lt</a> h\u2083)", [{"full_name": "Ordinal.opow_le_opow_right", "def_path": "Mathlib/SetTheory/Ordinal/Exponential.lean", "def_pos": [138, 9], "def_end_pos": [138, 27]}, {"full_name": "Ordinal.omega_pos", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [2423, 9], "def_end_pos": [2423, 18]}, {"full_name": "Order.succ_le_of_lt", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [221, 9], "def_end_pos": [221, 22]}]], "state_before": "case oadd'\no : ONote\nb\u271d : Ordinal.{0}\ne : ONote\nn : \u2115+\na : ONote\neb b : Ordinal.{0}\nh\u2081 : NFBelow e eb\nh\u2082 : NFBelow a (repr e)\nh\u2083 : repr e < b\na_ih\u271d : repr e < \u03c9 ^ eb\nIH : repr a < \u03c9 ^ repr e\n\u22a2 \u03c9 ^ succ (repr e) \u2264 \u03c9 ^ b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Quotient.lean", "full_name": "Submodule.quotEquivOfEq_mk", "start": [653, 1], "end": [656, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Order/ProjIcc.lean", "full_name": "Continuous.Icc_extend'", "start": [62, 11], "end": [64, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.sigmaToiUnion_surjective", "start": [2321, 1], "end": [2325, 22], "traced_tactics": [{"tactic": "simpa using hb", "annotated_tactic": ["simpa using hb", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nt : \u03b1 \u2192 Set \u03b2\nb : \u03b2\nhb : b \u2208 \u22c3 i, t i\n\u22a2 \u2203 a, b \u2208 t a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Sum/Basic.lean", "full_name": "Function.Injective.sum_elim", "start": [231, 1], "end": [236, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Variables.lean", "full_name": "MvPolynomial.vars_monomial_single", "start": [463, 1], "end": [465, 61], "traced_tactics": [{"tactic": "rw [vars_monomial hr, Finsupp.support_single_ne_zero _ he]", "annotated_tactic": ["rw [<a>vars_monomial</a> hr, <a>Finsupp.support_single_ne_zero</a> _ he]", [{"full_name": "MvPolynomial.vars_monomial", "def_path": "Mathlib/Data/MvPolynomial/Variables.lean", "def_pos": [290, 9], "def_end_pos": [290, 22]}, {"full_name": "Finsupp.support_single_ne_zero", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [358, 9], "def_end_pos": [358, 31]}]], "state_before": "R : Type u\nS : Type v\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nr\u271d : R\ne\u271d : \u2115\nn m : \u03c3\ns : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b9 : CommSemiring R\np q : MvPolynomial \u03c3 R\ninst\u271d : CommSemiring S\nf : R \u2192+* S\ni : \u03c3\ne : \u2115\nr : R\nhe : e \u2260 0\nhr : r \u2260 0\n\u22a2 vars (\u2191(monomial fun\u2080 | i => e) r) = {i}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/BigOperators/Basic.lean", "full_name": "RingHom.map_list_prod", "start": [230, 11], "end": [232, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "full_name": "ENNReal.essSup_const_mul", "start": [313, 1], "end": [314, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBSet.mkRBSet_eq", "start": [618, 9], "end": [618, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Ideal/Operations.lean", "full_name": "Ideal.IsPrime.radical_le_iff", "start": [1012, 1], "end": [1013, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean", "full_name": "jacobiSym.one_right", "start": [105, 1], "end": [106, 63], "traced_tactics": [{"tactic": "simp only [jacobiSym, factors_one, List.prod_nil, List.pmap]", "annotated_tactic": ["simp only [<a>jacobiSym</a>, <a>factors_one</a>, <a>List.prod_nil</a>, <a>List.pmap</a>]", [{"full_name": "jacobiSym", "def_path": "Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean", "def_pos": [80, 5], "def_end_pos": [80, 14]}, {"full_name": "Nat.factors_one", "def_path": "Mathlib/Data/Nat/Factors.lean", "def_pos": [47, 9], "def_end_pos": [47, 20]}, {"full_name": "List.prod_nil", "def_path": "Mathlib/Data/List/BigOperators/Basic.lean", "def_pos": [29, 9], "def_end_pos": [29, 17]}, {"full_name": "List.pmap", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [754, 13], "def_end_pos": [754, 17]}]], "state_before": "a : \u2124\n\u22a2 J(a | 1) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Star/NonUnitalSubalgebra.lean", "full_name": "NonUnitalStarSubalgebra.mem_toNonUnitalSubalgebra", "start": [133, 1], "end": [135, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/HNNExtension.lean", "full_name": "HNNExtension.induction_on", "start": [113, 1], "end": [129, 16], "traced_tactics": [{"tactic": "let S : Subgroup (HNNExtension G A B \u03c6) :=\n  { carrier := setOf motive\n    one_mem' := by simpa using of 1\n    mul_mem' := mul _ _\n    inv_mem' := inv _ }", "annotated_tactic": ["let S : <a>Subgroup</a> (<a>HNNExtension</a> G A B \u03c6) :=\n    { carrier := <a>setOf</a> motive\n      one_mem' := by simpa using of 1\n      mul_mem' := mul _ _\n      inv_mem' := inv _ }", [{"full_name": "Subgroup", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [363, 11], "def_end_pos": [363, 19]}, {"full_name": "HNNExtension", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [46, 5], "def_end_pos": [46, 17]}, {"full_name": "setOf", "def_path": "Mathlib/Init/Set.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b9 : Group H\nM : Type u_3\ninst\u271d : Monoid M\nmotive : HNNExtension G A B \u03c6 \u2192 Prop\nx : HNNExtension G A B \u03c6\nof : \u2200 (g : G), motive (\u2191HNNExtension.of g)\nt : motive HNNExtension.t\nmul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y)\ninv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9\n\u22a2 motive x", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b9 : Group H\nM : Type u_3\ninst\u271d : Monoid M\nmotive : HNNExtension G A B \u03c6 \u2192 Prop\nx : HNNExtension G A B \u03c6\nof : \u2200 (g : G), motive (\u2191HNNExtension.of g)\nt : motive HNNExtension.t\nmul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y)\ninv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9\nS : Subgroup (HNNExtension G A B \u03c6) :=\n  {\n    toSubmonoid :=\n      {\n        toSubsemigroup :=\n          { carrier := setOf motive,\n            mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) },\n        one_mem' :=\n          (_ :\n            1 \u2208\n              { carrier := setOf motive,\n                  mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) },\n    inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) }\n\u22a2 motive x"}, {"tactic": "let f : HNNExtension G A B \u03c6 \u2192* S :=\n  lift (HNNExtension.of.codRestrict S of)\n    \u27e8HNNExtension.t, t\u27e9 (by intro a; ext; simp [equiv_eq_conj, mul_assoc])", "annotated_tactic": ["let f : <a>HNNExtension</a> G A B \u03c6 \u2192* S :=\n    <a>lift</a> (HNNExtension.of.codRestrict S of)\n      \u27e8<a>HNNExtension.t</a>, t\u27e9 (by intro a; ext; simp [<a>equiv_eq_conj</a>, <a>mul_assoc</a>])", [{"full_name": "HNNExtension", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [46, 5], "def_end_pos": [46, 17]}, {"full_name": "HNNExtension.lift", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "HNNExtension.t", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [62, 5], "def_end_pos": [62, 6]}, {"full_name": "HNNExtension.equiv_eq_conj", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [73, 9], "def_end_pos": [73, 22]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b9 : Group H\nM : Type u_3\ninst\u271d : Monoid M\nmotive : HNNExtension G A B \u03c6 \u2192 Prop\nx : HNNExtension G A B \u03c6\nof : \u2200 (g : G), motive (\u2191HNNExtension.of g)\nt : motive HNNExtension.t\nmul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y)\ninv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9\nS : Subgroup (HNNExtension G A B \u03c6) :=\n  {\n    toSubmonoid :=\n      {\n        toSubsemigroup :=\n          { carrier := setOf motive,\n            mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) },\n        one_mem' :=\n          (_ :\n            1 \u2208\n              { carrier := setOf motive,\n                  mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) },\n    inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) }\n\u22a2 motive x", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b9 : Group H\nM : Type u_3\ninst\u271d : Monoid M\nmotive : HNNExtension G A B \u03c6 \u2192 Prop\nx : HNNExtension G A B \u03c6\nof : \u2200 (g : G), motive (\u2191HNNExtension.of g)\nt : motive HNNExtension.t\nmul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y)\ninv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9\nS : Subgroup (HNNExtension G A B \u03c6) :=\n  {\n    toSubmonoid :=\n      {\n        toSubsemigroup :=\n          { carrier := setOf motive,\n            mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) },\n        one_mem' :=\n          (_ :\n            1 \u2208\n              { carrier := setOf motive,\n                  mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) },\n    inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) }\nf : HNNExtension G A B \u03c6 \u2192* { x // x \u2208 S } :=\n  lift (MonoidHom.codRestrict HNNExtension.of S of) { val := HNNExtension.t, property := t }\n    (_ :\n      \u2200 (a : { x // x \u2208 A }),\n        { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a =\n          \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t })\n\u22a2 motive x"}, {"tactic": "have hf : S.subtype.comp f = MonoidHom.id _ :=\n  hom_ext (by ext; simp) (by simp)", "annotated_tactic": ["have hf : S.subtype.comp f = <a>MonoidHom.id</a> _ :=\n    <a>hom_ext</a> (by ext; simp) (by simp)", [{"full_name": "MonoidHom.id", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [1001, 5], "def_end_pos": [1001, 17]}, {"full_name": "HNNExtension.hom_ext", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [107, 9], "def_end_pos": [107, 16]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b9 : Group H\nM : Type u_3\ninst\u271d : Monoid M\nmotive : HNNExtension G A B \u03c6 \u2192 Prop\nx : HNNExtension G A B \u03c6\nof : \u2200 (g : G), motive (\u2191HNNExtension.of g)\nt : motive HNNExtension.t\nmul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y)\ninv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9\nS : Subgroup (HNNExtension G A B \u03c6) :=\n  {\n    toSubmonoid :=\n      {\n        toSubsemigroup :=\n          { carrier := setOf motive,\n            mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) },\n        one_mem' :=\n          (_ :\n            1 \u2208\n              { carrier := setOf motive,\n                  mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) },\n    inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) }\nf : HNNExtension G A B \u03c6 \u2192* { x // x \u2208 S } :=\n  lift (MonoidHom.codRestrict HNNExtension.of S of) { val := HNNExtension.t, property := t }\n    (_ :\n      \u2200 (a : { x // x \u2208 A }),\n        { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a =\n          \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t })\n\u22a2 motive x", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b9 : Group H\nM : Type u_3\ninst\u271d : Monoid M\nmotive : HNNExtension G A B \u03c6 \u2192 Prop\nx : HNNExtension G A B \u03c6\nof : \u2200 (g : G), motive (\u2191HNNExtension.of g)\nt : motive HNNExtension.t\nmul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y)\ninv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9\nS : Subgroup (HNNExtension G A B \u03c6) :=\n  {\n    toSubmonoid :=\n      {\n        toSubsemigroup :=\n          { carrier := setOf motive,\n            mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) },\n        one_mem' :=\n          (_ :\n            1 \u2208\n              { carrier := setOf motive,\n                  mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) },\n    inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) }\nf : HNNExtension G A B \u03c6 \u2192* { x // x \u2208 S } :=\n  lift (MonoidHom.codRestrict HNNExtension.of S of) { val := HNNExtension.t, property := t }\n    (_ :\n      \u2200 (a : { x // x \u2208 A }),\n        { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a =\n          \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t })\nhf : MonoidHom.comp (Subgroup.subtype S) f = MonoidHom.id (HNNExtension G A B \u03c6)\n\u22a2 motive x"}, {"tactic": "show motive (MonoidHom.id _ x)", "annotated_tactic": ["show motive (<a>MonoidHom.id</a> _ x)", [{"full_name": "MonoidHom.id", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [1001, 5], "def_end_pos": [1001, 17]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b9 : Group H\nM : Type u_3\ninst\u271d : Monoid M\nmotive : HNNExtension G A B \u03c6 \u2192 Prop\nx : HNNExtension G A B \u03c6\nof : \u2200 (g : G), motive (\u2191HNNExtension.of g)\nt : motive HNNExtension.t\nmul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y)\ninv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9\nS : Subgroup (HNNExtension G A B \u03c6) :=\n  {\n    toSubmonoid :=\n      {\n        toSubsemigroup :=\n          { carrier := setOf motive,\n            mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) },\n        one_mem' :=\n          (_ :\n            1 \u2208\n              { carrier := setOf motive,\n                  mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) },\n    inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) }\nf : HNNExtension G A B \u03c6 \u2192* { x // x \u2208 S } :=\n  lift (MonoidHom.codRestrict HNNExtension.of S of) { val := HNNExtension.t, property := t }\n    (_ :\n      \u2200 (a : { x // x \u2208 A }),\n        { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a =\n          \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t })\nhf : MonoidHom.comp (Subgroup.subtype S) f = MonoidHom.id (HNNExtension G A B \u03c6)\n\u22a2 motive x", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b9 : Group H\nM : Type u_3\ninst\u271d : Monoid M\nmotive : HNNExtension G A B \u03c6 \u2192 Prop\nx : HNNExtension G A B \u03c6\nof : \u2200 (g : G), motive (\u2191HNNExtension.of g)\nt : motive HNNExtension.t\nmul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y)\ninv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9\nS : Subgroup (HNNExtension G A B \u03c6) :=\n  {\n    toSubmonoid :=\n      {\n        toSubsemigroup :=\n          { carrier := setOf motive,\n            mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) },\n        one_mem' :=\n          (_ :\n            1 \u2208\n              { carrier := setOf motive,\n                  mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) },\n    inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) }\nf : HNNExtension G A B \u03c6 \u2192* { x // x \u2208 S } :=\n  lift (MonoidHom.codRestrict HNNExtension.of S of) { val := HNNExtension.t, property := t }\n    (_ :\n      \u2200 (a : { x // x \u2208 A }),\n        { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a =\n          \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t })\nhf : MonoidHom.comp (Subgroup.subtype S) f = MonoidHom.id (HNNExtension G A B \u03c6)\n\u22a2 motive (\u2191(MonoidHom.id (HNNExtension G A B \u03c6)) x)"}, {"tactic": "rw [\u2190 hf]", "annotated_tactic": ["rw [\u2190 hf]", []], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b9 : Group H\nM : Type u_3\ninst\u271d : Monoid M\nmotive : HNNExtension G A B \u03c6 \u2192 Prop\nx : HNNExtension G A B \u03c6\nof : \u2200 (g : G), motive (\u2191HNNExtension.of g)\nt : motive HNNExtension.t\nmul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y)\ninv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9\nS : Subgroup (HNNExtension G A B \u03c6) :=\n  {\n    toSubmonoid :=\n      {\n        toSubsemigroup :=\n          { carrier := setOf motive,\n            mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) },\n        one_mem' :=\n          (_ :\n            1 \u2208\n              { carrier := setOf motive,\n                  mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) },\n    inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) }\nf : HNNExtension G A B \u03c6 \u2192* { x // x \u2208 S } :=\n  lift (MonoidHom.codRestrict HNNExtension.of S of) { val := HNNExtension.t, property := t }\n    (_ :\n      \u2200 (a : { x // x \u2208 A }),\n        { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a =\n          \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t })\nhf : MonoidHom.comp (Subgroup.subtype S) f = MonoidHom.id (HNNExtension G A B \u03c6)\n\u22a2 motive (\u2191(MonoidHom.id (HNNExtension G A B \u03c6)) x)", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b9 : Group H\nM : Type u_3\ninst\u271d : Monoid M\nmotive : HNNExtension G A B \u03c6 \u2192 Prop\nx : HNNExtension G A B \u03c6\nof : \u2200 (g : G), motive (\u2191HNNExtension.of g)\nt : motive HNNExtension.t\nmul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y)\ninv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9\nS : Subgroup (HNNExtension G A B \u03c6) :=\n  {\n    toSubmonoid :=\n      {\n        toSubsemigroup :=\n          { carrier := setOf motive,\n            mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) },\n        one_mem' :=\n          (_ :\n            1 \u2208\n              { carrier := setOf motive,\n                  mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) },\n    inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) }\nf : HNNExtension G A B \u03c6 \u2192* { x // x \u2208 S } :=\n  lift (MonoidHom.codRestrict HNNExtension.of S of) { val := HNNExtension.t, property := t }\n    (_ :\n      \u2200 (a : { x // x \u2208 A }),\n        { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a =\n          \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t })\nhf : MonoidHom.comp (Subgroup.subtype S) f = MonoidHom.id (HNNExtension G A B \u03c6)\n\u22a2 motive (\u2191(MonoidHom.comp (Subgroup.subtype S) f) x)"}, {"tactic": "exact (f x).2", "annotated_tactic": ["exact (f x).2", []], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b9 : Group H\nM : Type u_3\ninst\u271d : Monoid M\nmotive : HNNExtension G A B \u03c6 \u2192 Prop\nx : HNNExtension G A B \u03c6\nof : \u2200 (g : G), motive (\u2191HNNExtension.of g)\nt : motive HNNExtension.t\nmul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y)\ninv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9\nS : Subgroup (HNNExtension G A B \u03c6) :=\n  {\n    toSubmonoid :=\n      {\n        toSubsemigroup :=\n          { carrier := setOf motive,\n            mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) },\n        one_mem' :=\n          (_ :\n            1 \u2208\n              { carrier := setOf motive,\n                  mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) },\n    inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) }\nf : HNNExtension G A B \u03c6 \u2192* { x // x \u2208 S } :=\n  lift (MonoidHom.codRestrict HNNExtension.of S of) { val := HNNExtension.t, property := t }\n    (_ :\n      \u2200 (a : { x // x \u2208 A }),\n        { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a =\n          \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t })\nhf : MonoidHom.comp (Subgroup.subtype S) f = MonoidHom.id (HNNExtension G A B \u03c6)\n\u22a2 motive (\u2191(MonoidHom.comp (Subgroup.subtype S) f) x)", "state_after": "no goals"}, {"tactic": "simpa using of 1", "annotated_tactic": ["simpa using of 1", []], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b9 : Group H\nM : Type u_3\ninst\u271d : Monoid M\nmotive : HNNExtension G A B \u03c6 \u2192 Prop\nx : HNNExtension G A B \u03c6\nof : \u2200 (g : G), motive (\u2191HNNExtension.of g)\nt : motive HNNExtension.t\nmul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y)\ninv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9\n\u22a2 1 \u2208\n    { carrier := setOf motive,\n        mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier", "state_after": "no goals"}, {"tactic": "intro a", "annotated_tactic": ["intro a", []], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b9 : Group H\nM : Type u_3\ninst\u271d : Monoid M\nmotive : HNNExtension G A B \u03c6 \u2192 Prop\nx : HNNExtension G A B \u03c6\nof : \u2200 (g : G), motive (\u2191HNNExtension.of g)\nt : motive HNNExtension.t\nmul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y)\ninv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9\nS : Subgroup (HNNExtension G A B \u03c6) :=\n  {\n    toSubmonoid :=\n      {\n        toSubsemigroup :=\n          { carrier := setOf motive,\n            mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) },\n        one_mem' :=\n          (_ :\n            1 \u2208\n              { carrier := setOf motive,\n                  mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) },\n    inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) }\n\u22a2 \u2200 (a : { x // x \u2208 A }),\n    { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a =\n      \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t }", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b9 : Group H\nM : Type u_3\ninst\u271d : Monoid M\nmotive : HNNExtension G A B \u03c6 \u2192 Prop\nx : HNNExtension G A B \u03c6\nof : \u2200 (g : G), motive (\u2191HNNExtension.of g)\nt : motive HNNExtension.t\nmul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y)\ninv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9\nS : Subgroup (HNNExtension G A B \u03c6) :=\n  {\n    toSubmonoid :=\n      {\n        toSubsemigroup :=\n          { carrier := setOf motive,\n            mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) },\n        one_mem' :=\n          (_ :\n            1 \u2208\n              { carrier := setOf motive,\n                  mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) },\n    inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) }\na : { x // x \u2208 A }\n\u22a2 { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a =\n    \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t }"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b9 : Group H\nM : Type u_3\ninst\u271d : Monoid M\nmotive : HNNExtension G A B \u03c6 \u2192 Prop\nx : HNNExtension G A B \u03c6\nof : \u2200 (g : G), motive (\u2191HNNExtension.of g)\nt : motive HNNExtension.t\nmul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y)\ninv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9\nS : Subgroup (HNNExtension G A B \u03c6) :=\n  {\n    toSubmonoid :=\n      {\n        toSubsemigroup :=\n          { carrier := setOf motive,\n            mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) },\n        one_mem' :=\n          (_ :\n            1 \u2208\n              { carrier := setOf motive,\n                  mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) },\n    inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) }\na : { x // x \u2208 A }\n\u22a2 { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a =\n    \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t }", "state_after": "case a\nG : Type u_1\ninst\u271d\u00b2 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b9 : Group H\nM : Type u_3\ninst\u271d : Monoid M\nmotive : HNNExtension G A B \u03c6 \u2192 Prop\nx : HNNExtension G A B \u03c6\nof : \u2200 (g : G), motive (\u2191HNNExtension.of g)\nt : motive HNNExtension.t\nmul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y)\ninv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9\nS : Subgroup (HNNExtension G A B \u03c6) :=\n  {\n    toSubmonoid :=\n      {\n        toSubsemigroup :=\n          { carrier := setOf motive,\n            mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) },\n        one_mem' :=\n          (_ :\n            1 \u2208\n              { carrier := setOf motive,\n                  mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) },\n    inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) }\na : { x // x \u2208 A }\n\u22a2 \u2191({ val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a) =\n    \u2191(\u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t })"}, {"tactic": "simp [equiv_eq_conj, mul_assoc]", "annotated_tactic": ["simp [<a>equiv_eq_conj</a>, <a>mul_assoc</a>]", [{"full_name": "HNNExtension.equiv_eq_conj", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [73, 9], "def_end_pos": [73, 22]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "case a\nG : Type u_1\ninst\u271d\u00b2 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b9 : Group H\nM : Type u_3\ninst\u271d : Monoid M\nmotive : HNNExtension G A B \u03c6 \u2192 Prop\nx : HNNExtension G A B \u03c6\nof : \u2200 (g : G), motive (\u2191HNNExtension.of g)\nt : motive HNNExtension.t\nmul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y)\ninv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9\nS : Subgroup (HNNExtension G A B \u03c6) :=\n  {\n    toSubmonoid :=\n      {\n        toSubsemigroup :=\n          { carrier := setOf motive,\n            mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) },\n        one_mem' :=\n          (_ :\n            1 \u2208\n              { carrier := setOf motive,\n                  mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) },\n    inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) }\na : { x // x \u2208 A }\n\u22a2 \u2191({ val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a) =\n    \u2191(\u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t })", "state_after": "no goals"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b9 : Group H\nM : Type u_3\ninst\u271d : Monoid M\nmotive : HNNExtension G A B \u03c6 \u2192 Prop\nx : HNNExtension G A B \u03c6\nof : \u2200 (g : G), motive (\u2191HNNExtension.of g)\nt : motive HNNExtension.t\nmul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y)\ninv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9\nS : Subgroup (HNNExtension G A B \u03c6) :=\n  {\n    toSubmonoid :=\n      {\n        toSubsemigroup :=\n          { carrier := setOf motive,\n            mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) },\n        one_mem' :=\n          (_ :\n            1 \u2208\n              { carrier := setOf motive,\n                  mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) },\n    inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) }\nf : HNNExtension G A B \u03c6 \u2192* { x // x \u2208 S } :=\n  lift (MonoidHom.codRestrict HNNExtension.of S of) { val := HNNExtension.t, property := t }\n    (_ :\n      \u2200 (a : { x // x \u2208 A }),\n        { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a =\n          \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t })\n\u22a2 MonoidHom.comp (MonoidHom.comp (Subgroup.subtype S) f) HNNExtension.of =\n    MonoidHom.comp (MonoidHom.id (HNNExtension G A B \u03c6)) HNNExtension.of", "state_after": "case h\nG : Type u_1\ninst\u271d\u00b2 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b9 : Group H\nM : Type u_3\ninst\u271d : Monoid M\nmotive : HNNExtension G A B \u03c6 \u2192 Prop\nx : HNNExtension G A B \u03c6\nof : \u2200 (g : G), motive (\u2191HNNExtension.of g)\nt : motive HNNExtension.t\nmul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y)\ninv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9\nS : Subgroup (HNNExtension G A B \u03c6) :=\n  {\n    toSubmonoid :=\n      {\n        toSubsemigroup :=\n          { carrier := setOf motive,\n            mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) },\n        one_mem' :=\n          (_ :\n            1 \u2208\n              { carrier := setOf motive,\n                  mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) },\n    inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) }\nf : HNNExtension G A B \u03c6 \u2192* { x // x \u2208 S } :=\n  lift (MonoidHom.codRestrict HNNExtension.of S of) { val := HNNExtension.t, property := t }\n    (_ :\n      \u2200 (a : { x // x \u2208 A }),\n        { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a =\n          \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t })\nx\u271d : G\n\u22a2 \u2191(MonoidHom.comp (MonoidHom.comp (Subgroup.subtype S) f) HNNExtension.of) x\u271d =\n    \u2191(MonoidHom.comp (MonoidHom.id (HNNExtension G A B \u03c6)) HNNExtension.of) x\u271d"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\nG : Type u_1\ninst\u271d\u00b2 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b9 : Group H\nM : Type u_3\ninst\u271d : Monoid M\nmotive : HNNExtension G A B \u03c6 \u2192 Prop\nx : HNNExtension G A B \u03c6\nof : \u2200 (g : G), motive (\u2191HNNExtension.of g)\nt : motive HNNExtension.t\nmul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y)\ninv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9\nS : Subgroup (HNNExtension G A B \u03c6) :=\n  {\n    toSubmonoid :=\n      {\n        toSubsemigroup :=\n          { carrier := setOf motive,\n            mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) },\n        one_mem' :=\n          (_ :\n            1 \u2208\n              { carrier := setOf motive,\n                  mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) },\n    inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) }\nf : HNNExtension G A B \u03c6 \u2192* { x // x \u2208 S } :=\n  lift (MonoidHom.codRestrict HNNExtension.of S of) { val := HNNExtension.t, property := t }\n    (_ :\n      \u2200 (a : { x // x \u2208 A }),\n        { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a =\n          \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t })\nx\u271d : G\n\u22a2 \u2191(MonoidHom.comp (MonoidHom.comp (Subgroup.subtype S) f) HNNExtension.of) x\u271d =\n    \u2191(MonoidHom.comp (MonoidHom.id (HNNExtension G A B \u03c6)) HNNExtension.of) x\u271d", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\nA B : Subgroup G\n\u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B }\nH : Type u_2\ninst\u271d\u00b9 : Group H\nM : Type u_3\ninst\u271d : Monoid M\nmotive : HNNExtension G A B \u03c6 \u2192 Prop\nx : HNNExtension G A B \u03c6\nof : \u2200 (g : G), motive (\u2191HNNExtension.of g)\nt : motive HNNExtension.t\nmul : \u2200 (x y : HNNExtension G A B \u03c6), motive x \u2192 motive y \u2192 motive (x * y)\ninv : \u2200 (x : HNNExtension G A B \u03c6), motive x \u2192 motive x\u207b\u00b9\nS : Subgroup (HNNExtension G A B \u03c6) :=\n  {\n    toSubmonoid :=\n      {\n        toSubsemigroup :=\n          { carrier := setOf motive,\n            mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) },\n        one_mem' :=\n          (_ :\n            1 \u2208\n              { carrier := setOf motive,\n                  mul_mem' := (_ : \u2200 {a b : HNNExtension G A B \u03c6}, motive a \u2192 motive b \u2192 motive (a * b)) }.carrier) },\n    inv_mem' := (_ : \u2200 {x : HNNExtension G A B \u03c6}, motive x \u2192 motive x\u207b\u00b9) }\nf : HNNExtension G A B \u03c6 \u2192* { x // x \u2208 S } :=\n  lift (MonoidHom.codRestrict HNNExtension.of S of) { val := HNNExtension.t, property := t }\n    (_ :\n      \u2200 (a : { x // x \u2208 A }),\n        { val := HNNExtension.t, property := t } * \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191a =\n          \u2191(MonoidHom.codRestrict HNNExtension.of S of) \u2191(\u2191\u03c6 a) * { val := HNNExtension.t, property := t })\n\u22a2 \u2191(MonoidHom.comp (Subgroup.subtype S) f) HNNExtension.t = \u2191(MonoidHom.id (HNNExtension G A B \u03c6)) HNNExtension.t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.image_iInter_subset", "start": [1614, 1], "end": [1615, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Cardinal/Finite.lean", "full_name": "Nat.card_of_isEmpty", "start": [97, 1], "end": [97, 64], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : IsEmpty \u03b1\n\u22a2 Nat.card \u03b1 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Connected/PathConnected.lean", "full_name": "pathComponent_symm", "start": [900, 1], "end": [901, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Int.floor_sub_int", "start": [795, 1], "end": [796, 70], "traced_tactics": [{"tactic": "rw [Int.cast_neg, sub_eq_add_neg]", "annotated_tactic": ["rw [<a>Int.cast_neg</a>, <a>sub_eq_add_neg</a>]", [{"full_name": "Int.cast_neg", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 17]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz\u271d : \u2124\na\u271d a : \u03b1\nz : \u2124\n\u22a2 \u230aa - \u2191z\u230b = \u230aa + \u2191(-z)\u230b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "full_name": "MeasureTheory.ProbabilityMeasure.toFiniteMeasure_embedding", "start": [265, 1], "end": [269, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Convex/Combination.lean", "full_name": "Finset.mem_convexHull", "start": [374, 1], "end": [376, 46], "traced_tactics": [{"tactic": "rw [Finset.convexHull_eq, Set.mem_setOf_eq]", "annotated_tactic": ["rw [<a>Finset.convexHull_eq</a>, <a>Set.mem_setOf_eq</a>]", [{"full_name": "Finset.convexHull_eq", "def_path": "Mathlib/Analysis/Convex/Combination.lean", "def_pos": [352, 9], "def_end_pos": [352, 29]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\n\u03b9 : Type u_5\n\u03b9' : Type u_6\n\u03b1 : Type u_7\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : LinearOrderedField R'\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns\u271d : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\ns : Finset E\nx : E\n\u22a2 x \u2208 \u2191(convexHull R) \u2191s \u2194 \u2203 w x_1 x_2, centerMass s w id = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Real/NNReal.lean", "full_name": "NNReal.coe_eq_one", "start": [217, 11], "end": [218, 39], "traced_tactics": [{"tactic": "rw [\u2190 NNReal.coe_one, NNReal.coe_eq]", "annotated_tactic": ["rw [\u2190 <a>NNReal.coe_one</a>, <a>NNReal.coe_eq</a>]", [{"full_name": "NNReal.coe_one", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [176, 19], "def_end_pos": [176, 26]}, {"full_name": "NNReal.coe_eq", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [166, 19], "def_end_pos": [166, 25]}]], "state_before": "r : \u211d\u22650\n\u22a2 \u2191r = 1 \u2194 r = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/Hindman.lean", "full_name": "Hindman.exists_FP_of_finite_cover", "start": [232, 1], "end": [237, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/BigOperators/Finprod.lean", "full_name": "mul_finprod_cond_ne", "start": [1122, 1], "end": [1135, 35], "traced_tactics": [{"tactic": "rw [finprod_eq_prod _ hf]", "annotated_tactic": ["rw [<a>finprod_eq_prod</a> _ hf]", [{"full_name": "finprod_eq_prod", "def_path": "Mathlib/Algebra/BigOperators/Finprod.lean", "def_pos": [433, 9], "def_end_pos": [433, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf g : \u03b1 \u2192 M\na\u271d b : \u03b1\ns t : Set \u03b1\na : \u03b1\nhf : Set.Finite (mulSupport f)\n\u22a2 f a * \u220f\u1da0 (i : \u03b1) (_ : i \u2260 a), f i = \u220f\u1da0 (i : \u03b1), f i", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf g : \u03b1 \u2192 M\na\u271d b : \u03b1\ns t : Set \u03b1\na : \u03b1\nhf : Set.Finite (mulSupport f)\n\u22a2 f a * \u220f\u1da0 (i : \u03b1) (_ : i \u2260 a), f i = \u220f i in Finite.toFinset hf, f i"}, {"tactic": "have h : \u2200 x : \u03b1, f x \u2260 1 \u2192 (x \u2260 a \u2194 x \u2208 hf.toFinset \\ {a}) := by\n  intro x hx\n  rw [Finset.mem_sdiff, Finset.mem_singleton, Finite.mem_toFinset, mem_mulSupport]\n  exact \u27e8fun h => And.intro hx h, fun h => h.2\u27e9", "annotated_tactic": ["have h : \u2200 x : \u03b1, f x \u2260 1 \u2192 (x \u2260 a \u2194 x \u2208 hf.toFinset \\ {a}) := by\n      intro x hx\n      rw [<a>Finset.mem_sdiff</a>, <a>Finset.mem_singleton</a>, <a>Finite.mem_toFinset</a>, <a>mem_mulSupport</a>]\n      exact \u27e8fun h => <a>And.intro</a> hx h, fun h => h.2\u27e9", [{"full_name": "Finset.mem_sdiff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2077, 9], "def_end_pos": [2077, 18]}, {"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 22]}, {"full_name": "Set.Finite.mem_toFinset", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [173, 19], "def_end_pos": [173, 31]}, {"full_name": "Function.mem_mulSupport", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [66, 9], "def_end_pos": [66, 23]}, {"full_name": "And.intro", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [501, 3], "def_end_pos": [501, 8]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf g : \u03b1 \u2192 M\na\u271d b : \u03b1\ns t : Set \u03b1\na : \u03b1\nhf : Set.Finite (mulSupport f)\n\u22a2 f a * \u220f\u1da0 (i : \u03b1) (_ : i \u2260 a), f i = \u220f i in Finite.toFinset hf, f i", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf g : \u03b1 \u2192 M\na\u271d b : \u03b1\ns t : Set \u03b1\na : \u03b1\nhf : Set.Finite (mulSupport f)\nh : \u2200 (x : \u03b1), f x \u2260 1 \u2192 (x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a})\n\u22a2 f a * \u220f\u1da0 (i : \u03b1) (_ : i \u2260 a), f i = \u220f i in Finite.toFinset hf, f i"}, {"tactic": "rw [finprod_cond_eq_prod_of_cond_iff f (fun hx => h _ hx), Finset.sdiff_singleton_eq_erase]", "annotated_tactic": ["rw [<a>finprod_cond_eq_prod_of_cond_iff</a> f (fun hx => h _ hx), <a>Finset.sdiff_singleton_eq_erase</a>]", [{"full_name": "finprod_cond_eq_prod_of_cond_iff", "def_path": "Mathlib/Algebra/BigOperators/Finprod.lean", "def_pos": [445, 9], "def_end_pos": [445, 41]}, {"full_name": "Finset.sdiff_singleton_eq_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2258, 9], "def_end_pos": [2258, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf g : \u03b1 \u2192 M\na\u271d b : \u03b1\ns t : Set \u03b1\na : \u03b1\nhf : Set.Finite (mulSupport f)\nh : \u2200 (x : \u03b1), f x \u2260 1 \u2192 (x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a})\n\u22a2 f a * \u220f\u1da0 (i : \u03b1) (_ : i \u2260 a), f i = \u220f i in Finite.toFinset hf, f i", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf g : \u03b1 \u2192 M\na\u271d b : \u03b1\ns t : Set \u03b1\na : \u03b1\nhf : Set.Finite (mulSupport f)\nh : \u2200 (x : \u03b1), f x \u2260 1 \u2192 (x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a})\n\u22a2 f a * \u220f i in Finset.erase (Finite.toFinset hf) a, f i = \u220f i in Finite.toFinset hf, f i"}, {"tactic": "by_cases ha : a \u2208 mulSupport f", "annotated_tactic": ["by_cases ha : a \u2208 <a>mulSupport</a> f", [{"full_name": "Function.mulSupport", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [43, 5], "def_end_pos": [43, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf g : \u03b1 \u2192 M\na\u271d b : \u03b1\ns t : Set \u03b1\na : \u03b1\nhf : Set.Finite (mulSupport f)\nh : \u2200 (x : \u03b1), f x \u2260 1 \u2192 (x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a})\n\u22a2 f a * \u220f i in Finset.erase (Finite.toFinset hf) a, f i = \u220f i in Finite.toFinset hf, f i", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf g : \u03b1 \u2192 M\na\u271d b : \u03b1\ns t : Set \u03b1\na : \u03b1\nhf : Set.Finite (mulSupport f)\nh : \u2200 (x : \u03b1), f x \u2260 1 \u2192 (x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a})\nha : a \u2208 mulSupport f\n\u22a2 f a * \u220f i in Finset.erase (Finite.toFinset hf) a, f i = \u220f i in Finite.toFinset hf, f i\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf g : \u03b1 \u2192 M\na\u271d b : \u03b1\ns t : Set \u03b1\na : \u03b1\nhf : Set.Finite (mulSupport f)\nh : \u2200 (x : \u03b1), f x \u2260 1 \u2192 (x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a})\nha : \u00aca \u2208 mulSupport f\n\u22a2 f a * \u220f i in Finset.erase (Finite.toFinset hf) a, f i = \u220f i in Finite.toFinset hf, f i"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf g : \u03b1 \u2192 M\na\u271d b : \u03b1\ns t : Set \u03b1\na : \u03b1\nhf : Set.Finite (mulSupport f)\n\u22a2 \u2200 (x : \u03b1), f x \u2260 1 \u2192 (x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a})", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf g : \u03b1 \u2192 M\na\u271d b : \u03b1\ns t : Set \u03b1\na : \u03b1\nhf : Set.Finite (mulSupport f)\nx : \u03b1\nhx : f x \u2260 1\n\u22a2 x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a}"}, {"tactic": "rw [Finset.mem_sdiff, Finset.mem_singleton, Finite.mem_toFinset, mem_mulSupport]", "annotated_tactic": ["rw [<a>Finset.mem_sdiff</a>, <a>Finset.mem_singleton</a>, <a>Finite.mem_toFinset</a>, <a>mem_mulSupport</a>]", [{"full_name": "Finset.mem_sdiff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2077, 9], "def_end_pos": [2077, 18]}, {"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 22]}, {"full_name": "Set.Finite.mem_toFinset", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [173, 19], "def_end_pos": [173, 31]}, {"full_name": "Function.mem_mulSupport", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [66, 9], "def_end_pos": [66, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf g : \u03b1 \u2192 M\na\u271d b : \u03b1\ns t : Set \u03b1\na : \u03b1\nhf : Set.Finite (mulSupport f)\nx : \u03b1\nhx : f x \u2260 1\n\u22a2 x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf g : \u03b1 \u2192 M\na\u271d b : \u03b1\ns t : Set \u03b1\na : \u03b1\nhf : Set.Finite (mulSupport f)\nx : \u03b1\nhx : f x \u2260 1\n\u22a2 x \u2260 a \u2194 f x \u2260 1 \u2227 \u00acx = a"}, {"tactic": "exact \u27e8fun h => And.intro hx h, fun h => h.2\u27e9", "annotated_tactic": ["exact \u27e8fun h => <a>And.intro</a> hx h, fun h => h.2\u27e9", [{"full_name": "And.intro", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [501, 3], "def_end_pos": [501, 8]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf g : \u03b1 \u2192 M\na\u271d b : \u03b1\ns t : Set \u03b1\na : \u03b1\nhf : Set.Finite (mulSupport f)\nx : \u03b1\nhx : f x \u2260 1\n\u22a2 x \u2260 a \u2194 f x \u2260 1 \u2227 \u00acx = a", "state_after": "no goals"}, {"tactic": "apply Finset.mul_prod_erase _ _ ((Finite.mem_toFinset _).mpr ha)", "annotated_tactic": ["apply <a>Finset.mul_prod_erase</a> _ _ ((<a>Finite.mem_toFinset</a> _).<a>mpr</a> ha)", [{"full_name": "Finset.mul_prod_erase", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1663, 9], "def_end_pos": [1663, 23]}, {"full_name": "Set.Finite.mem_toFinset", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [173, 19], "def_end_pos": [173, 31]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf g : \u03b1 \u2192 M\na\u271d b : \u03b1\ns t : Set \u03b1\na : \u03b1\nhf : Set.Finite (mulSupport f)\nh : \u2200 (x : \u03b1), f x \u2260 1 \u2192 (x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a})\nha : a \u2208 mulSupport f\n\u22a2 f a * \u220f i in Finset.erase (Finite.toFinset hf) a, f i = \u220f i in Finite.toFinset hf, f i", "state_after": "no goals"}, {"tactic": "rw [mem_mulSupport, not_not] at ha", "annotated_tactic": ["rw [<a>mem_mulSupport</a>, <a>not_not</a>] at ha", [{"full_name": "Function.mem_mulSupport", "def_path": "Mathlib/Algebra/Support.lean", "def_pos": [66, 9], "def_end_pos": [66, 23]}, {"full_name": "Classical.not_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [683, 24], "def_end_pos": [683, 31]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf g : \u03b1 \u2192 M\na\u271d b : \u03b1\ns t : Set \u03b1\na : \u03b1\nhf : Set.Finite (mulSupport f)\nh : \u2200 (x : \u03b1), f x \u2260 1 \u2192 (x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a})\nha : \u00aca \u2208 mulSupport f\n\u22a2 f a * \u220f i in Finset.erase (Finite.toFinset hf) a, f i = \u220f i in Finite.toFinset hf, f i", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf g : \u03b1 \u2192 M\na\u271d b : \u03b1\ns t : Set \u03b1\na : \u03b1\nhf : Set.Finite (mulSupport f)\nh : \u2200 (x : \u03b1), f x \u2260 1 \u2192 (x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a})\nha : f a = 1\n\u22a2 f a * \u220f i in Finset.erase (Finite.toFinset hf) a, f i = \u220f i in Finite.toFinset hf, f i"}, {"tactic": "rw [ha, one_mul]", "annotated_tactic": ["rw [ha, <a>one_mul</a>]", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf g : \u03b1 \u2192 M\na\u271d b : \u03b1\ns t : Set \u03b1\na : \u03b1\nhf : Set.Finite (mulSupport f)\nh : \u2200 (x : \u03b1), f x \u2260 1 \u2192 (x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a})\nha : f a = 1\n\u22a2 f a * \u220f i in Finset.erase (Finite.toFinset hf) a, f i = \u220f i in Finite.toFinset hf, f i", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf g : \u03b1 \u2192 M\na\u271d b : \u03b1\ns t : Set \u03b1\na : \u03b1\nhf : Set.Finite (mulSupport f)\nh : \u2200 (x : \u03b1), f x \u2260 1 \u2192 (x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a})\nha : f a = 1\n\u22a2 \u220f i in Finset.erase (Finite.toFinset hf) a, f i = \u220f i in Finite.toFinset hf, f i"}, {"tactic": "apply Finset.prod_erase _ ha", "annotated_tactic": ["apply <a>Finset.prod_erase</a> _ ha", [{"full_name": "Finset.prod_erase", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1680, 9], "def_end_pos": [1680, 19]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : CommMonoid N\nf g : \u03b1 \u2192 M\na\u271d b : \u03b1\ns t : Set \u03b1\na : \u03b1\nhf : Set.Finite (mulSupport f)\nh : \u2200 (x : \u03b1), f x \u2260 1 \u2192 (x \u2260 a \u2194 x \u2208 Finite.toFinset hf \\ {a})\nha : f a = 1\n\u22a2 \u220f i in Finset.erase (Finite.toFinset hf) a, f i = \u220f i in Finite.toFinset hf, f i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Monoid.lean", "full_name": "Set.Ioi_add_bij", "start": [36, 1], "end": [42, 43], "traced_tactics": [{"tactic": "refine'\n  \u27e8fun x h => add_lt_add_right (mem_Ioi.mp h) _, fun _ _ _ _ h => add_right_cancel h, fun _ h =>\n    _\u27e9", "annotated_tactic": ["refine'\n    \u27e8fun x h => <a>add_lt_add_right</a> (mem_Ioi.mp h) _, fun _ _ _ _ h => <a>add_right_cancel</a> h, fun _ h =>\n      _\u27e9", [{"full_name": "add_lt_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [135, 15], "def_end_pos": [135, 31]}, {"full_name": "add_right_cancel", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [206, 3], "def_end_pos": [206, 14]}]], "state_before": "M : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 BijOn (fun x => x + d) (Ioi a) (Ioi (a + d))", "state_after": "M : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d x\u271d : M\nh : x\u271d \u2208 Ioi (a + d)\n\u22a2 x\u271d \u2208 (fun x => x + d) '' Ioi a"}, {"tactic": "obtain \u27e8c, rfl\u27e9 := exists_add_of_le (mem_Ioi.mp h).le", "annotated_tactic": ["obtain \u27e8c, rfl\u27e9 := <a>exists_add_of_le</a> (mem_Ioi.mp h).<a>le</a>", [{"full_name": "ExistsAddOfLE.exists_add_of_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 19]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "M : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d x\u271d : M\nh : x\u271d \u2208 Ioi (a + d)\n\u22a2 x\u271d \u2208 (fun x => x + d) '' Ioi a", "state_after": "case intro\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c\u271d d c : M\nh : a + d + c \u2208 Ioi (a + d)\n\u22a2 a + d + c \u2208 (fun x => x + d) '' Ioi a"}, {"tactic": "rw [mem_Ioi, add_right_comm, add_lt_add_iff_right] at h", "annotated_tactic": ["rw [<a>mem_Ioi</a>, <a>add_right_comm</a>, <a>add_lt_add_iff_right</a>] at h", [{"full_name": "Set.mem_Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 16]}, {"full_name": "add_right_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [101, 3], "def_end_pos": [101, 14]}, {"full_name": "add_lt_add_iff_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [112, 3], "def_end_pos": [112, 14]}]], "state_before": "case intro\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c\u271d d c : M\nh : a + d + c \u2208 Ioi (a + d)\n\u22a2 a + d + c \u2208 (fun x => x + d) '' Ioi a", "state_after": "case intro\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c\u271d d c : M\nh : a < a + c\n\u22a2 a + d + c \u2208 (fun x => x + d) '' Ioi a"}, {"tactic": "exact \u27e8a + c, h, by rw [add_right_comm]\u27e9", "annotated_tactic": ["exact \u27e8a + c, h, by rw [<a>add_right_comm</a>]\u27e9", [{"full_name": "add_right_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [101, 3], "def_end_pos": [101, 14]}]], "state_before": "case intro\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c\u271d d c : M\nh : a < a + c\n\u22a2 a + d + c \u2208 (fun x => x + d) '' Ioi a", "state_after": "no goals"}, {"tactic": "rw [add_right_comm]", "annotated_tactic": ["rw [<a>add_right_comm</a>]", [{"full_name": "add_right_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [101, 3], "def_end_pos": [101, 14]}]], "state_before": "M : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c\u271d d c : M\nh : a < a + c\n\u22a2 (fun x => x + d) (a + c) = a + d + c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean", "full_name": "CategoryTheory.tensorLeftHomEquiv_symm_naturality", "start": [378, 1], "end": [383, 45], "traced_tactics": [{"tactic": "dsimp [tensorLeftHomEquiv]", "annotated_tactic": ["dsimp [<a>tensorLeftHomEquiv</a>]", [{"full_name": "CategoryTheory.tensorLeftHomEquiv", "def_path": "Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean", "def_pos": [301, 5], "def_end_pos": [301, 23]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\nX X' Y Y' Z : C\ninst\u271d : ExactPairing Y Y'\nf : X \u27f6 X'\ng : X' \u27f6 Y \u2297 Z\n\u22a2 \u2191(tensorLeftHomEquiv X Y Y' Z).symm (f \u226b g) = (\ud835\udfd9 Y' \u2297 f) \u226b \u2191(tensorLeftHomEquiv X' Y Y' Z).symm g", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\nX X' Y Y' Z : C\ninst\u271d : ExactPairing Y Y'\nf : X \u27f6 X'\ng : X' \u27f6 Y \u2297 Z\n\u22a2 (\ud835\udfd9 Y' \u2297 f \u226b g) \u226b (\u03b1_ Y' Y Z).inv \u226b (\u03b5_ Y Y' \u2297 \ud835\udfd9 Z) \u226b (\u03bb_ Z).hom =\n    (\ud835\udfd9 Y' \u2297 f) \u226b (\ud835\udfd9 Y' \u2297 g) \u226b (\u03b1_ Y' Y Z).inv \u226b (\u03b5_ Y Y' \u2297 \ud835\udfd9 Z) \u226b (\u03bb_ Z).hom"}, {"tactic": "simp only [id_tensor_comp, Category.assoc]", "annotated_tactic": ["simp only [<a>id_tensor_comp</a>, <a>Category.assoc</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.id_tensor_comp", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [270, 9], "def_end_pos": [270, 23]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [159, 3], "def_end_pos": [159, 8]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\nX X' Y Y' Z : C\ninst\u271d : ExactPairing Y Y'\nf : X \u27f6 X'\ng : X' \u27f6 Y \u2297 Z\n\u22a2 (\ud835\udfd9 Y' \u2297 f \u226b g) \u226b (\u03b1_ Y' Y Z).inv \u226b (\u03b5_ Y Y' \u2297 \ud835\udfd9 Z) \u226b (\u03bb_ Z).hom =\n    (\ud835\udfd9 Y' \u2297 f) \u226b (\ud835\udfd9 Y' \u2297 g) \u226b (\u03b1_ Y' Y Z).inv \u226b (\u03b5_ Y Y' \u2297 \ud835\udfd9 Z) \u226b (\u03bb_ Z).hom", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/Quiver/Covering.lean", "full_name": "Prefunctor.pathStar_apply", "start": [212, 1], "end": [214, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "full_name": "Set.Nonempty.of_mul_left", "start": [385, 1], "end": [386, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finsupp/Defs.lean", "full_name": "Finsupp.single_apply_eq_zero", "start": [381, 1], "end": [382, 33], "traced_tactics": [{"tactic": "simp [single_eq_set_indicator]", "annotated_tactic": ["simp [<a>single_eq_set_indicator</a>]", [{"full_name": "Finsupp.single_eq_set_indicator", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [317, 9], "def_end_pos": [317, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\na\u271d a' : \u03b1\nb\u271d : M\na x : \u03b1\nb : M\n\u22a2 \u2191(single a b) x = 0 \u2194 x = a \u2192 b = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "full_name": "Nat.bit0_lt_bit1", "start": [331, 11], "end": [332, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "full_name": "PowerSeries.coeff_zero_eq_constantCoeff_apply", "start": [1439, 1], "end": [1440, 38], "traced_tactics": [{"tactic": "rw [coeff_zero_eq_constantCoeff]", "annotated_tactic": ["rw [<a>coeff_zero_eq_constantCoeff</a>]", [{"full_name": "PowerSeries.coeff_zero_eq_constantCoeff", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [1434, 9], "def_end_pos": [1434, 36]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\n\u03c6 : R\u27e6X\u27e7\n\u22a2 \u2191(coeff R 0) \u03c6 = \u2191(constantCoeff R) \u03c6", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Submonoid/Inverses.lean", "full_name": "Submonoid.mul_leftInvEquiv_symm", "start": [213, 1], "end": [215, 7], "traced_tactics": [{"tactic": "convert S.leftInvEquiv_mul hS ((S.leftInvEquiv hS).symm x)", "annotated_tactic": ["convert S.leftInvEquiv_mul hS ((S.leftInvEquiv hS).<a>symm</a> x)", [{"full_name": "MulEquiv.symm", "def_path": "Mathlib/Algebra/Hom/Equiv/Basic.lean", "def_pos": [294, 5], "def_end_pos": [294, 9]}]], "state_before": "M : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nx : { x // x \u2208 S }\n\u22a2 \u2191x * \u2191(\u2191(MulEquiv.symm (leftInvEquiv S hS)) x) = 1", "state_after": "case h.e'_2.h.e'_5.h.e'_3\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nx : { x // x \u2208 S }\n\u22a2 x = \u2191(leftInvEquiv S hS) (\u2191(MulEquiv.symm (leftInvEquiv S hS)) x)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_2.h.e'_5.h.e'_3\nM : Type u_1\ninst\u271d : CommMonoid M\nS : Submonoid M\nhS : S \u2264 IsUnit.submonoid M\nx : { x // x \u2208 S }\n\u22a2 x = \u2191(leftInvEquiv S hS) (\u2191(MulEquiv.symm (leftInvEquiv S hS)) x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Real/Hyperreal.lean", "full_name": "Hyperreal.InfinitePos.pos", "start": [434, 1], "end": [434, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Euclidean/MongePoint.lean", "full_name": "Affine.Simplex.mongePoint_eq_of_range_eq", "start": [105, 1], "end": [108, 35], "traced_tactics": [{"tactic": "simp_rw [mongePoint_eq_smul_vsub_vadd_circumcenter, centroid_eq_of_range_eq h,\n  circumcenter_eq_of_range_eq h]", "annotated_tactic": ["simp_rw [<a>mongePoint_eq_smul_vsub_vadd_circumcenter</a>, <a>centroid_eq_of_range_eq</a> h,\n    <a>circumcenter_eq_of_range_eq</a> h]", [{"full_name": "Affine.Simplex.mongePoint_eq_smul_vsub_vadd_circumcenter", "def_path": "Mathlib/Geometry/Euclidean/MongePoint.lean", "def_pos": [89, 9], "def_end_pos": [89, 50]}, {"full_name": "Affine.Simplex.centroid_eq_of_range_eq", "def_path": "Mathlib/LinearAlgebra/AffineSpace/Independent.lean", "def_pos": [964, 9], "def_end_pos": [964, 32]}, {"full_name": "Affine.Simplex.circumcenter_eq_of_range_eq", "def_path": "Mathlib/Geometry/Euclidean/Circumcenter.lean", "def_pos": [501, 9], "def_end_pos": [501, 36]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ns\u2081 s\u2082 : Simplex \u211d P n\nh : Set.range s\u2081.points = Set.range s\u2082.points\n\u22a2 mongePoint s\u2081 = mongePoint s\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/Laurent.lean", "full_name": "LaurentPolynomial.commute_T", "start": [332, 1], "end": [336, 22], "traced_tactics": [{"tactic": "rw [T, T, \u2190 single_eq_C, single_mul_single, single_mul_single, single_mul_single]", "annotated_tactic": ["rw [<a>T</a>, <a>T</a>, \u2190 <a>single_eq_C</a>, <a>single_mul_single</a>, <a>single_mul_single</a>, <a>single_mul_single</a>]", [{"full_name": "LaurentPolynomial.T", "def_path": "Mathlib/Data/Polynomial/Laurent.lean", "def_pos": [173, 5], "def_end_pos": [173, 6]}, {"full_name": "LaurentPolynomial.T", "def_path": "Mathlib/Data/Polynomial/Laurent.lean", "def_pos": [173, 5], "def_end_pos": [173, 6]}, {"full_name": "LaurentPolynomial.single_eq_C", "def_path": "Mathlib/Data/Polynomial/Laurent.lean", "def_pos": [161, 9], "def_end_pos": [161, 20]}, {"full_name": "AddMonoidAlgebra.single_mul_single", "def_path": "Mathlib/Algebra/MonoidAlgebra/Basic.lean", "def_pos": [1553, 9], "def_end_pos": [1553, 26]}, {"full_name": "AddMonoidAlgebra.single_mul_single", "def_path": "Mathlib/Algebra/MonoidAlgebra/Basic.lean", "def_pos": [1553, 9], "def_end_pos": [1553, 26]}, {"full_name": "AddMonoidAlgebra.single_mul_single", "def_path": "Mathlib/Algebra/MonoidAlgebra/Basic.lean", "def_pos": [1553, 9], "def_end_pos": [1553, 26]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\nn : \u2124\nf : R[T;T\u207b\u00b9]\nm : \u2124\na : R\n\u22a2 T n * (\u2191C a * T m) = \u2191C a * T m * T n", "state_after": "R : Type u_1\ninst\u271d : Semiring R\nn : \u2124\nf : R[T;T\u207b\u00b9]\nm : \u2124\na : R\n\u22a2 AddMonoidAlgebra.single (n + (0 + m)) (1 * (a * 1)) = AddMonoidAlgebra.single (0 + m + n) (a * 1 * 1)"}, {"tactic": "simp [add_comm]", "annotated_tactic": ["simp [<a>add_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\nn : \u2124\nf : R[T;T\u207b\u00b9]\nm : \u2124\na : R\n\u22a2 AddMonoidAlgebra.single (n + (0 + m)) (1 * (a * 1)) = AddMonoidAlgebra.single (0 + m + n) (a * 1 * 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean", "full_name": "CliffordAlgebra.EquivEven.Q'_apply", "start": [56, 1], "end": [57, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Normed/Group/SemiNormedGroupCat/Kernels.lean", "full_name": "SemiNormedGroupCat.explicitCokernelDesc_zero", "start": [297, 1], "end": [299, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Hom/Set.lean", "full_name": "StrictMono.orderIsoOfSurjective_symm_apply_self", "start": [123, 1], "end": [125, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Hom/Basic.lean", "full_name": "OrderIso.toFun_eq_coe", "start": [790, 1], "end": [791, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Submonoid/Pointwise.lean", "full_name": "Submonoid.pow_smul_mem_closure_smul", "start": [99, 1], "end": [108, 24], "traced_tactics": [{"tactic": "refine' @closure_induction N _ s (fun x : N => \u2203 n : \u2115, r ^ n \u2022 x \u2208 closure (r \u2022 s)) _ hx _ _ _", "annotated_tactic": ["refine' @<a>closure_induction</a> N _ s (fun x : N => \u2203 n : \u2115, r ^ n \u2022 x \u2208 <a>closure</a> (r \u2022 s)) _ hx _ _ _", [{"full_name": "Submonoid.closure_induction", "def_path": "Mathlib/GroupTheory/Submonoid/Basic.lean", "def_pos": [446, 9], "def_end_pos": [446, 26]}, {"full_name": "Submonoid.closure", "def_path": "Mathlib/GroupTheory/Submonoid/Basic.lean", "def_pos": [386, 5], "def_end_pos": [386, 12]}]], "state_before": "\u03b1 : Type u_1\nG : Type u_2\nM : Type u_3\nR : Type u_4\nA : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : AddMonoid A\ns\u271d t u : Set M\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : MulAction M N\ninst\u271d : IsScalarTower M N N\nr : M\ns : Set N\nx : N\nhx : x \u2208 closure s\n\u22a2 \u2203 n, r ^ n \u2022 x \u2208 closure (r \u2022 s)", "state_after": "case refine'_1\n\u03b1 : Type u_1\nG : Type u_2\nM : Type u_3\nR : Type u_4\nA : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : AddMonoid A\ns\u271d t u : Set M\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : MulAction M N\ninst\u271d : IsScalarTower M N N\nr : M\ns : Set N\nx : N\nhx : x \u2208 closure s\n\u22a2 \u2200 (x : N), x \u2208 s \u2192 (fun x => \u2203 n, r ^ n \u2022 x \u2208 closure (r \u2022 s)) x\n\ncase refine'_2\n\u03b1 : Type u_1\nG : Type u_2\nM : Type u_3\nR : Type u_4\nA : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : AddMonoid A\ns\u271d t u : Set M\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : MulAction M N\ninst\u271d : IsScalarTower M N N\nr : M\ns : Set N\nx : N\nhx : x \u2208 closure s\n\u22a2 (fun x => \u2203 n, r ^ n \u2022 x \u2208 closure (r \u2022 s)) 1\n\ncase refine'_3\n\u03b1 : Type u_1\nG : Type u_2\nM : Type u_3\nR : Type u_4\nA : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : AddMonoid A\ns\u271d t u : Set M\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : MulAction M N\ninst\u271d : IsScalarTower M N N\nr : M\ns : Set N\nx : N\nhx : x \u2208 closure s\n\u22a2 \u2200 (x y : N),\n    (fun x => \u2203 n, r ^ n \u2022 x \u2208 closure (r \u2022 s)) x \u2192\n      (fun x => \u2203 n, r ^ n \u2022 x \u2208 closure (r \u2022 s)) y \u2192 (fun x => \u2203 n, r ^ n \u2022 x \u2208 closure (r \u2022 s)) (x * y)"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\nG : Type u_2\nM : Type u_3\nR : Type u_4\nA : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : AddMonoid A\ns\u271d t u : Set M\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : MulAction M N\ninst\u271d : IsScalarTower M N N\nr : M\ns : Set N\nx : N\nhx : x \u2208 closure s\n\u22a2 \u2200 (x : N), x \u2208 s \u2192 (fun x => \u2203 n, r ^ n \u2022 x \u2208 closure (r \u2022 s)) x", "state_after": "case refine'_1\n\u03b1 : Type u_1\nG : Type u_2\nM : Type u_3\nR : Type u_4\nA : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : AddMonoid A\ns\u271d t u : Set M\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : MulAction M N\ninst\u271d : IsScalarTower M N N\nr : M\ns : Set N\nx\u271d : N\nhx\u271d : x\u271d \u2208 closure s\nx : N\nhx : x \u2208 s\n\u22a2 \u2203 n, r ^ n \u2022 x \u2208 closure (r \u2022 s)"}, {"tactic": "exact \u27e81, subset_closure \u27e8_, hx, by rw [pow_one]\u27e9\u27e9", "annotated_tactic": ["exact \u27e81, <a>subset_closure</a> \u27e8_, hx, by rw [<a>pow_one</a>]\u27e9\u27e9", [{"full_name": "Submonoid.subset_closure", "def_path": "Mathlib/GroupTheory/Submonoid/Basic.lean", "def_pos": [400, 9], "def_end_pos": [400, 23]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nG : Type u_2\nM : Type u_3\nR : Type u_4\nA : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : AddMonoid A\ns\u271d t u : Set M\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : MulAction M N\ninst\u271d : IsScalarTower M N N\nr : M\ns : Set N\nx\u271d : N\nhx\u271d : x\u271d \u2208 closure s\nx : N\nhx : x \u2208 s\n\u22a2 \u2203 n, r ^ n \u2022 x \u2208 closure (r \u2022 s)", "state_after": "no goals"}, {"tactic": "rw [pow_one]", "annotated_tactic": ["rw [<a>pow_one</a>]", [{"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}]], "state_before": "\u03b1 : Type u_1\nG : Type u_2\nM : Type u_3\nR : Type u_4\nA : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : AddMonoid A\ns\u271d t u : Set M\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : MulAction M N\ninst\u271d : IsScalarTower M N N\nr : M\ns : Set N\nx\u271d : N\nhx\u271d : x\u271d \u2208 closure s\nx : N\nhx : x \u2208 s\n\u22a2 (fun x => r \u2022 x) x = r ^ 1 \u2022 x", "state_after": "no goals"}, {"tactic": "exact \u27e80, by simpa using one_mem _\u27e9", "annotated_tactic": ["exact \u27e80, by simpa using <a>one_mem</a> _\u27e9", [{"full_name": "OneMemClass.one_mem", "def_path": "Mathlib/GroupTheory/Submonoid/Basic.lean", "def_pos": [73, 3], "def_end_pos": [73, 10]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nG : Type u_2\nM : Type u_3\nR : Type u_4\nA : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : AddMonoid A\ns\u271d t u : Set M\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : MulAction M N\ninst\u271d : IsScalarTower M N N\nr : M\ns : Set N\nx : N\nhx : x \u2208 closure s\n\u22a2 (fun x => \u2203 n, r ^ n \u2022 x \u2208 closure (r \u2022 s)) 1", "state_after": "no goals"}, {"tactic": "simpa using one_mem _", "annotated_tactic": ["simpa using <a>one_mem</a> _", [{"full_name": "OneMemClass.one_mem", "def_path": "Mathlib/GroupTheory/Submonoid/Basic.lean", "def_pos": [73, 3], "def_end_pos": [73, 10]}]], "state_before": "\u03b1 : Type u_1\nG : Type u_2\nM : Type u_3\nR : Type u_4\nA : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : AddMonoid A\ns\u271d t u : Set M\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : MulAction M N\ninst\u271d : IsScalarTower M N N\nr : M\ns : Set N\nx : N\nhx : x \u2208 closure s\n\u22a2 r ^ 0 \u2022 1 \u2208 closure (r \u2022 s)", "state_after": "no goals"}, {"tactic": "rintro x y \u27e8nx, hx\u27e9 \u27e8ny, hy\u27e9", "annotated_tactic": ["rintro x y \u27e8nx, hx\u27e9 \u27e8ny, hy\u27e9", []], "state_before": "case refine'_3\n\u03b1 : Type u_1\nG : Type u_2\nM : Type u_3\nR : Type u_4\nA : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : AddMonoid A\ns\u271d t u : Set M\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : MulAction M N\ninst\u271d : IsScalarTower M N N\nr : M\ns : Set N\nx : N\nhx : x \u2208 closure s\n\u22a2 \u2200 (x y : N),\n    (fun x => \u2203 n, r ^ n \u2022 x \u2208 closure (r \u2022 s)) x \u2192\n      (fun x => \u2203 n, r ^ n \u2022 x \u2208 closure (r \u2022 s)) y \u2192 (fun x => \u2203 n, r ^ n \u2022 x \u2208 closure (r \u2022 s)) (x * y)", "state_after": "case refine'_3.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nM : Type u_3\nR : Type u_4\nA : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : AddMonoid A\ns\u271d t u : Set M\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : MulAction M N\ninst\u271d : IsScalarTower M N N\nr : M\ns : Set N\nx\u271d : N\nhx\u271d : x\u271d \u2208 closure s\nx y : N\nnx : \u2115\nhx : r ^ nx \u2022 x \u2208 closure (r \u2022 s)\nny : \u2115\nhy : r ^ ny \u2022 y \u2208 closure (r \u2022 s)\n\u22a2 \u2203 n, r ^ n \u2022 (x * y) \u2208 closure (r \u2022 s)"}, {"tactic": "use ny + nx", "annotated_tactic": ["use ny + nx", []], "state_before": "case refine'_3.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nM : Type u_3\nR : Type u_4\nA : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : AddMonoid A\ns\u271d t u : Set M\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : MulAction M N\ninst\u271d : IsScalarTower M N N\nr : M\ns : Set N\nx\u271d : N\nhx\u271d : x\u271d \u2208 closure s\nx y : N\nnx : \u2115\nhx : r ^ nx \u2022 x \u2208 closure (r \u2022 s)\nny : \u2115\nhy : r ^ ny \u2022 y \u2208 closure (r \u2022 s)\n\u22a2 \u2203 n, r ^ n \u2022 (x * y) \u2208 closure (r \u2022 s)", "state_after": "case h\n\u03b1 : Type u_1\nG : Type u_2\nM : Type u_3\nR : Type u_4\nA : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : AddMonoid A\ns\u271d t u : Set M\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : MulAction M N\ninst\u271d : IsScalarTower M N N\nr : M\ns : Set N\nx\u271d : N\nhx\u271d : x\u271d \u2208 closure s\nx y : N\nnx : \u2115\nhx : r ^ nx \u2022 x \u2208 closure (r \u2022 s)\nny : \u2115\nhy : r ^ ny \u2022 y \u2208 closure (r \u2022 s)\n\u22a2 r ^ (ny + nx) \u2022 (x * y) \u2208 closure (r \u2022 s)"}, {"tactic": "rw [pow_add, mul_smul, \u2190 smul_mul_assoc, mul_comm, \u2190 smul_mul_assoc]", "annotated_tactic": ["rw [<a>pow_add</a>, <a>mul_smul</a>, \u2190 <a>smul_mul_assoc</a>, <a>mul_comm</a>, \u2190 <a>smul_mul_assoc</a>]", [{"full_name": "pow_add", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 16]}, {"full_name": "MulAction.mul_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [112, 3], "def_end_pos": [112, 11]}, {"full_name": "smul_mul_assoc", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [429, 9], "def_end_pos": [429, 23]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "smul_mul_assoc", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [429, 9], "def_end_pos": [429, 23]}]], "state_before": "case h\n\u03b1 : Type u_1\nG : Type u_2\nM : Type u_3\nR : Type u_4\nA : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : AddMonoid A\ns\u271d t u : Set M\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : MulAction M N\ninst\u271d : IsScalarTower M N N\nr : M\ns : Set N\nx\u271d : N\nhx\u271d : x\u271d \u2208 closure s\nx y : N\nnx : \u2115\nhx : r ^ nx \u2022 x \u2208 closure (r \u2022 s)\nny : \u2115\nhy : r ^ ny \u2022 y \u2208 closure (r \u2022 s)\n\u22a2 r ^ (ny + nx) \u2022 (x * y) \u2208 closure (r \u2022 s)", "state_after": "case h\n\u03b1 : Type u_1\nG : Type u_2\nM : Type u_3\nR : Type u_4\nA : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : AddMonoid A\ns\u271d t u : Set M\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : MulAction M N\ninst\u271d : IsScalarTower M N N\nr : M\ns : Set N\nx\u271d : N\nhx\u271d : x\u271d \u2208 closure s\nx y : N\nnx : \u2115\nhx : r ^ nx \u2022 x \u2208 closure (r \u2022 s)\nny : \u2115\nhy : r ^ ny \u2022 y \u2208 closure (r \u2022 s)\n\u22a2 r ^ ny \u2022 y * r ^ nx \u2022 x \u2208 closure (r \u2022 s)"}, {"tactic": "exact mul_mem hy hx", "annotated_tactic": ["exact <a>mul_mem</a> hy hx", [{"full_name": "MulMemClass.mul_mem", "def_path": "Mathlib/GroupTheory/Subsemigroup/Basic.lean", "def_pos": [64, 3], "def_end_pos": [64, 10]}]], "state_before": "case h\n\u03b1 : Type u_1\nG : Type u_2\nM : Type u_3\nR : Type u_4\nA : Type u_5\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : AddMonoid A\ns\u271d t u : Set M\nN : Type u_6\ninst\u271d\u00b2 : CommMonoid N\ninst\u271d\u00b9 : MulAction M N\ninst\u271d : IsScalarTower M N N\nr : M\ns : Set N\nx\u271d : N\nhx\u271d : x\u271d \u2208 closure s\nx y : N\nnx : \u2115\nhx : r ^ nx \u2022 x \u2208 closure (r \u2022 s)\nny : \u2115\nhy : r ^ ny \u2022 y \u2208 closure (r \u2022 s)\n\u22a2 r ^ ny \u2022 y * r ^ nx \u2022 x \u2208 closure (r \u2022 s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.union_get_eq", "start": [856, 1], "end": [858, 29], "traced_tactics": [{"tactic": "simp [union_def]", "annotated_tactic": ["simp [<a>union_def</a>]", [{"full_name": "Part.union_def", "def_path": "Mathlib/Data/Part.lean", "def_pos": [708, 9], "def_end_pos": [708, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Union \u03b1\na b : Part \u03b1\nhab : (a \u222a b).Dom\n\u22a2 get (a \u222a b) hab = get a (_ : a.Dom) \u222a get b (_ : b.Dom)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Union \u03b1\na b : Part \u03b1\nhab : (a \u222a b).Dom\n\u22a2 get (Part.bind a fun y => map (fun x => y \u222a x) b) (_ : (Part.bind a fun y => map (fun x => y \u222a x) b).Dom) =\n    get a (_ : a.Dom) \u222a get b (_ : b.Dom)"}, {"tactic": "aesop", "annotated_tactic": ["aesop", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : Union \u03b1\na b : Part \u03b1\nhab : (a \u222a b).Dom\n\u22a2 get (Part.bind a fun y => map (fun x => y \u222a x) b) (_ : (Part.bind a fun y => map (fun x => y \u222a x) b).Dom) =\n    get a (_ : a.Dom) \u222a get b (_ : b.Dom)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Algebra/Bilinear.lean", "full_name": "LinearMap.mulRight_toAddMonoidHom", "start": [66, 1], "end": [67, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Hom/Aut.lean", "full_name": "MulAut.mul_apply", "start": [89, 1], "end": [90, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "full_name": "WithBot.le_of_add_le_add_right", "start": [718, 11], "end": [720, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.list_tail", "start": [1048, 1], "end": [1049, 85], "traced_tactics": [{"tactic": "cases l <;> rfl", "annotated_tactic": ["cases l <;> rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\ninst\u271d\u00b3 : Primcodable \u03b1\ninst\u271d\u00b2 : Primcodable \u03b2\ninst\u271d\u00b9 : Primcodable \u03b3\ninst\u271d : Primcodable \u03c3\nl : List \u03b1\n\u22a2 (List.casesOn (id l) [] fun b l_1 => (l, b, l_1).2.2) = List.tail l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.range_dcomp", "start": [768, 1], "end": [774, 63], "traced_tactics": [{"tactic": "refine Subset.antisymm ?_ fun x hx => ?_", "annotated_tactic": ["refine <a>Subset.antisymm</a> ?_ fun x hx => ?_", [{"full_name": "Set.Subset.antisymm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [370, 9], "def_end_pos": [370, 24]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\n\u22a2 (range fun g i => f i (g i)) = pi univ fun i => range (f i)", "state_after": "case refine_1\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\n\u22a2 (range fun g i => f i (g i)) \u2286 pi univ fun i => range (f i)\n\ncase refine_2\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\nx : (i : \u03b9) \u2192 \u03b2 i\nhx : x \u2208 pi univ fun i => range (f i)\n\u22a2 x \u2208 range fun g i => f i (g i)"}, {"tactic": "rintro _ \u27e8x, rfl\u27e9 i -", "annotated_tactic": ["rintro _ \u27e8x, rfl\u27e9 i -", []], "state_before": "case refine_1\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\n\u22a2 (range fun g i => f i (g i)) \u2286 pi univ fun i => range (f i)", "state_after": "case refine_1.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d : \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\nx : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\n\u22a2 (fun g i => f i (g i)) x i \u2208 (fun i => range (f i)) i"}, {"tactic": "exact \u27e8x i, rfl\u27e9", "annotated_tactic": ["exact \u27e8x i, <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case refine_1.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni\u271d : \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\nx : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\n\u22a2 (fun g i => f i (g i)) x i \u2208 (fun i => range (f i)) i", "state_after": "no goals"}, {"tactic": "choose y hy using hx", "annotated_tactic": ["choose y hy using hx", []], "state_before": "case refine_2\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\nx : (i : \u03b9) \u2192 \u03b2 i\nhx : x \u2208 pi univ fun i => range (f i)\n\u22a2 x \u2208 range fun g i => f i (g i)", "state_after": "case refine_2\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\nx : (i : \u03b9) \u2192 \u03b2 i\ny : (i : \u03b9) \u2192 i \u2208 univ \u2192 \u03b1 i\nhy : \u2200 (i : \u03b9) (a : i \u2208 univ), f i (y i a) = x i\n\u22a2 x \u2208 range fun g i => f i (g i)"}, {"tactic": "exact \u27e8fun i => y i trivial, funext fun i => hy i trivial\u27e9", "annotated_tactic": ["exact \u27e8fun i => y i <a>trivial</a>, <a>funext</a> fun i => hy i <a>trivial</a>\u27e9", [{"full_name": "trivial", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [514, 31], "def_end_pos": [514, 38]}, {"full_name": "funext", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1555, 9], "def_end_pos": [1555, 15]}, {"full_name": "trivial", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [514, 31], "def_end_pos": [514, 38]}]], "state_before": "case refine_2\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\nx : (i : \u03b9) \u2192 \u03b2 i\ny : (i : \u03b9) \u2192 i \u2208 univ \u2192 \u03b1 i\nhy : \u2200 (i : \u03b9) (a : i \u2208 univ), f i (y i a) = x i\n\u22a2 x \u2208 range fun g i => f i (g i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean", "full_name": "ExistsContDiffBumpBase.u_smooth", "start": [254, 1], "end": [255, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/FieldTheory/IntermediateField.lean", "full_name": "IntermediateField.coe_nat_mem", "start": [273, 1], "end": [273, 76], "traced_tactics": [{"tactic": "simpa using coe_int_mem S n", "annotated_tactic": ["simpa using <a>coe_int_mem</a> S n", [{"full_name": "coe_int_mem", "def_path": "Mathlib/RingTheory/Subring/Basic.lean", "def_pos": [90, 9], "def_end_pos": [90, 20]}]], "state_before": "K : Type u_1\nL : Type u_2\nL' : Type u_3\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field L'\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K L'\nS : IntermediateField K L\nn : \u2115\n\u22a2 \u2191n \u2208 S", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "StrictMonoOn.comp_strictAntiOn", "start": [1594, 1], "end": [1597, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecialFunctions/Integrals.lean", "full_name": "integral_pow", "start": [378, 1], "end": [379, 96], "traced_tactics": [{"tactic": "simpa only [\u2190 Int.ofNat_succ, zpow_ofNat] using integral_zpow (Or.inl (Int.coe_nat_nonneg n))", "annotated_tactic": ["simpa only [\u2190 <a>Int.ofNat_succ</a>, <a>zpow_ofNat</a>] using <a>integral_zpow</a> (<a>Or.inl</a> (<a>Int.coe_nat_nonneg</a> n))", [{"full_name": "Int.ofNat_succ", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [37, 9], "def_end_pos": [37, 19]}, {"full_name": "zpow_ofNat", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [948, 9], "def_end_pos": [948, 19]}, {"full_name": "integral_zpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Integrals.lean", "def_pos": [371, 9], "def_end_pos": [371, 22]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "Int.coe_nat_nonneg", "def_path": "Mathlib/Data/Int/Basic.lean", "def_pos": [132, 9], "def_end_pos": [132, 23]}]], "state_before": "a b : \u211d\nn : \u2115\n\u22a2 \u222b (x : \u211d) in a..b, x ^ n = (b ^ (n + 1) - a ^ (n + 1)) / (\u2191n + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Hom/Centroid.lean", "full_name": "CentroidHom.toAddMonoidHom_eq_coe", "start": [124, 1], "end": [125, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Semiring.lean", "full_name": "SetSemiring.up_down", "start": [66, 11], "end": [67, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Atoms.lean", "full_name": "IsCoatom.le_iff", "start": [181, 1], "end": [182, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "full_name": "IsCompact.totallyBounded", "start": [634, 11], "end": [635, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean", "full_name": "isBoundedBilinearMap_smulRight", "start": [471, 1], "end": [474, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Matrix/Adjugate.lean", "full_name": "Matrix.det_eq_sum_mul_adjugate_row", "start": [428, 1], "end": [439, 86], "traced_tactics": [{"tactic": "haveI : Nonempty n := \u27e8i\u27e9", "annotated_tactic": ["haveI : <a>Nonempty</a> n := \u27e8i\u27e9", [{"full_name": "Nonempty", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [686, 17], "def_end_pos": [686, 25]}]], "state_before": "m : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni : n\n\u22a2 det A = \u2211 j : n, A i j * adjugate A j i", "state_after": "m : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni : n\nthis : Nonempty n\n\u22a2 det A = \u2211 j : n, A i j * adjugate A j i"}, {"tactic": "obtain \u27e8n', hn'\u27e9 := Nat.exists_eq_succ_of_ne_zero (Fintype.card_ne_zero : Fintype.card n \u2260 0)", "annotated_tactic": ["obtain \u27e8n', hn'\u27e9 := <a>Nat.exists_eq_succ_of_ne_zero</a> (<a>Fintype.card_ne_zero</a> : <a>Fintype.card</a> n \u2260 0)", [{"full_name": "Nat.exists_eq_succ_of_ne_zero", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [238, 9], "def_end_pos": [238, 34]}, {"full_name": "Fintype.card_ne_zero", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [563, 9], "def_end_pos": [563, 21]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}]], "state_before": "m : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni : n\nthis : Nonempty n\n\u22a2 det A = \u2211 j : n, A i j * adjugate A j i", "state_after": "case intro\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni : n\nthis : Nonempty n\nn' : \u2115\nhn' : Fintype.card n = Nat.succ n'\n\u22a2 det A = \u2211 j : n, A i j * adjugate A j i"}, {"tactic": "obtain \u27e8e\u27e9 := Fintype.truncEquivFinOfCardEq hn'", "annotated_tactic": ["obtain \u27e8e\u27e9 := <a>Fintype.truncEquivFinOfCardEq</a> hn'", [{"full_name": "Fintype.truncEquivFinOfCardEq", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [167, 5], "def_end_pos": [167, 26]}]], "state_before": "case intro\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni : n\nthis : Nonempty n\nn' : \u2115\nhn' : Fintype.card n = Nat.succ n'\n\u22a2 det A = \u2211 j : n, A i j * adjugate A j i", "state_after": "case intro.mk\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni : n\nthis : Nonempty n\nn' : \u2115\nhn' : Fintype.card n = Nat.succ n'\nx\u271d : Trunc (n \u2243 Fin (Nat.succ n'))\ne : n \u2243 Fin (Nat.succ n')\n\u22a2 det A = \u2211 j : n, A i j * adjugate A j i"}, {"tactic": "let A' := reindex e e A", "annotated_tactic": ["let A' := <a>reindex</a> e e A", [{"full_name": "Matrix.reindex", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [2574, 5], "def_end_pos": [2574, 12]}]], "state_before": "case intro.mk\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni : n\nthis : Nonempty n\nn' : \u2115\nhn' : Fintype.card n = Nat.succ n'\nx\u271d : Trunc (n \u2243 Fin (Nat.succ n'))\ne : n \u2243 Fin (Nat.succ n')\n\u22a2 det A = \u2211 j : n, A i j * adjugate A j i", "state_after": "case intro.mk\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni : n\nthis : Nonempty n\nn' : \u2115\nhn' : Fintype.card n = Nat.succ n'\nx\u271d : Trunc (n \u2243 Fin (Nat.succ n'))\ne : n \u2243 Fin (Nat.succ n')\nA' : (fun x => Matrix (Fin (Nat.succ n')) (Fin (Nat.succ n')) \u03b1) A := \u2191(reindex e e) A\n\u22a2 det A = \u2211 j : n, A i j * adjugate A j i"}, {"tactic": "suffices det A' = \u2211 j : Fin n'.succ, A' (e i) j * adjugate A' j (e i) by\n  simp_rw [det_reindex_self, adjugate_reindex, reindex_apply, submatrix_apply, \u2190 e.sum_comp,\n    Equiv.symm_apply_apply] at this\n  exact this", "annotated_tactic": ["suffices <a>det</a> A' = \u2211 j : <a>Fin</a> n'.succ, A' (e i) j * <a>adjugate</a> A' j (e i) by\n    simp_rw [<a>det_reindex_self</a>, <a>adjugate_reindex</a>, <a>reindex_apply</a>, <a>submatrix_apply</a>, \u2190 e.sum_comp,\n      <a>Equiv.symm_apply_apply</a>] at this\n    exact this", [{"full_name": "Matrix.det", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [65, 8], "def_end_pos": [65, 11]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Matrix.adjugate", "def_path": "Mathlib/LinearAlgebra/Matrix/Adjugate.lean", "def_pos": [206, 5], "def_end_pos": [206, 13]}, {"full_name": "Matrix.det_reindex_self", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [264, 9], "def_end_pos": [264, 25]}, {"full_name": "Matrix.adjugate_reindex", "def_path": "Mathlib/LinearAlgebra/Matrix/Adjugate.lean", "def_pos": [261, 9], "def_end_pos": [261, 25]}, {"full_name": "Matrix.reindex_apply", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [2582, 9], "def_end_pos": [2582, 22]}, {"full_name": "Matrix.submatrix_apply", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [2413, 9], "def_end_pos": [2413, 24]}, {"full_name": "Equiv.symm_apply_apply", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [283, 17], "def_end_pos": [283, 33]}]], "state_before": "case intro.mk\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni : n\nthis : Nonempty n\nn' : \u2115\nhn' : Fintype.card n = Nat.succ n'\nx\u271d : Trunc (n \u2243 Fin (Nat.succ n'))\ne : n \u2243 Fin (Nat.succ n')\nA' : (fun x => Matrix (Fin (Nat.succ n')) (Fin (Nat.succ n')) \u03b1) A := \u2191(reindex e e) A\n\u22a2 det A = \u2211 j : n, A i j * adjugate A j i", "state_after": "case intro.mk\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni : n\nthis : Nonempty n\nn' : \u2115\nhn' : Fintype.card n = Nat.succ n'\nx\u271d : Trunc (n \u2243 Fin (Nat.succ n'))\ne : n \u2243 Fin (Nat.succ n')\nA' : (fun x => Matrix (Fin (Nat.succ n')) (Fin (Nat.succ n')) \u03b1) A := \u2191(reindex e e) A\n\u22a2 det A' = \u2211 j : Fin (Nat.succ n'), A' (\u2191e i) j * adjugate A' j (\u2191e i)"}, {"tactic": "rw [det_succ_row A' (e i)]", "annotated_tactic": ["rw [<a>det_succ_row</a> A' (e i)]", [{"full_name": "Matrix.det_succ_row", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [751, 9], "def_end_pos": [751, 21]}]], "state_before": "case intro.mk\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni : n\nthis : Nonempty n\nn' : \u2115\nhn' : Fintype.card n = Nat.succ n'\nx\u271d : Trunc (n \u2243 Fin (Nat.succ n'))\ne : n \u2243 Fin (Nat.succ n')\nA' : (fun x => Matrix (Fin (Nat.succ n')) (Fin (Nat.succ n')) \u03b1) A := \u2191(reindex e e) A\n\u22a2 det A' = \u2211 j : Fin (Nat.succ n'), A' (\u2191e i) j * adjugate A' j (\u2191e i)", "state_after": "case intro.mk\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni : n\nthis : Nonempty n\nn' : \u2115\nhn' : Fintype.card n = Nat.succ n'\nx\u271d : Trunc (n \u2243 Fin (Nat.succ n'))\ne : n \u2243 Fin (Nat.succ n')\nA' : (fun x => Matrix (Fin (Nat.succ n')) (Fin (Nat.succ n')) \u03b1) A := \u2191(reindex e e) A\n\u22a2 \u2211 j : Fin (Nat.succ n'),\n      (-1) ^ (\u2191(\u2191e i) + \u2191j) * A' (\u2191e i) j * det (submatrix A' (Fin.succAbove (\u2191e i)) (Fin.succAbove j)) =\n    \u2211 j : Fin (Nat.succ n'), A' (\u2191e i) j * adjugate A' j (\u2191e i)"}, {"tactic": "simp_rw [mul_assoc, mul_left_comm _ (A' _ _), \u2190 adjugate_fin_succ_eq_det_submatrix]", "annotated_tactic": ["simp_rw [<a>mul_assoc</a>, <a>mul_left_comm</a> _ (A' _ _), \u2190 <a>adjugate_fin_succ_eq_det_submatrix</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [96, 9], "def_end_pos": [96, 22]}, {"full_name": "Matrix.adjugate_fin_succ_eq_det_submatrix", "def_path": "Mathlib/LinearAlgebra/Matrix/Adjugate.lean", "def_pos": [421, 9], "def_end_pos": [421, 43]}]], "state_before": "case intro.mk\nm : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni : n\nthis : Nonempty n\nn' : \u2115\nhn' : Fintype.card n = Nat.succ n'\nx\u271d : Trunc (n \u2243 Fin (Nat.succ n'))\ne : n \u2243 Fin (Nat.succ n')\nA' : (fun x => Matrix (Fin (Nat.succ n')) (Fin (Nat.succ n')) \u03b1) A := \u2191(reindex e e) A\n\u22a2 \u2211 j : Fin (Nat.succ n'),\n      (-1) ^ (\u2191(\u2191e i) + \u2191j) * A' (\u2191e i) j * det (submatrix A' (Fin.succAbove (\u2191e i)) (Fin.succAbove j)) =\n    \u2211 j : Fin (Nat.succ n'), A' (\u2191e i) j * adjugate A' j (\u2191e i)", "state_after": "no goals"}, {"tactic": "simp_rw [det_reindex_self, adjugate_reindex, reindex_apply, submatrix_apply, \u2190 e.sum_comp,\n  Equiv.symm_apply_apply] at this", "annotated_tactic": ["simp_rw [<a>det_reindex_self</a>, <a>adjugate_reindex</a>, <a>reindex_apply</a>, <a>submatrix_apply</a>, \u2190 e.sum_comp,\n      <a>Equiv.symm_apply_apply</a>] at this", [{"full_name": "Matrix.det_reindex_self", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [264, 9], "def_end_pos": [264, 25]}, {"full_name": "Matrix.adjugate_reindex", "def_path": "Mathlib/LinearAlgebra/Matrix/Adjugate.lean", "def_pos": [261, 9], "def_end_pos": [261, 25]}, {"full_name": "Matrix.reindex_apply", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [2582, 9], "def_end_pos": [2582, 22]}, {"full_name": "Matrix.submatrix_apply", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [2413, 9], "def_end_pos": [2413, 24]}, {"full_name": "Equiv.symm_apply_apply", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [283, 17], "def_end_pos": [283, 33]}]], "state_before": "m : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni : n\nthis\u271d : Nonempty n\nn' : \u2115\nhn' : Fintype.card n = Nat.succ n'\nx\u271d : Trunc (n \u2243 Fin (Nat.succ n'))\ne : n \u2243 Fin (Nat.succ n')\nA' : (fun x => Matrix (Fin (Nat.succ n')) (Fin (Nat.succ n')) \u03b1) A := \u2191(reindex e e) A\nthis : det A' = \u2211 j : Fin (Nat.succ n'), A' (\u2191e i) j * adjugate A' j (\u2191e i)\n\u22a2 det A = \u2211 j : n, A i j * adjugate A j i", "state_after": "m : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni : n\nthis\u271d : Nonempty n\nn' : \u2115\nhn' : Fintype.card n = Nat.succ n'\nx\u271d : Trunc (n \u2243 Fin (Nat.succ n'))\ne : n \u2243 Fin (Nat.succ n')\nA' : (fun x => Matrix (Fin (Nat.succ n')) (Fin (Nat.succ n')) \u03b1) A := \u2191(reindex e e) A\nthis : det A = \u2211 x : n, A i x * adjugate A x i\n\u22a2 det A = \u2211 j : n, A i j * adjugate A j i"}, {"tactic": "exact this", "annotated_tactic": ["exact this", []], "state_before": "m : Type u\nn : Type v\n\u03b1 : Type w\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\ni : n\nthis\u271d : Nonempty n\nn' : \u2115\nhn' : Fintype.card n = Nat.succ n'\nx\u271d : Trunc (n \u2243 Fin (Nat.succ n'))\ne : n \u2243 Fin (Nat.succ n')\nA' : (fun x => Matrix (Fin (Nat.succ n')) (Fin (Nat.succ n')) \u03b1) A := \u2191(reindex e e) A\nthis : det A = \u2211 x : n, A i x * adjugate A x i\n\u22a2 det A = \u2211 j : n, A i j * adjugate A j i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Basic.lean", "full_name": "IsMax.Ici_eq", "start": [966, 1], "end": [967, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Basic.lean", "full_name": "dist_le_pi_dist", "start": [2102, 1], "end": [2103, 72], "traced_tactics": [{"tactic": "simp only [dist_nndist, NNReal.coe_le_coe, nndist_le_pi_nndist f g b]", "annotated_tactic": ["simp only [<a>dist_nndist</a>, <a>NNReal.coe_le_coe</a>, <a>nndist_le_pi_nndist</a> f g b]", [{"full_name": "dist_nndist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 20]}, {"full_name": "NNReal.coe_le_coe", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [366, 19], "def_end_pos": [366, 29]}, {"full_name": "nndist_le_pi_nndist", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2097, 9], "def_end_pos": [2097, 28]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PseudoMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_3\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b)\nf g : (b : \u03b2) \u2192 \u03c0 b\nb : \u03b2\n\u22a2 dist (f b) (g b) \u2264 dist f g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Index.lean", "full_name": "Subgroup.nat_card_dvd_of_surjective", "start": [310, 1], "end": [313, 61], "traced_tactics": [{"tactic": "rw [\u2190 Nat.card_congr (QuotientGroup.quotientKerEquivOfSurjective f hf).toEquiv]", "annotated_tactic": ["rw [\u2190 <a>Nat.card_congr</a> (<a>QuotientGroup.quotientKerEquivOfSurjective</a> f hf).<a>toEquiv</a>]", [{"full_name": "Nat.card_congr", "def_path": "Mathlib/SetTheory/Cardinal/Finite.lean", "def_pos": [55, 9], "def_end_pos": [55, 19]}, {"full_name": "QuotientGroup.quotientKerEquivOfSurjective", "def_path": "Mathlib/GroupTheory/QuotientGroup.lean", "def_pos": [441, 19], "def_end_pos": [441, 47]}, {"full_name": "MulEquiv.toEquiv", "def_path": "Mathlib/Algebra/Hom/Equiv/Basic.lean", "def_pos": [101, 14], "def_end_pos": [101, 30]}]], "state_before": "G\u271d : Type u_1\ninst\u271d\u00b2 : Group G\u271d\nH\u271d K L : Subgroup G\u271d\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192* H\nhf : Function.Surjective \u2191f\n\u22a2 Nat.card H \u2223 Nat.card G", "state_after": "G\u271d : Type u_1\ninst\u271d\u00b2 : Group G\u271d\nH\u271d K L : Subgroup G\u271d\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192* H\nhf : Function.Surjective \u2191f\n\u22a2 Nat.card (G \u29f8 MonoidHom.ker f) \u2223 Nat.card G"}, {"tactic": "exact Dvd.intro_left (Nat.card f.ker) f.ker.card_mul_index", "annotated_tactic": ["exact <a>Dvd.intro_left</a> (<a>Nat.card</a> f.ker) f.ker.card_mul_index", [{"full_name": "Dvd.intro_left", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [147, 9], "def_end_pos": [147, 23]}, {"full_name": "Nat.card", "def_path": "Mathlib/SetTheory/Cardinal/Finite.lean", "def_pos": [37, 15], "def_end_pos": [37, 19]}]], "state_before": "G\u271d : Type u_1\ninst\u271d\u00b2 : Group G\u271d\nH\u271d K L : Subgroup G\u271d\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192* H\nhf : Function.Surjective \u2191f\n\u22a2 Nat.card (G \u29f8 MonoidHom.ker f) \u2223 Nat.card G", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.forall_of_forall_insert", "start": [3143, 1], "end": [3145, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Equiv.lean", "full_name": "ContinuousLinearEquiv.uniqueDiffOn_preimage_iff", "start": [530, 1], "end": [532, 65], "traced_tactics": [{"tactic": "rw [\u2190 e.image_symm_eq_preimage, e.symm.uniqueDiffOn_image_iff]", "annotated_tactic": ["rw [\u2190 e.image_symm_eq_preimage, e.symm.uniqueDiffOn_image_iff]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\ns : Set E\nf' : E \u2192L[\ud835\udd5c] F\ne : F \u2243L[\ud835\udd5c] E\n\u22a2 UniqueDiffOn \ud835\udd5c (\u2191e \u207b\u00b9' s) \u2194 UniqueDiffOn \ud835\udd5c s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Localization/Basic.lean", "full_name": "IsLocalization.map_smul", "start": [670, 1], "end": [671, 67], "traced_tactics": [{"tactic": "rw [Algebra.smul_def, Algebra.smul_def, RingHom.map_mul, map_eq]", "annotated_tactic": ["rw [<a>Algebra.smul_def</a>, <a>Algebra.smul_def</a>, <a>RingHom.map_mul</a>, <a>map_eq</a>]", [{"full_name": "Algebra.smul_def", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [347, 9], "def_end_pos": [347, 17]}, {"full_name": "Algebra.smul_def", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [347, 9], "def_end_pos": [347, 17]}, {"full_name": "RingHom.map_mul", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [569, 19], "def_end_pos": [569, 26]}, {"full_name": "IsLocalization.map_eq", "def_path": "Mathlib/RingTheory/Localization/Basic.lean", "def_pos": [621, 9], "def_end_pos": [621, 15]}]], "state_before": "R : Type u_1\ninst\u271d\u2077 : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2076 : CommSemiring S\ninst\u271d\u2075 : Algebra R S\nP : Type u_3\ninst\u271d\u2074 : CommSemiring P\ninst\u271d\u00b3 : IsLocalization M S\ng : R \u2192+* P\nhg : \u2200 (y : { x // x \u2208 M }), IsUnit (\u2191g \u2191y)\nT : Submonoid P\nQ : Type u_4\ninst\u271d\u00b2 : CommSemiring Q\nhy : M \u2264 Submonoid.comap g T\ninst\u271d\u00b9 : Algebra P Q\ninst\u271d : IsLocalization T Q\nx : S\nz : R\n\u22a2 \u2191(map Q g hy) (z \u2022 x) = \u2191g z \u2022 \u2191(map Q g hy) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Interval.lean", "full_name": "Int.card_fintype_Icc", "start": [152, 1], "end": [153, 41], "traced_tactics": [{"tactic": "rw [\u2190 card_Icc, Fintype.card_ofFinset]", "annotated_tactic": ["rw [\u2190 <a>card_Icc</a>, <a>Fintype.card_ofFinset</a>]", [{"full_name": "Int.card_Icc", "def_path": "Mathlib/Data/Int/Interval.lean", "def_pos": [109, 9], "def_end_pos": [109, 17]}, {"full_name": "Fintype.card_ofFinset", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [132, 9], "def_end_pos": [132, 22]}]], "state_before": "a b : \u2124\n\u22a2 Fintype.card \u2191(Set.Icc a b) = toNat (b + 1 - a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Convex/Quasiconvex.lean", "full_name": "MonotoneOn.quasiconvexOn", "start": [182, 1], "end": [183, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/FieldTheory/Subfield.lean", "full_name": "Subfield.closure_le", "start": [723, 1], "end": [724, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "full_name": "PowerSeries.rescale_mk", "start": [1815, 1], "end": [1817, 41], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "R : Type u_1\ninst\u271d : CommSemiring R\nf : \u2115 \u2192 R\na : R\n\u22a2 \u2191(rescale a) (mk f) = mk fun n => a ^ n * f n", "state_after": "case h\nR : Type u_1\ninst\u271d : CommSemiring R\nf : \u2115 \u2192 R\na : R\nn\u271d : \u2115\n\u22a2 \u2191(coeff R n\u271d) (\u2191(rescale a) (mk f)) = \u2191(coeff R n\u271d) (mk fun n => a ^ n * f n)"}, {"tactic": "rw [coeff_rescale, coeff_mk, coeff_mk]", "annotated_tactic": ["rw [<a>coeff_rescale</a>, <a>coeff_mk</a>, <a>coeff_mk</a>]", [{"full_name": "PowerSeries.coeff_rescale", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [1793, 9], "def_end_pos": [1793, 22]}, {"full_name": "PowerSeries.coeff_mk", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [1382, 9], "def_end_pos": [1382, 17]}, {"full_name": "PowerSeries.coeff_mk", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [1382, 9], "def_end_pos": [1382, 17]}]], "state_before": "case h\nR : Type u_1\ninst\u271d : CommSemiring R\nf : \u2115 \u2192 R\na : R\nn\u271d : \u2115\n\u22a2 \u2191(coeff R n\u271d) (\u2191(rescale a) (mk f)) = \u2191(coeff R n\u271d) (mk fun n => a ^ n * f n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean", "full_name": "UpperHalfPlane.SL_neg_smul", "start": [391, 1], "end": [392, 66], "traced_tactics": [{"tactic": "simp only [coe_GLPos_neg, sl_moeb, coe_int_neg, neg_smul, coe']", "annotated_tactic": ["simp only [<a>coe_GLPos_neg</a>, <a>sl_moeb</a>, <a>coe_int_neg</a>, <a>neg_smul</a>, <a>coe'</a>]", [{"full_name": "Matrix.SpecialLinearGroup.coe_GLPos_neg", "def_path": "Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup.lean", "def_pos": [290, 9], "def_end_pos": [290, 22]}, {"full_name": "UpperHalfPlane.sl_moeb", "def_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean", "def_pos": [377, 9], "def_end_pos": [377, 16]}, {"full_name": "Matrix.SpecialLinearGroup.coe_int_neg", "def_path": "Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean", "def_pos": [275, 9], "def_end_pos": [275, 20]}, {"full_name": "UpperHalfPlane.neg_smul", "def_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean", "def_pos": [364, 9], "def_end_pos": [364, 17]}, {"full_name": "UpperHalfPlane.coe'", "def_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean", "def_pos": [266, 5], "def_end_pos": [266, 9]}]], "state_before": "g\u271d : SL(2, \u2124)\nz\u271d : \u210d\n\u0393 : Subgroup SL(2, \u2124)\ng : SL(2, \u2124)\nz : \u210d\n\u22a2 -g \u2022 z = g \u2022 z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/NumberField/Units.lean", "full_name": "NumberField.Units.dirichletUnitTheorem.seq_norm_ne_zero", "start": [370, 1], "end": [371, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Module/Alternating.lean", "full_name": "ContinuousMultilinearMap.alternatization_apply_toAlternatingMap", "start": [611, 1], "end": [614, 84], "traced_tactics": [{"tactic": "ext v", "annotated_tactic": ["ext v", []], "state_before": "R : Type u_1\nM : Type u_2\nN : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : AddCommGroup N\ninst\u271d\u2074 : Module R N\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : TopologicalAddGroup N\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nf g : ContinuousMultilinearMap R (fun x => M) N\n\u22a2 ContinuousAlternatingMap.toAlternatingMap (\u2191alternatization f) = \u2191MultilinearMap.alternatization f.toMultilinearMap", "state_after": "case H\nR : Type u_1\nM : Type u_2\nN : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : TopologicalSpace M\ninst\u271d\u2075 : AddCommGroup N\ninst\u271d\u2074 : Module R N\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : TopologicalAddGroup N\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nf g : ContinuousMultilinearMap R (fun x => M) N\nv : \u03b9 \u2192 M\n\u22a2 \u2191(ContinuousAlternatingMap.toAlternatingMap (\u2191alternatization f)) v =\n    \u2191(\u2191MultilinearMap.alternatization f.toMultilinearMap) v"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Ultrafilter.lean", "full_name": "Ultrafilter.em", "start": [183, 11], "end": [184, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Alternating/Basic.lean", "full_name": "AlternatingMap.curryLeft_add", "start": [1025, 1], "end": [1027, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/Young/YoungDiagram.lean", "full_name": "YoungDiagram.mk_mem_col_iff", "start": [352, 1], "end": [352, 101], "traced_tactics": [{"tactic": "simp [col]", "annotated_tactic": ["simp [<a>col</a>]", [{"full_name": "YoungDiagram.col", "def_path": "Mathlib/Combinatorics/Young/YoungDiagram.lean", "def_pos": [344, 5], "def_end_pos": [344, 8]}]], "state_before": "\u03bc : YoungDiagram\ni j : \u2115\n\u22a2 (i, j) \u2208 col \u03bc j \u2194 (i, j) \u2208 \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Sheaves/Presheaf.lean", "full_name": "TopCat.Presheaf.Pushforward.id_inv_app'", "start": [258, 1], "end": [260, 29], "traced_tactics": [{"tactic": "dsimp [id]", "annotated_tactic": ["dsimp [<a>id</a>]", [{"full_name": "TopCat.Presheaf.Pushforward.id", "def_path": "Mathlib/Topology/Sheaves/Presheaf.lean", "def_pos": [222, 5], "def_end_pos": [222, 7]}]], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u2131 : Presheaf C X\nU : Set \u2191X\np : IsOpen U\n\u22a2 (id \u2131).inv.app (op { carrier := U, is_open' := p }) = \u2131.map (\ud835\udfd9 (op { carrier := U, is_open' := p }))", "state_after": "C : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u2131 : Presheaf C X\nU : Set \u2191X\np : IsOpen U\n\u22a2 ((Functor.leftUnitor \u2131).inv \u226b whiskerRight (NatTrans.op (Opens.mapId X).hom) \u2131).app\n      (op { carrier := U, is_open' := p }) =\n    \u2131.map (\ud835\udfd9 (op { carrier := U, is_open' := p }))"}, {"tactic": "simp [CategoryStruct.comp]", "annotated_tactic": ["simp [<a>CategoryStruct.comp</a>]", [{"full_name": "CategoryTheory.CategoryStruct.comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [100, 3], "def_end_pos": [100, 7]}]], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX : TopCat\n\u2131 : Presheaf C X\nU : Set \u2191X\np : IsOpen U\n\u22a2 ((Functor.leftUnitor \u2131).inv \u226b whiskerRight (NatTrans.op (Opens.mapId X).hom) \u2131).app\n      (op { carrier := U, is_open' := p }) =\n    \u2131.map (\ud835\udfd9 (op { carrier := U, is_open' := p }))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "full_name": "pi_nnnorm_const'", "start": [2592, 1], "end": [2593, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Init/Data/Int/CompLemmas.lean", "full_name": "Int.nonneg_of_pos", "start": [80, 11], "end": [81, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/ContDiffDef.lean", "full_name": "contDiffWithinAt_inter", "start": [511, 1], "end": [513, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Polish.lean", "full_name": "ClosedEmbedding.polishSpace", "start": [152, 1], "end": [160, 17], "traced_tactics": [{"tactic": "letI := upgradePolishSpace \u03b2", "annotated_tactic": ["letI := <a>upgradePolishSpace</a> \u03b2", [{"full_name": "upgradePolishSpace", "def_path": "Mathlib/Topology/MetricSpace/Polish.lean", "def_pos": [96, 5], "def_end_pos": [96, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\n\u22a2 PolishSpace \u03b1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\nthis : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2\n\u22a2 PolishSpace \u03b1"}, {"tactic": "letI : MetricSpace \u03b1 := hf.toEmbedding.comapMetricSpace f", "annotated_tactic": ["letI : <a>MetricSpace</a> \u03b1 := hf.toEmbedding.comapMetricSpace f", [{"full_name": "MetricSpace", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2892, 7], "def_end_pos": [2892, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\nthis : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2\n\u22a2 PolishSpace \u03b1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\nthis\u271d : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2\nthis : MetricSpace \u03b1 := Embedding.comapMetricSpace f (_ : _root_.Embedding f)\n\u22a2 PolishSpace \u03b1"}, {"tactic": "haveI : SecondCountableTopology \u03b1 := hf.toEmbedding.secondCountableTopology", "annotated_tactic": ["haveI : <a>SecondCountableTopology</a> \u03b1 := hf.toEmbedding.secondCountableTopology", [{"full_name": "TopologicalSpace.SecondCountableTopology", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [665, 7], "def_end_pos": [665, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\nthis\u271d : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2\nthis : MetricSpace \u03b1 := Embedding.comapMetricSpace f (_ : _root_.Embedding f)\n\u22a2 PolishSpace \u03b1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2\nthis\u271d : MetricSpace \u03b1 := Embedding.comapMetricSpace f (_ : _root_.Embedding f)\nthis : SecondCountableTopology \u03b1\n\u22a2 PolishSpace \u03b1"}, {"tactic": "have : CompleteSpace \u03b1 := by\n  rw [completeSpace_iff_isComplete_range hf.toEmbedding.to_isometry.uniformInducing]\n  exact hf.closed_range.isComplete", "annotated_tactic": ["have : <a>CompleteSpace</a> \u03b1 := by\n    rw [<a>completeSpace_iff_isComplete_range</a> hf.toEmbedding.to_isometry.uniformInducing]\n    exact hf.closed_range.isComplete", [{"full_name": "CompleteSpace", "def_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "def_pos": [397, 7], "def_end_pos": [397, 20]}, {"full_name": "completeSpace_iff_isComplete_range", "def_path": "Mathlib/Topology/UniformSpace/UniformEmbedding.lean", "def_pos": [304, 9], "def_end_pos": [304, 43]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2\nthis\u271d : MetricSpace \u03b1 := Embedding.comapMetricSpace f (_ : _root_.Embedding f)\nthis : SecondCountableTopology \u03b1\n\u22a2 PolishSpace \u03b1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\nthis\u271d\u00b2 : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2\nthis\u271d\u00b9 : MetricSpace \u03b1 := Embedding.comapMetricSpace f (_ : _root_.Embedding f)\nthis\u271d : SecondCountableTopology \u03b1\nthis : CompleteSpace \u03b1\n\u22a2 PolishSpace \u03b1"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\nthis\u271d\u00b2 : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2\nthis\u271d\u00b9 : MetricSpace \u03b1 := Embedding.comapMetricSpace f (_ : _root_.Embedding f)\nthis\u271d : SecondCountableTopology \u03b1\nthis : CompleteSpace \u03b1\n\u22a2 PolishSpace \u03b1", "state_after": "no goals"}, {"tactic": "rw [completeSpace_iff_isComplete_range hf.toEmbedding.to_isometry.uniformInducing]", "annotated_tactic": ["rw [<a>completeSpace_iff_isComplete_range</a> hf.toEmbedding.to_isometry.uniformInducing]", [{"full_name": "completeSpace_iff_isComplete_range", "def_path": "Mathlib/Topology/UniformSpace/UniformEmbedding.lean", "def_pos": [304, 9], "def_end_pos": [304, 43]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2\nthis\u271d : MetricSpace \u03b1 := Embedding.comapMetricSpace f (_ : _root_.Embedding f)\nthis : SecondCountableTopology \u03b1\n\u22a2 CompleteSpace \u03b1", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2\nthis\u271d : MetricSpace \u03b1 := Embedding.comapMetricSpace f (_ : _root_.Embedding f)\nthis : SecondCountableTopology \u03b1\n\u22a2 IsComplete (range f)"}, {"tactic": "exact hf.closed_range.isComplete", "annotated_tactic": ["exact hf.closed_range.isComplete", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b2 := upgradePolishSpace \u03b2\nthis\u271d : MetricSpace \u03b1 := Embedding.comapMetricSpace f (_ : _root_.Embedding f)\nthis : SecondCountableTopology \u03b1\n\u22a2 IsComplete (range f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finsupp/Defs.lean", "full_name": "Finsupp.support_neg", "start": [1308, 1], "end": [1313, 8], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Fin/Tuple/Basic.lean", "full_name": "Fin.repeat_one", "start": [389, 1], "end": [394, 42], "traced_tactics": [{"tactic": "generalize_proofs h", "annotated_tactic": ["generalize_proofs h", []], "state_before": "m n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d 0\nq : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (succ i)\ni : Fin n\ny : \u03b1\u271d (succ i)\nz : \u03b1\u271d 0\n\u03b1 : Type u_1\na : Fin n \u2192 \u03b1\n\u22a2 repeat 1 a = a \u2218 cast (_ : 1 * n = n)", "state_after": "m n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d 0\nq : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (succ i)\ni : Fin n\ny : \u03b1\u271d (succ i)\nz : \u03b1\u271d 0\n\u03b1 : Type u_1\na : Fin n \u2192 \u03b1\nh : 1 * n = n\n\u22a2 repeat 1 a = a \u2218 cast h"}, {"tactic": "apply funext", "annotated_tactic": ["apply <a>funext</a>", [{"full_name": "funext", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1555, 9], "def_end_pos": [1555, 15]}]], "state_before": "m n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d 0\nq : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (succ i)\ni : Fin n\ny : \u03b1\u271d (succ i)\nz : \u03b1\u271d 0\n\u03b1 : Type u_1\na : Fin n \u2192 \u03b1\nh : 1 * n = n\n\u22a2 repeat 1 a = a \u2218 cast h", "state_after": "case h\nm n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d 0\nq : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (succ i)\ni : Fin n\ny : \u03b1\u271d (succ i)\nz : \u03b1\u271d 0\n\u03b1 : Type u_1\na : Fin n \u2192 \u03b1\nh : 1 * n = n\n\u22a2 \u2200 (x : Fin (1 * n)), repeat 1 a x = (a \u2218 cast h) x"}, {"tactic": "rw [(Fin.rightInverse_cast h.symm).surjective.forall]", "annotated_tactic": ["rw [(<a>Fin.rightInverse_cast</a> h.symm).surjective.forall]", [{"full_name": "Fin.rightInverse_cast", "def_path": "Mathlib/Data/Fin/Basic.lean", "def_pos": [799, 9], "def_end_pos": [799, 26]}]], "state_before": "case h\nm n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d 0\nq : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (succ i)\ni : Fin n\ny : \u03b1\u271d (succ i)\nz : \u03b1\u271d 0\n\u03b1 : Type u_1\na : Fin n \u2192 \u03b1\nh : 1 * n = n\n\u22a2 \u2200 (x : Fin (1 * n)), repeat 1 a x = (a \u2218 cast h) x", "state_after": "case h\nm n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d 0\nq : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (succ i)\ni : Fin n\ny : \u03b1\u271d (succ i)\nz : \u03b1\u271d 0\n\u03b1 : Type u_1\na : Fin n \u2192 \u03b1\nh : 1 * n = n\n\u22a2 \u2200 (x : Fin n), repeat 1 a (cast (_ : n = 1 * n) x) = (a \u2218 cast h) (cast (_ : n = 1 * n) x)"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "case h\nm n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d 0\nq : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (succ i)\ni : Fin n\ny : \u03b1\u271d (succ i)\nz : \u03b1\u271d 0\n\u03b1 : Type u_1\na : Fin n \u2192 \u03b1\nh : 1 * n = n\n\u22a2 \u2200 (x : Fin n), repeat 1 a (cast (_ : n = 1 * n) x) = (a \u2218 cast h) (cast (_ : n = 1 * n) x)", "state_after": "case h\nm n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d 0\nq : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (succ i)\ni\u271d : Fin n\ny : \u03b1\u271d (succ i\u271d)\nz : \u03b1\u271d 0\n\u03b1 : Type u_1\na : Fin n \u2192 \u03b1\nh : 1 * n = n\ni : Fin n\n\u22a2 repeat 1 a (cast (_ : n = 1 * n) i) = (a \u2218 cast h) (cast (_ : n = 1 * n) i)"}, {"tactic": "simp [modNat, Nat.mod_eq_of_lt i.is_lt]", "annotated_tactic": ["simp [<a>modNat</a>, <a>Nat.mod_eq_of_lt</a> i.is_lt]", [{"full_name": "Fin.modNat", "def_path": "Mathlib/Data/Fin/Basic.lean", "def_pos": [1078, 5], "def_end_pos": [1078, 11]}, {"full_name": "Nat.mod_eq_of_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [111, 9], "def_end_pos": [111, 21]}]], "state_before": "case h\nm n : \u2115\n\u03b1\u271d : Fin (n + 1) \u2192 Type u\nx : \u03b1\u271d 0\nq : (i : Fin (n + 1)) \u2192 \u03b1\u271d i\np : (i : Fin n) \u2192 \u03b1\u271d (succ i)\ni\u271d : Fin n\ny : \u03b1\u271d (succ i\u271d)\nz : \u03b1\u271d 0\n\u03b1 : Type u_1\na : Fin n \u2192 \u03b1\nh : 1 * n = n\ni : Fin n\n\u22a2 repeat 1 a (cast (_ : n = 1 * n) i) = (a \u2218 cast h) (cast (_ : n = 1 * n) i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/Infix.lean", "full_name": "List.IsPrefix.filter_map", "start": [291, 1], "end": [300, 23], "traced_tactics": [{"tactic": "induction' l\u2081 with hd\u2081 tl\u2081 hl generalizing l\u2082", "annotated_tactic": ["induction' l\u2081 with hd\u2081 tl\u2081 hl generalizing l\u2082", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 l\u2083 : List \u03b1\na b : \u03b1\nm n : \u2115\nh : l\u2081 <+: l\u2082\nf : \u03b1 \u2192 Option \u03b2\n\u22a2 filterMap f l\u2081 <+: filterMap f l\u2082", "state_after": "case nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082\u271d l\u2083 : List \u03b1\na b : \u03b1\nm n : \u2115\nh\u271d : l\u2081 <+: l\u2082\u271d\nf : \u03b1 \u2192 Option \u03b2\nl\u2082 : List \u03b1\nh : [] <+: l\u2082\n\u22a2 filterMap f [] <+: filterMap f l\u2082\n\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082\u271d l\u2083 : List \u03b1\na b : \u03b1\nm n : \u2115\nh\u271d : l\u2081 <+: l\u2082\u271d\nf : \u03b1 \u2192 Option \u03b2\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhl : \u2200 {l\u2082 : List \u03b1}, tl\u2081 <+: l\u2082 \u2192 filterMap f tl\u2081 <+: filterMap f l\u2082\nl\u2082 : List \u03b1\nh : hd\u2081 :: tl\u2081 <+: l\u2082\n\u22a2 filterMap f (hd\u2081 :: tl\u2081) <+: filterMap f l\u2082"}, {"tactic": "simp only [nil_prefix, filterMap_nil]", "annotated_tactic": ["simp only [<a>nil_prefix</a>, <a>filterMap_nil</a>]", [{"full_name": "List.nil_prefix", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1746, 9], "def_end_pos": [1746, 19]}, {"full_name": "List.filterMap_nil", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1272, 17], "def_end_pos": [1272, 30]}]], "state_before": "case nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082\u271d l\u2083 : List \u03b1\na b : \u03b1\nm n : \u2115\nh\u271d : l\u2081 <+: l\u2082\u271d\nf : \u03b1 \u2192 Option \u03b2\nl\u2082 : List \u03b1\nh : [] <+: l\u2082\n\u22a2 filterMap f [] <+: filterMap f l\u2082", "state_after": "no goals"}, {"tactic": "cases' l\u2082 with hd\u2082 tl\u2082", "annotated_tactic": ["cases' l\u2082 with hd\u2082 tl\u2082", []], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082\u271d l\u2083 : List \u03b1\na b : \u03b1\nm n : \u2115\nh\u271d : l\u2081 <+: l\u2082\u271d\nf : \u03b1 \u2192 Option \u03b2\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhl : \u2200 {l\u2082 : List \u03b1}, tl\u2081 <+: l\u2082 \u2192 filterMap f tl\u2081 <+: filterMap f l\u2082\nl\u2082 : List \u03b1\nh : hd\u2081 :: tl\u2081 <+: l\u2082\n\u22a2 filterMap f (hd\u2081 :: tl\u2081) <+: filterMap f l\u2082", "state_after": "case cons.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 l\u2083 : List \u03b1\na b : \u03b1\nm n : \u2115\nh\u271d : l\u2081 <+: l\u2082\nf : \u03b1 \u2192 Option \u03b2\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhl : \u2200 {l\u2082 : List \u03b1}, tl\u2081 <+: l\u2082 \u2192 filterMap f tl\u2081 <+: filterMap f l\u2082\nh : hd\u2081 :: tl\u2081 <+: []\n\u22a2 filterMap f (hd\u2081 :: tl\u2081) <+: filterMap f []\n\ncase cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 l\u2083 : List \u03b1\na b : \u03b1\nm n : \u2115\nh\u271d : l\u2081 <+: l\u2082\nf : \u03b1 \u2192 Option \u03b2\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhl : \u2200 {l\u2082 : List \u03b1}, tl\u2081 <+: l\u2082 \u2192 filterMap f tl\u2081 <+: filterMap f l\u2082\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh : hd\u2081 :: tl\u2081 <+: hd\u2082 :: tl\u2082\n\u22a2 filterMap f (hd\u2081 :: tl\u2081) <+: filterMap f (hd\u2082 :: tl\u2082)"}, {"tactic": "simpa only using eq_nil_of_prefix_nil h", "annotated_tactic": ["simpa only using <a>eq_nil_of_prefix_nil</a> h", [{"full_name": "List.eq_nil_of_prefix_nil", "def_path": "Mathlib/Data/List/Infix.lean", "def_pos": [107, 8], "def_end_pos": [107, 28]}]], "state_before": "case cons.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 l\u2083 : List \u03b1\na b : \u03b1\nm n : \u2115\nh\u271d : l\u2081 <+: l\u2082\nf : \u03b1 \u2192 Option \u03b2\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhl : \u2200 {l\u2082 : List \u03b1}, tl\u2081 <+: l\u2082 \u2192 filterMap f tl\u2081 <+: filterMap f l\u2082\nh : hd\u2081 :: tl\u2081 <+: []\n\u22a2 filterMap f (hd\u2081 :: tl\u2081) <+: filterMap f []", "state_after": "no goals"}, {"tactic": "rw [cons_prefix_iff] at h", "annotated_tactic": ["rw [<a>cons_prefix_iff</a>] at h", [{"full_name": "List.cons_prefix_iff", "def_path": "Mathlib/Data/List/Infix.lean", "def_pos": [272, 9], "def_end_pos": [272, 24]}]], "state_before": "case cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 l\u2083 : List \u03b1\na b : \u03b1\nm n : \u2115\nh\u271d : l\u2081 <+: l\u2082\nf : \u03b1 \u2192 Option \u03b2\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhl : \u2200 {l\u2082 : List \u03b1}, tl\u2081 <+: l\u2082 \u2192 filterMap f tl\u2081 <+: filterMap f l\u2082\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh : hd\u2081 :: tl\u2081 <+: hd\u2082 :: tl\u2082\n\u22a2 filterMap f (hd\u2081 :: tl\u2081) <+: filterMap f (hd\u2082 :: tl\u2082)", "state_after": "case cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 l\u2083 : List \u03b1\na b : \u03b1\nm n : \u2115\nh\u271d : l\u2081 <+: l\u2082\nf : \u03b1 \u2192 Option \u03b2\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhl : \u2200 {l\u2082 : List \u03b1}, tl\u2081 <+: l\u2082 \u2192 filterMap f tl\u2081 <+: filterMap f l\u2082\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh : hd\u2081 = hd\u2082 \u2227 tl\u2081 <+: tl\u2082\n\u22a2 filterMap f (hd\u2081 :: tl\u2081) <+: filterMap f (hd\u2082 :: tl\u2082)"}, {"tactic": "rw [\u2190 @singleton_append _ hd\u2081 _, \u2190 @singleton_append _ hd\u2082 _, filterMap_append,\n  filterMap_append, h.left, prefix_append_right_inj]", "annotated_tactic": ["rw [\u2190 @<a>singleton_append</a> _ hd\u2081 _, \u2190 @<a>singleton_append</a> _ hd\u2082 _, <a>filterMap_append</a>,\n        <a>filterMap_append</a>, h.left, <a>prefix_append_right_inj</a>]", [{"full_name": "List.singleton_append", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [55, 22], "def_end_pos": [55, 38]}, {"full_name": "List.singleton_append", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [55, 22], "def_end_pos": [55, 38]}, {"full_name": "List.filterMap_append", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1286, 9], "def_end_pos": [1286, 25]}, {"full_name": "List.filterMap_append", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1286, 9], "def_end_pos": [1286, 25]}, {"full_name": "List.prefix_append_right_inj", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1883, 9], "def_end_pos": [1883, 32]}]], "state_before": "case cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 l\u2083 : List \u03b1\na b : \u03b1\nm n : \u2115\nh\u271d : l\u2081 <+: l\u2082\nf : \u03b1 \u2192 Option \u03b2\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhl : \u2200 {l\u2082 : List \u03b1}, tl\u2081 <+: l\u2082 \u2192 filterMap f tl\u2081 <+: filterMap f l\u2082\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh : hd\u2081 = hd\u2082 \u2227 tl\u2081 <+: tl\u2082\n\u22a2 filterMap f (hd\u2081 :: tl\u2081) <+: filterMap f (hd\u2082 :: tl\u2082)", "state_after": "case cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 l\u2083 : List \u03b1\na b : \u03b1\nm n : \u2115\nh\u271d : l\u2081 <+: l\u2082\nf : \u03b1 \u2192 Option \u03b2\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhl : \u2200 {l\u2082 : List \u03b1}, tl\u2081 <+: l\u2082 \u2192 filterMap f tl\u2081 <+: filterMap f l\u2082\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh : hd\u2081 = hd\u2082 \u2227 tl\u2081 <+: tl\u2082\n\u22a2 filterMap f tl\u2081 <+: filterMap f tl\u2082"}, {"tactic": "exact hl h.right", "annotated_tactic": ["exact hl h.right", []], "state_before": "case cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081 l\u2082 l\u2083 : List \u03b1\na b : \u03b1\nm n : \u2115\nh\u271d : l\u2081 <+: l\u2082\nf : \u03b1 \u2192 Option \u03b2\nhd\u2081 : \u03b1\ntl\u2081 : List \u03b1\nhl : \u2200 {l\u2082 : List \u03b1}, tl\u2081 <+: l\u2082 \u2192 filterMap f tl\u2081 <+: filterMap f l\u2082\nhd\u2082 : \u03b1\ntl\u2082 : List \u03b1\nh : hd\u2081 = hd\u2082 \u2227 tl\u2081 <+: tl\u2082\n\u22a2 filterMap f tl\u2081 <+: filterMap f tl\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/ModularForms/JacobiTheta/Basic.lean", "full_name": "norm_exp_mul_sq_le", "start": [36, 1], "end": [50, 95], "traced_tactics": [{"tactic": "let y := rexp (-\u03c0 * z.im)", "annotated_tactic": ["let y := rexp (-\u03c0 * z.im)", []], "state_before": "z : \u2102\nhz : 0 < z.im\nn : \u2124\n\u22a2 \u2016cexp (\u2191\u03c0 * I * \u2191n ^ 2 * z)\u2016 \u2264 rexp (-\u03c0 * z.im) ^ Int.natAbs n", "state_after": "z : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\n\u22a2 \u2016cexp (\u2191\u03c0 * I * \u2191n ^ 2 * z)\u2016 \u2264 rexp (-\u03c0 * z.im) ^ Int.natAbs n"}, {"tactic": "have h : y < 1 := exp_lt_one_iff.mpr (mul_neg_of_neg_of_pos (neg_lt_zero.mpr pi_pos) hz)", "annotated_tactic": ["have h : y < 1 := exp_lt_one_iff.mpr (<a>mul_neg_of_neg_of_pos</a> (neg_lt_zero.mpr <a>pi_pos</a>) hz)", [{"full_name": "mul_neg_of_neg_of_pos", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [364, 9], "def_end_pos": [364, 30]}, {"full_name": "Real.pi_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "def_pos": [163, 9], "def_end_pos": [163, 15]}]], "state_before": "z : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\n\u22a2 \u2016cexp (\u2191\u03c0 * I * \u2191n ^ 2 * z)\u2016 \u2264 rexp (-\u03c0 * z.im) ^ Int.natAbs n", "state_after": "z : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\n\u22a2 \u2016cexp (\u2191\u03c0 * I * \u2191n ^ 2 * z)\u2016 \u2264 rexp (-\u03c0 * z.im) ^ Int.natAbs n"}, {"tactic": "refine' (le_of_eq _).trans (_ : y ^ n ^ 2 \u2264 _)", "annotated_tactic": ["refine' (<a>le_of_eq</a> _).<a>trans</a> (_ : y ^ n ^ 2 \u2264 _)", [{"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "z : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\n\u22a2 \u2016cexp (\u2191\u03c0 * I * \u2191n ^ 2 * z)\u2016 \u2264 rexp (-\u03c0 * z.im) ^ Int.natAbs n", "state_after": "case refine'_1\nz : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\n\u22a2 \u2016cexp (\u2191\u03c0 * I * \u2191n ^ 2 * z)\u2016 = y ^ n ^ 2\n\ncase refine'_2\nz : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\n\u22a2 y ^ n ^ 2 \u2264 rexp (-\u03c0 * z.im) ^ Int.natAbs n"}, {"tactic": "rw [Complex.norm_eq_abs, Complex.abs_exp]", "annotated_tactic": ["rw [<a>Complex.norm_eq_abs</a>, <a>Complex.abs_exp</a>]", [{"full_name": "Complex.norm_eq_abs", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [51, 9], "def_end_pos": [51, 20]}, {"full_name": "Complex.abs_exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [2048, 9], "def_end_pos": [2048, 16]}]], "state_before": "case refine'_1\nz : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\n\u22a2 \u2016cexp (\u2191\u03c0 * I * \u2191n ^ 2 * z)\u2016 = y ^ n ^ 2", "state_after": "case refine'_1\nz : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\n\u22a2 rexp (\u2191\u03c0 * I * \u2191n ^ 2 * z).re = y ^ n ^ 2"}, {"tactic": "have : (\u2191\u03c0 * I * (n : \u2102) ^ 2 * z).re = -\u03c0 * z.im * (n : \u211d) ^ 2 := by\n  rw [(by push_cast; ring : \u2191\u03c0 * I * (n : \u2102) ^ 2 * z = \u2191(\u03c0 * (n : \u211d) ^ 2) * (z * I)),\n    ofReal_mul_re, mul_I_re]\n  ring", "annotated_tactic": ["have : (\u2191\u03c0 * <a>I</a> * (n : \u2102) ^ 2 * z).<a>re</a> = -\u03c0 * z.im * (n : \u211d) ^ 2 := by\n      rw [(by push_cast; ring : \u2191\u03c0 * I * (n : \u2102) ^ 2 * z = \u2191(\u03c0 * (n : \u211d) ^ 2) * (z * I)),\n        <a>ofReal_mul_re</a>, <a>mul_I_re</a>]\n      ring", [{"full_name": "Complex.I", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [293, 5], "def_end_pos": [293, 6]}, {"full_name": "Complex.re", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [29, 3], "def_end_pos": [29, 5]}, {"full_name": "Complex.ofReal_mul_re", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [279, 9], "def_end_pos": [279, 22]}, {"full_name": "Complex.mul_I_re", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [336, 9], "def_end_pos": [336, 17]}]], "state_before": "case refine'_1\nz : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\n\u22a2 rexp (\u2191\u03c0 * I * \u2191n ^ 2 * z).re = y ^ n ^ 2", "state_after": "case refine'_1\nz : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\nthis : (\u2191\u03c0 * I * \u2191n ^ 2 * z).re = -\u03c0 * z.im * \u2191n ^ 2\n\u22a2 rexp (\u2191\u03c0 * I * \u2191n ^ 2 * z).re = y ^ n ^ 2"}, {"tactic": "obtain \u27e8m, hm\u27e9 := Int.eq_ofNat_of_zero_le (sq_nonneg n)", "annotated_tactic": ["obtain \u27e8m, hm\u27e9 := <a>Int.eq_ofNat_of_zero_le</a> (<a>sq_nonneg</a> n)", [{"full_name": "Int.eq_ofNat_of_zero_le", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [584, 9], "def_end_pos": [584, 28]}, {"full_name": "sq_nonneg", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [645, 9], "def_end_pos": [645, 18]}]], "state_before": "case refine'_1\nz : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\nthis : (\u2191\u03c0 * I * \u2191n ^ 2 * z).re = -\u03c0 * z.im * \u2191n ^ 2\n\u22a2 rexp (\u2191\u03c0 * I * \u2191n ^ 2 * z).re = y ^ n ^ 2", "state_after": "case refine'_1.intro\nz : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\nthis : (\u2191\u03c0 * I * \u2191n ^ 2 * z).re = -\u03c0 * z.im * \u2191n ^ 2\nm : \u2115\nhm : n ^ 2 = \u2191m\n\u22a2 rexp (\u2191\u03c0 * I * \u2191n ^ 2 * z).re = y ^ n ^ 2"}, {"tactic": "rw [this, exp_mul, \u2190 Int.cast_pow, rpow_int_cast, hm, zpow_ofNat]", "annotated_tactic": ["rw [this, <a>exp_mul</a>, \u2190 <a>Int.cast_pow</a>, <a>rpow_int_cast</a>, hm, <a>zpow_ofNat</a>]", [{"full_name": "Real.exp_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [58, 9], "def_end_pos": [58, 16]}, {"full_name": "Int.cast_pow", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [624, 9], "def_end_pos": [624, 21]}, {"full_name": "Real.rpow_int_cast", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [351, 9], "def_end_pos": [351, 22]}, {"full_name": "zpow_ofNat", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [948, 9], "def_end_pos": [948, 19]}]], "state_before": "case refine'_1.intro\nz : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\nthis : (\u2191\u03c0 * I * \u2191n ^ 2 * z).re = -\u03c0 * z.im * \u2191n ^ 2\nm : \u2115\nhm : n ^ 2 = \u2191m\n\u22a2 rexp (\u2191\u03c0 * I * \u2191n ^ 2 * z).re = y ^ n ^ 2", "state_after": "no goals"}, {"tactic": "rw [(by push_cast; ring : \u2191\u03c0 * I * (n : \u2102) ^ 2 * z = \u2191(\u03c0 * (n : \u211d) ^ 2) * (z * I)),\n  ofReal_mul_re, mul_I_re]", "annotated_tactic": ["rw [(by push_cast; ring : \u2191\u03c0 * I * (n : \u2102) ^ 2 * z = \u2191(\u03c0 * (n : \u211d) ^ 2) * (z * I)),\n        <a>ofReal_mul_re</a>, <a>mul_I_re</a>]", [{"full_name": "Complex.ofReal_mul_re", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [279, 9], "def_end_pos": [279, 22]}, {"full_name": "Complex.mul_I_re", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [336, 9], "def_end_pos": [336, 17]}]], "state_before": "z : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\n\u22a2 (\u2191\u03c0 * I * \u2191n ^ 2 * z).re = -\u03c0 * z.im * \u2191n ^ 2", "state_after": "z : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\n\u22a2 \u03c0 * \u2191n ^ 2 * -z.im = -\u03c0 * z.im * \u2191n ^ 2"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "z : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\n\u22a2 \u03c0 * \u2191n ^ 2 * -z.im = -\u03c0 * z.im * \u2191n ^ 2", "state_after": "no goals"}, {"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "state_before": "z : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\n\u22a2 \u2191\u03c0 * I * \u2191n ^ 2 * z = \u2191(\u03c0 * \u2191n ^ 2) * (z * I)", "state_after": "z : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\n\u22a2 \u2191\u03c0 * I * \u2191n ^ 2 * z = \u2191\u03c0 * \u2191n ^ 2 * (z * I)"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "z : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\n\u22a2 \u2191\u03c0 * I * \u2191n ^ 2 * z = \u2191\u03c0 * \u2191n ^ 2 * (z * I)", "state_after": "no goals"}, {"tactic": "have : n ^ 2 = \u2191(n.natAbs ^ 2) := by rw [Nat.cast_pow, Int.natAbs_sq]", "annotated_tactic": ["have : n ^ 2 = \u2191(n.natAbs ^ 2) := by rw [<a>Nat.cast_pow</a>, <a>Int.natAbs_sq</a>]", [{"full_name": "Nat.cast_pow", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [533, 9], "def_end_pos": [533, 21]}, {"full_name": "Int.natAbs_sq", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [772, 7], "def_end_pos": [772, 16]}]], "state_before": "case refine'_2\nz : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\n\u22a2 y ^ n ^ 2 \u2264 rexp (-\u03c0 * z.im) ^ Int.natAbs n", "state_after": "case refine'_2\nz : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\nthis : n ^ 2 = \u2191(Int.natAbs n ^ 2)\n\u22a2 y ^ n ^ 2 \u2264 rexp (-\u03c0 * z.im) ^ Int.natAbs n"}, {"tactic": "rw [this, zpow_ofNat]", "annotated_tactic": ["rw [this, <a>zpow_ofNat</a>]", [{"full_name": "zpow_ofNat", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [948, 9], "def_end_pos": [948, 19]}]], "state_before": "case refine'_2\nz : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\nthis : n ^ 2 = \u2191(Int.natAbs n ^ 2)\n\u22a2 y ^ n ^ 2 \u2264 rexp (-\u03c0 * z.im) ^ Int.natAbs n", "state_after": "case refine'_2\nz : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\nthis : n ^ 2 = \u2191(Int.natAbs n ^ 2)\n\u22a2 y ^ Int.natAbs n ^ 2 \u2264 rexp (-\u03c0 * z.im) ^ Int.natAbs n"}, {"tactic": "exact pow_le_pow_of_le_one (exp_pos _).le h.le ((sq n.natAbs).symm \u25b8 n.natAbs.le_mul_self)", "annotated_tactic": ["exact <a>pow_le_pow_of_le_one</a> (<a>exp_pos</a> _).<a>le</a> h.le ((<a>sq</a> n.natAbs).<a>symm</a> \u25b8 n.natAbs.le_mul_self)", [{"full_name": "pow_le_pow_of_le_one", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [670, 9], "def_end_pos": [670, 29]}, {"full_name": "Real.exp_pos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1499, 9], "def_end_pos": [1499, 16]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}, {"full_name": "sq", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [106, 7], "def_end_pos": [106, 9]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case refine'_2\nz : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\nthis : n ^ 2 = \u2191(Int.natAbs n ^ 2)\n\u22a2 y ^ Int.natAbs n ^ 2 \u2264 rexp (-\u03c0 * z.im) ^ Int.natAbs n", "state_after": "no goals"}, {"tactic": "rw [Nat.cast_pow, Int.natAbs_sq]", "annotated_tactic": ["rw [<a>Nat.cast_pow</a>, <a>Int.natAbs_sq</a>]", [{"full_name": "Nat.cast_pow", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [533, 9], "def_end_pos": [533, 21]}, {"full_name": "Int.natAbs_sq", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [772, 7], "def_end_pos": [772, 16]}]], "state_before": "z : \u2102\nhz : 0 < z.im\nn : \u2124\ny : \u211d := rexp (-\u03c0 * z.im)\nh : y < 1\n\u22a2 n ^ 2 = \u2191(Int.natAbs n ^ 2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "full_name": "MeasureTheory.Measure.rnDeriv_lt_top", "start": [212, 1], "end": [219, 45], "traced_tactics": [{"tactic": "suffices \u2200 n, \u2200\u1d50 x \u2202\u03bd, x \u2208 spanningSets \u03bc n \u2192 \u03bc.rnDeriv \u03bd x < \u221e by\n  filter_upwards [ae_all_iff.2 this] with _ hx using hx _ (mem_spanningSetsIndex _ _)", "annotated_tactic": ["suffices \u2200 n, \u2200\u1d50 x \u2202\u03bd, x \u2208 <a>spanningSets</a> \u03bc n \u2192 \u03bc.rnDeriv \u03bd x < \u221e by\n    filter_upwards [<a>ae_all_iff</a>.2 this] with _ hx using hx _ (<a>mem_spanningSetsIndex</a> _ _)", [{"full_name": "MeasureTheory.spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3316, 5], "def_end_pos": [3316, 17]}, {"full_name": "MeasureTheory.ae_all_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [422, 9], "def_end_pos": [422, 19]}, {"full_name": "MeasureTheory.mem_spanningSetsIndex", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3368, 9], "def_end_pos": [3368, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bd, rnDeriv \u03bc \u03bd x < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u22a2 \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bd, x \u2208 spanningSets \u03bc n \u2192 rnDeriv \u03bc \u03bd x < \u22a4"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u22a2 \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bd, x \u2208 spanningSets \u03bc n \u2192 rnDeriv \u03bc \u03bd x < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nn : \u2115\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bd, x \u2208 spanningSets \u03bc n \u2192 rnDeriv \u03bc \u03bd x < \u22a4"}, {"tactic": "rw [\u2190 ae_restrict_iff' (measurable_spanningSets _ _)]", "annotated_tactic": ["rw [\u2190 <a>ae_restrict_iff'</a> (<a>measurable_spanningSets</a> _ _)]", [{"full_name": "MeasureTheory.ae_restrict_iff'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [2572, 9], "def_end_pos": [2572, 25]}, {"full_name": "MeasureTheory.measurable_spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3324, 9], "def_end_pos": [3324, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nn : \u2115\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bd, x \u2208 spanningSets \u03bc n \u2192 rnDeriv \u03bc \u03bd x < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nn : \u2115\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202restrict \u03bd (spanningSets \u03bc n), rnDeriv \u03bc \u03bd x < \u22a4"}, {"tactic": "apply ae_lt_top (measurable_rnDeriv _ _)", "annotated_tactic": ["apply <a>ae_lt_top</a> (<a>measurable_rnDeriv</a> _ _)", [{"full_name": "MeasureTheory.ae_lt_top", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1522, 9], "def_end_pos": [1522, 18]}, {"full_name": "MeasureTheory.Measure.measurable_rnDeriv", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [124, 9], "def_end_pos": [124, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nn : \u2115\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202restrict \u03bd (spanningSets \u03bc n), rnDeriv \u03bc \u03bd x < \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nn : \u2115\n\u22a2 \u222b\u207b (x : \u03b1) in spanningSets \u03bc n, rnDeriv \u03bc \u03bd x \u2202\u03bd \u2260 \u22a4"}, {"tactic": "refine' (lintegral_rnDeriv_lt_top_of_measure_ne_top _ _).ne", "annotated_tactic": ["refine' (<a>lintegral_rnDeriv_lt_top_of_measure_ne_top</a> _ _).<a>ne</a>", [{"full_name": "MeasureTheory.Measure.lintegral_rnDeriv_lt_top_of_measure_ne_top", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [185, 9], "def_end_pos": [185, 51]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nn : \u2115\n\u22a2 \u222b\u207b (x : \u03b1) in spanningSets \u03bc n, rnDeriv \u03bc \u03bd x \u2202\u03bd \u2260 \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (spanningSets \u03bc n) \u2260 \u22a4"}, {"tactic": "exact (measure_spanningSets_lt_top _ _).ne", "annotated_tactic": ["exact (<a>measure_spanningSets_lt_top</a> _ _).<a>ne</a>", [{"full_name": "MeasureTheory.measure_spanningSets_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3329, 9], "def_end_pos": [3329, 36]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nn : \u2115\n\u22a2 \u2191\u2191\u03bc (spanningSets \u03bc n) \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "filter_upwards [ae_all_iff.2 this] with _ hx using hx _ (mem_spanningSetsIndex _ _)", "annotated_tactic": ["filter_upwards [<a>ae_all_iff</a>.2 this] with _ hx using hx _ (<a>mem_spanningSetsIndex</a> _ _)", [{"full_name": "MeasureTheory.ae_all_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [422, 9], "def_end_pos": [422, 19]}, {"full_name": "MeasureTheory.mem_spanningSetsIndex", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3368, 9], "def_end_pos": [3368, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d \u03bc \u03bd : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\nthis : \u2200 (n : \u2115), \u2200\u1d50 (x : \u03b1) \u2202\u03bd, x \u2208 spanningSets \u03bc n \u2192 rnDeriv \u03bc \u03bd x < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bd, rnDeriv \u03bc \u03bd x < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Determinant.lean", "full_name": "Basis.det_unitsSMul", "start": [664, 1], "end": [673, 12], "traced_tactics": [{"tactic": "ext f", "annotated_tactic": ["ext f", []], "state_before": "R : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\ne : Basis \u03b9 R M\nw : \u03b9 \u2192 R\u02e3\n\u22a2 det (unitsSMul e w) = \u2191(\u220f i : \u03b9, w i)\u207b\u00b9 \u2022 det e", "state_after": "case H\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\ne : Basis \u03b9 R M\nw : \u03b9 \u2192 R\u02e3\nf : \u03b9 \u2192 M\n\u22a2 \u2191(det (unitsSMul e w)) f = \u2191(\u2191(\u220f i : \u03b9, w i)\u207b\u00b9 \u2022 det e) f"}, {"tactic": "change\n  (Matrix.det fun i j => (e.unitsSMul w).repr (f j) i) =\n    (\u2191(\u220f i, w i)\u207b\u00b9 : R) \u2022 Matrix.det fun i j => e.repr (f j) i", "annotated_tactic": ["change\n    (<a>Matrix.det</a> fun i j => (e.unitsSMul w).<a>repr</a> (f j) i) =\n      (\u2191(\u220f i, w i)\u207b\u00b9 : R) \u2022 <a>Matrix.det</a> fun i j => e.repr (f j) i", [{"full_name": "Matrix.det", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [65, 8], "def_end_pos": [65, 11]}, {"full_name": "Basis.repr", "def_path": "Mathlib/LinearAlgebra/Basis.lean", "def_pos": [96, 5], "def_end_pos": [96, 9]}, {"full_name": "Matrix.det", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [65, 8], "def_end_pos": [65, 11]}]], "state_before": "case H\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\ne : Basis \u03b9 R M\nw : \u03b9 \u2192 R\u02e3\nf : \u03b9 \u2192 M\n\u22a2 \u2191(det (unitsSMul e w)) f = \u2191(\u2191(\u220f i : \u03b9, w i)\u207b\u00b9 \u2022 det e) f", "state_after": "case H\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\ne : Basis \u03b9 R M\nw : \u03b9 \u2192 R\u02e3\nf : \u03b9 \u2192 M\n\u22a2 (Matrix.det fun i j => \u2191(\u2191(unitsSMul e w).repr (f j)) i) =\n    \u2191(\u220f i : \u03b9, w i)\u207b\u00b9 \u2022 Matrix.det fun i j => \u2191(\u2191e.repr (f j)) i"}, {"tactic": "simp only [e.repr_unitsSMul]", "annotated_tactic": ["simp only [e.repr_unitsSMul]", []], "state_before": "case H\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\ne : Basis \u03b9 R M\nw : \u03b9 \u2192 R\u02e3\nf : \u03b9 \u2192 M\n\u22a2 (Matrix.det fun i j => \u2191(\u2191(unitsSMul e w).repr (f j)) i) =\n    \u2191(\u220f i : \u03b9, w i)\u207b\u00b9 \u2022 Matrix.det fun i j => \u2191(\u2191e.repr (f j)) i", "state_after": "case H\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\ne : Basis \u03b9 R M\nw : \u03b9 \u2192 R\u02e3\nf : \u03b9 \u2192 M\n\u22a2 (Matrix.det fun i j => (w i)\u207b\u00b9 \u2022 \u2191(\u2191e.repr (f j)) i) = \u2191(\u220f i : \u03b9, w i)\u207b\u00b9 \u2022 Matrix.det fun i j => \u2191(\u2191e.repr (f j)) i"}, {"tactic": "convert Matrix.det_mul_column (fun i => (\u2191(w i)\u207b\u00b9 : R)) fun i j => e.repr (f j) i", "annotated_tactic": ["convert <a>Matrix.det_mul_column</a> (fun i => (\u2191(w i)\u207b\u00b9 : R)) fun i j => e.repr (f j) i", [{"full_name": "Matrix.det_mul_column", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [313, 9], "def_end_pos": [313, 23]}]], "state_before": "case H\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\ne : Basis \u03b9 R M\nw : \u03b9 \u2192 R\u02e3\nf : \u03b9 \u2192 M\n\u22a2 (Matrix.det fun i j => (w i)\u207b\u00b9 \u2022 \u2191(\u2191e.repr (f j)) i) = \u2191(\u220f i : \u03b9, w i)\u207b\u00b9 \u2022 Matrix.det fun i j => \u2191(\u2191e.repr (f j)) i", "state_after": "case h.e'_3.h.e'_1\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\ne : Basis \u03b9 R M\nw : \u03b9 \u2192 R\u02e3\nf : \u03b9 \u2192 M\n\u22a2 \u2191(\u220f i : \u03b9, w i)\u207b\u00b9 = \u220f i : \u03b9, \u2191(w i)\u207b\u00b9"}, {"tactic": "simp only [\u2190 Finset.prod_inv_distrib]", "annotated_tactic": ["simp only [\u2190 <a>Finset.prod_inv_distrib</a>]", [{"full_name": "Finset.prod_inv_distrib", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1815, 9], "def_end_pos": [1815, 25]}]], "state_before": "case h.e'_3.h.e'_1\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\ne : Basis \u03b9 R M\nw : \u03b9 \u2192 R\u02e3\nf : \u03b9 \u2192 M\n\u22a2 \u2191(\u220f i : \u03b9, w i)\u207b\u00b9 = \u220f i : \u03b9, \u2191(w i)\u207b\u00b9", "state_after": "case h.e'_3.h.e'_1\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\ne : Basis \u03b9 R M\nw : \u03b9 \u2192 R\u02e3\nf : \u03b9 \u2192 M\n\u22a2 \u2191(\u220f x : \u03b9, (w x)\u207b\u00b9) = \u220f i : \u03b9, \u2191(w i)\u207b\u00b9"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "case h.e'_3.h.e'_1\nR : Type u_1\ninst\u271d\u2078 : CommRing R\nM : Type u_2\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\nM' : Type u_3\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b3 : DecidableEq \u03b9\ninst\u271d\u00b2 : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\nA : Type u_5\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Module A M\ne : Basis \u03b9 R M\nw : \u03b9 \u2192 R\u02e3\nf : \u03b9 \u2192 M\n\u22a2 \u2191(\u220f x : \u03b9, (w x)\u207b\u00b9) = \u220f i : \u03b9, \u2191(w i)\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "full_name": "List.bind_id", "start": [110, 9], "end": [110, 93], "traced_tactics": [{"tactic": "simp [List.bind]", "annotated_tactic": ["simp [<a>List.bind</a>]", [{"full_name": "List.bind", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [643, 25], "def_end_pos": [643, 29]}]], "state_before": "\u03b1 : Type u_1\nl : List (List \u03b1)\n\u22a2 List.bind l id = join l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Basic.lean", "full_name": "Metric.hasBasis_cobounded_compl_closedBall", "start": [2429, 1], "end": [2431, 91], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr : \u211d\nc : \u03b1\nx\u271d : Set \u03b1\n\u22a2 (\u2203 r, x\u271d \u2286 closedBall c r) \u2194 \u2203 i, True \u2227 (closedBall c i)\u1d9c \u2286 x\u271d\u1d9c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Complex/Basic.lean", "full_name": "Complex.dist_self_conj", "start": [137, 1], "end": [137, 99], "traced_tactics": [{"tactic": "rw [dist_comm, dist_conj_self]", "annotated_tactic": ["rw [<a>dist_comm</a>, <a>dist_conj_self</a>]", [{"full_name": "dist_comm", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [188, 9], "def_end_pos": [188, 18]}, {"full_name": "Complex.dist_conj_self", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [128, 9], "def_end_pos": [128, 23]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nz : \u2102\n\u22a2 dist z (\u2191(starRingEnd \u2102) z) = 2 * |z.im|", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Game/Domineering.lean", "full_name": "SetTheory.PGame.Domineering.moveRight_card", "start": [119, 1], "end": [124, 52], "traced_tactics": [{"tactic": "dsimp [moveRight]", "annotated_tactic": ["dsimp [<a>moveRight</a>]", [{"full_name": "SetTheory.PGame.Domineering.moveRight", "def_path": "Mathlib/SetTheory/Game/Domineering.lean", "def_pos": [77, 5], "def_end_pos": [77, 14]}]], "state_before": "b : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 right b\n\u22a2 Finset.card (moveRight b m) + 2 = Finset.card b", "state_after": "b : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 right b\n\u22a2 Finset.card (Finset.erase (Finset.erase b m) (m.1 - 1, m.2)) + 2 = Finset.card b"}, {"tactic": "rw [Finset.card_erase_of_mem (fst_pred_mem_erase_of_mem_right h)]", "annotated_tactic": ["rw [<a>Finset.card_erase_of_mem</a> (<a>fst_pred_mem_erase_of_mem_right</a> h)]", [{"full_name": "Finset.card_erase_of_mem", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [135, 9], "def_end_pos": [135, 26]}, {"full_name": "SetTheory.PGame.Domineering.fst_pred_mem_erase_of_mem_right", "def_path": "Mathlib/SetTheory/Game/Domineering.lean", "def_pos": [81, 9], "def_end_pos": [81, 40]}]], "state_before": "b : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 right b\n\u22a2 Finset.card (Finset.erase (Finset.erase b m) (m.1 - 1, m.2)) + 2 = Finset.card b", "state_after": "b : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 right b\n\u22a2 Finset.card (Finset.erase b m) - 1 + 2 = Finset.card b"}, {"tactic": "rw [Finset.card_erase_of_mem (Finset.mem_of_mem_inter_left h)]", "annotated_tactic": ["rw [<a>Finset.card_erase_of_mem</a> (<a>Finset.mem_of_mem_inter_left</a> h)]", [{"full_name": "Finset.card_erase_of_mem", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [135, 9], "def_end_pos": [135, 26]}, {"full_name": "Finset.mem_of_mem_inter_left", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1605, 9], "def_end_pos": [1605, 30]}]], "state_before": "b : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 right b\n\u22a2 Finset.card (Finset.erase b m) - 1 + 2 = Finset.card b", "state_after": "b : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 right b\n\u22a2 Finset.card b - 1 - 1 + 2 = Finset.card b"}, {"tactic": "exact tsub_add_cancel_of_le (card_of_mem_right h)", "annotated_tactic": ["exact <a>tsub_add_cancel_of_le</a> (<a>card_of_mem_right</a> h)", [{"full_name": "tsub_add_cancel_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [30, 9], "def_end_pos": [30, 30]}, {"full_name": "SetTheory.PGame.Domineering.card_of_mem_right", "def_path": "Mathlib/SetTheory/Game/Domineering.lean", "def_pos": [103, 9], "def_end_pos": [103, 26]}]], "state_before": "b : Board\nm : \u2124 \u00d7 \u2124\nh : m \u2208 right b\n\u22a2 Finset.card b - 1 - 1 + 2 = Finset.card b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.univ_pi_subset_univ_pi_iff", "start": [886, 1], "end": [887, 29], "traced_tactics": [{"tactic": "simp [pi_subset_pi_iff]", "annotated_tactic": ["simp [<a>pi_subset_pi_iff</a>]", [{"full_name": "Set.pi_subset_pi_iff", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [877, 9], "def_end_pos": [877, 25]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : \u03b9 \u2192 Type u_3\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\ni : \u03b9\n\u22a2 pi univ t\u2081 \u2286 pi univ t\u2082 \u2194 (\u2200 (i : \u03b9), t\u2081 i \u2286 t\u2082 i) \u2228 \u2203 i, t\u2081 i = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/EraseLead.lean", "full_name": "Polynomial.eraseLead_card_support", "start": [114, 1], "end": [118, 88], "traced_tactics": [{"tactic": "by_cases f0 : f = 0", "annotated_tactic": ["by_cases f0 : f = 0", []], "state_before": "R : Type u_1\ninst\u271d : Semiring R\nf : R[X]\nc : \u2115\nfc : card (support f) = c\n\u22a2 card (support (eraseLead f)) = c - 1", "state_after": "case pos\nR : Type u_1\ninst\u271d : Semiring R\nf : R[X]\nc : \u2115\nfc : card (support f) = c\nf0 : f = 0\n\u22a2 card (support (eraseLead f)) = c - 1\n\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\nf : R[X]\nc : \u2115\nfc : card (support f) = c\nf0 : 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"def_end_pos": [30, 10]}, {"full_name": "Ordering.swap", "def_path": "lake-packages/std/Std/Data/Ord.lean", "def_pos": [12, 5], "def_end_pos": [12, 9]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : IsTotal \u03b1 fun x x_1 => x \u2264 x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nx y : \u03b1\n\u22a2 Ordering.swap (cmpLE x y) = cmpLE y x", "state_after": "case neg\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : LE \u03b1\ninst\u271d\u00b9 : IsTotal \u03b1 fun x x_1 => x \u2264 x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2264 x_1\nx y : \u03b1\nxy : \u00acx \u2264 y\nyx : \u00acy \u2264 x\n\u22a2 False"}, {"tactic": "cases not_or_of_not xy yx (total_of _ _ _)", "annotated_tactic": ["cases <a>not_or_of_not</a> xy yx (<a>total_of</a> _ _ _)", [{"full_name": "not_or_of_not", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [193, 9], "def_end_pos": [193, 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"rw [\u2190 get?_eq_get, eq, get?_append_right (h \u25b8 Nat.le_refl _), h, Nat.sub_self]", "annotated_tactic": ["rw [\u2190 <a>get?_eq_get</a>, eq, <a>get?_append_right</a> (h \u25b8 <a>Nat.le_refl</a> _), h, <a>Nat.sub_self</a>]", [{"full_name": "List.get?_eq_get", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [581, 9], "def_end_pos": [581, 20]}, {"full_name": "List.get?_append_right", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [682, 9], "def_end_pos": [682, 26]}, {"full_name": "Nat.le_refl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1605, 19], "def_end_pos": [1605, 30]}, {"full_name": "Nat.sub_self", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [250, 27], "def_end_pos": [250, 35]}]], "state_before": "\u03b1 : Type u_1\nl\u2081 : List \u03b1\na : \u03b1\nl\u2082 : List \u03b1\nn : Nat\nl : List \u03b1\neq : l = l\u2081 ++ a :: l\u2082\nh : length l\u2081 = n\n\u22a2 some (get l { val := n, isLt := (_ : n < length l) }) = some a", "state_after": "\u03b1 : Type u_1\nl\u2081 : List \u03b1\na : \u03b1\nl\u2082 : List \u03b1\nn : Nat\nl : List \u03b1\neq : l = l\u2081 ++ a :: l\u2082\nh : length l\u2081 = n\n\u22a2 get? (a :: l\u2082) 0 = some a"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\nl\u2081 : List \u03b1\na : \u03b1\nl\u2082 : List \u03b1\nn : Nat\nl : List \u03b1\neq : l = l\u2081 ++ a :: l\u2082\nh : length l\u2081 = n\n\u22a2 get? (a :: l\u2082) 0 = some a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Connected/Basic.lean", "full_name": "isPreconnected_iff_subset_of_disjoint_closed", "start": [1074, 1], "end": [1099, 35], "traced_tactics": [{"tactic": "constructor <;> intro h", "annotated_tactic": ["constructor <;> intro h", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 IsPreconnected s \u2194 \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v", "state_after": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : IsPreconnected s\n\u22a2 \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\n\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\n\u22a2 IsPreconnected s"}, {"tactic": "intro u v hu hv hs huv", "annotated_tactic": ["intro u v hu hv hs huv", []], "state_before": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : IsPreconnected s\n\u22a2 \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v", "state_after": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d : Set \u03b1\nh : IsPreconnected s\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhuv : s \u2229 (u \u2229 v) = \u2205\n\u22a2 s \u2286 u \u2228 s \u2286 v"}, {"tactic": "rw [isPreconnected_closed_iff] at h", "annotated_tactic": ["rw [<a>isPreconnected_closed_iff</a>] at h", [{"full_name": "isPreconnected_closed_iff", "def_path": "Mathlib/Topology/Connected/Basic.lean", "def_pos": [337, 9], "def_end_pos": [337, 34]}]], "state_before": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d : Set \u03b1\nh : IsPreconnected s\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhuv : s \u2229 (u \u2229 v) = \u2205\n\u22a2 s \u2286 u \u2228 s \u2286 v", "state_after": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d : Set \u03b1\nh :\n  \u2200 (t t' : Set \u03b1),\n    IsClosed t \u2192 IsClosed t' \u2192 s \u2286 t \u222a t' \u2192 Set.Nonempty (s \u2229 t) \u2192 Set.Nonempty (s \u2229 t') \u2192 Set.Nonempty (s \u2229 (t \u2229 t'))\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhuv : s \u2229 (u \u2229 v) = \u2205\n\u22a2 s \u2286 u \u2228 s \u2286 v"}, {"tactic": "specialize h u v hu hv hs", "annotated_tactic": ["specialize h u v hu hv hs", []], "state_before": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d : Set \u03b1\nh :\n  \u2200 (t t' : Set \u03b1),\n    IsClosed t \u2192 IsClosed t' \u2192 s \u2286 t \u222a t' \u2192 Set.Nonempty (s \u2229 t) \u2192 Set.Nonempty (s \u2229 t') \u2192 Set.Nonempty (s \u2229 (t \u2229 t'))\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhuv : s \u2229 (u \u2229 v) = \u2205\n\u22a2 s \u2286 u \u2228 s \u2286 v", "state_after": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhuv : s \u2229 (u \u2229 v) = \u2205\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\n\u22a2 s \u2286 u \u2228 s \u2286 v"}, {"tactic": "contrapose! huv", "annotated_tactic": ["contrapose! huv", []], "state_before": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhuv : s \u2229 (u \u2229 v) = \u2205\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\n\u22a2 s \u2286 u \u2228 s \u2286 v", "state_after": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nhuv : \u00acs \u2286 u \u2227 \u00acs \u2286 v\n\u22a2 s \u2229 (u \u2229 v) \u2260 \u2205"}, {"tactic": "rw [\u2190 nonempty_iff_ne_empty]", "annotated_tactic": ["rw [\u2190 <a>nonempty_iff_ne_empty</a>]", [{"full_name": "Set.nonempty_iff_ne_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [610, 9], "def_end_pos": [610, 30]}]], "state_before": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nhuv : \u00acs \u2286 u \u2227 \u00acs \u2286 v\n\u22a2 s \u2229 (u \u2229 v) \u2260 \u2205", "state_after": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nhuv : \u00acs \u2286 u \u2227 \u00acs \u2286 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))"}, {"tactic": "simp [not_subset] at huv", "annotated_tactic": ["simp [<a>not_subset</a>] at huv", [{"full_name": "Set.not_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [391, 9], "def_end_pos": [391, 19]}]], "state_before": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nhuv : \u00acs \u2286 u \u2227 \u00acs \u2286 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))", "state_after": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nhuv : (\u2203 a, a \u2208 s \u2227 \u00aca \u2208 u) \u2227 \u2203 a, a \u2208 s \u2227 \u00aca \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))"}, {"tactic": "rcases huv with \u27e8\u27e8x, hxs, hxu\u27e9, \u27e8y, hys, hyv\u27e9\u27e9", "annotated_tactic": ["rcases huv with \u27e8\u27e8x, hxs, hxu\u27e9, \u27e8y, hys, hyv\u27e9\u27e9", []], "state_before": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nhuv : (\u2203 a, a \u2208 s \u2227 \u00aca \u2208 u) \u2227 \u2203 a, a \u2208 s \u2227 \u00aca \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))", "state_after": "case mp.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nx : \u03b1\nhxs : x \u2208 s\nhxu : \u00acx \u2208 u\ny : \u03b1\nhys : y \u2208 s\nhyv : \u00acy \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))"}, {"tactic": "have hxv : x \u2208 v := or_iff_not_imp_left.mp (hs hxs) hxu", "annotated_tactic": ["have hxv : x \u2208 v := or_iff_not_imp_left.mp (hs hxs) hxu", []], "state_before": "case mp.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nx : \u03b1\nhxs : x \u2208 s\nhxu : \u00acx \u2208 u\ny : \u03b1\nhys : y \u2208 s\nhyv : \u00acy \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))", "state_after": "case mp.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nx : \u03b1\nhxs : x \u2208 s\nhxu : \u00acx \u2208 u\ny : \u03b1\nhys : y \u2208 s\nhyv : \u00acy \u2208 v\nhxv : x \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))"}, {"tactic": "have hyu : y \u2208 u := or_iff_not_imp_right.mp (hs hys) hyv", "annotated_tactic": ["have hyu : y \u2208 u := or_iff_not_imp_right.mp (hs hys) hyv", []], "state_before": "case mp.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nx : \u03b1\nhxs : x \u2208 s\nhxu : \u00acx \u2208 u\ny : \u03b1\nhys : y \u2208 s\nhyv : \u00acy \u2208 v\nhxv : x \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))", "state_after": "case mp.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nx : \u03b1\nhxs : x \u2208 s\nhxu : \u00acx \u2208 u\ny : \u03b1\nhys : y \u2208 s\nhyv : \u00acy \u2208 v\nhxv : x \u2208 v\nhyu : y \u2208 u\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))"}, {"tactic": "exact h \u27e8y, hys, hyu\u27e9 \u27e8x, hxs, hxv\u27e9", "annotated_tactic": ["exact h \u27e8y, hys, hyu\u27e9 \u27e8x, hxs, hxv\u27e9", []], "state_before": "case mp.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nx : \u03b1\nhxs : x \u2208 s\nhxu : \u00acx \u2208 u\ny : \u03b1\nhys : y \u2208 s\nhyv : \u00acy \u2208 v\nhxv : x \u2208 v\nhyu : y \u2208 u\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))", "state_after": "no goals"}, {"tactic": "rw [isPreconnected_closed_iff]", "annotated_tactic": ["rw [<a>isPreconnected_closed_iff</a>]", [{"full_name": "isPreconnected_closed_iff", "def_path": "Mathlib/Topology/Connected/Basic.lean", "def_pos": [337, 9], "def_end_pos": [337, 34]}]], "state_before": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\n\u22a2 IsPreconnected s", "state_after": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\n\u22a2 \u2200 (t t' : Set \u03b1),\n    IsClosed t \u2192 IsClosed t' \u2192 s \u2286 t \u222a t' \u2192 Set.Nonempty (s \u2229 t) \u2192 Set.Nonempty (s \u2229 t') \u2192 Set.Nonempty (s \u2229 (t \u2229 t'))"}, {"tactic": "intro u v hu hv hs hsu hsv", "annotated_tactic": ["intro u v hu hv hs hsu hsv", []], "state_before": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\n\u22a2 \u2200 (t t' : Set \u03b1),\n    IsClosed t \u2192 IsClosed t' \u2192 s \u2286 t \u222a t' \u2192 Set.Nonempty (s \u2229 t) \u2192 Set.Nonempty (s \u2229 t') \u2192 Set.Nonempty (s \u2229 (t \u2229 t'))", "state_after": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d : Set \u03b1\nh : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))"}, {"tactic": "rw [nonempty_iff_ne_empty]", "annotated_tactic": ["rw [<a>nonempty_iff_ne_empty</a>]", [{"full_name": "Set.nonempty_iff_ne_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [610, 9], "def_end_pos": [610, 30]}]], "state_before": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d : Set \u03b1\nh : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))", "state_after": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d : Set \u03b1\nh : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\n\u22a2 s \u2229 (u \u2229 v) \u2260 \u2205"}, {"tactic": "intro H", "annotated_tactic": ["intro H", []], "state_before": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d : Set \u03b1\nh : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\n\u22a2 s \u2229 (u \u2229 v) \u2260 \u2205", "state_after": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d : Set \u03b1\nh : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nH : s \u2229 (u \u2229 v) = \u2205\n\u22a2 False"}, {"tactic": "specialize h u v hu hv hs H", "annotated_tactic": ["specialize h u v hu hv hs H", []], "state_before": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d : Set \u03b1\nh : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nH : s \u2229 (u \u2229 v) = \u2205\n\u22a2 False", "state_after": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nH : s \u2229 (u \u2229 v) = \u2205\nh : s \u2286 u \u2228 s \u2286 v\n\u22a2 False"}, {"tactic": "contrapose H", "annotated_tactic": ["contrapose H", []], "state_before": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nH : s \u2229 (u \u2229 v) = \u2205\nh : s \u2286 u \u2228 s \u2286 v\n\u22a2 False", "state_after": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nh : s \u2286 u \u2228 s \u2286 v\nH : \u00acFalse\n\u22a2 \u00acs \u2229 (u \u2229 v) = \u2205"}, {"tactic": "apply Nonempty.ne_empty", "annotated_tactic": ["apply <a>Nonempty.ne_empty</a>", [{"full_name": "Set.Nonempty.ne_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [622, 8], "def_end_pos": [622, 25]}]], "state_before": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nh : s \u2286 u \u2228 s \u2286 v\nH : \u00acFalse\n\u22a2 \u00acs \u2229 (u \u2229 v) = \u2205", "state_after": "case mpr.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nh : s \u2286 u \u2228 s \u2286 v\nH : \u00acFalse\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))"}, {"tactic": "cases' h with h h", "annotated_tactic": ["cases' h with h h", []], "state_before": "case mpr.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nh : s \u2286 u \u2228 s \u2286 v\nH : \u00acFalse\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))", "state_after": "case mpr.a.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nH : \u00acFalse\nh : s \u2286 u\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n\ncase mpr.a.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nH : \u00acFalse\nh : s \u2286 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))"}, {"tactic": "rcases hsv with \u27e8x, hxs, hxv\u27e9", "annotated_tactic": ["rcases hsv with \u27e8x, hxs, hxv\u27e9", []], "state_before": "case mpr.a.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nH : \u00acFalse\nh : s \u2286 u\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))", "state_after": "case mpr.a.inl.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nH : \u00acFalse\nh : s \u2286 u\nx : \u03b1\nhxs : x \u2208 s\nhxv : x \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))"}, {"tactic": "exact \u27e8x, hxs, \u27e8h hxs, hxv\u27e9\u27e9", "annotated_tactic": ["exact \u27e8x, hxs, \u27e8h hxs, hxv\u27e9\u27e9", []], "state_before": "case mpr.a.inl.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nH : \u00acFalse\nh : s \u2286 u\nx : \u03b1\nhxs : x \u2208 s\nhxv : x \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))", "state_after": "no goals"}, {"tactic": "rcases hsu with \u27e8x, hxs, hxu\u27e9", "annotated_tactic": ["rcases hsu with \u27e8x, hxs, hxu\u27e9", []], "state_before": "case mpr.a.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nH : \u00acFalse\nh : s \u2286 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))", "state_after": "case mpr.a.inr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsv : Set.Nonempty (s \u2229 v)\nH : \u00acFalse\nh : s \u2286 v\nx : \u03b1\nhxs : x \u2208 s\nhxu : x \u2208 u\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))"}, {"tactic": "exact \u27e8x, hxs, \u27e8hxu, h hxs\u27e9\u27e9", "annotated_tactic": ["exact \u27e8x, hxs, \u27e8hxu, h hxs\u27e9\u27e9", []], "state_before": "case mpr.a.inr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsv : Set.Nonempty (s \u2229 v)\nH : \u00acFalse\nh : s \u2286 v\nx : \u03b1\nhxs : x \u2208 s\nhxu : x \u2208 u\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Logic/Basic.lean", "full_name": "xor_comm", "start": [291, 1], "end": [291, 82], "traced_tactics": [{"tactic": "simp [Xor', and_comm, or_comm]", "annotated_tactic": ["simp [<a>Xor'</a>, <a>and_comm</a>, <a>or_comm</a>]", [{"full_name": "Xor'", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [103, 5], "def_end_pos": [103, 9]}, {"full_name": "and_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [159, 9], "def_end_pos": [159, 17]}, {"full_name": "or_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [263, 9], "def_end_pos": [263, 16]}]], "state_before": "a b : Prop\n\u22a2 Xor' a b = Xor' b a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Real/ENNReal.lean", "full_name": "ENNReal.coe_lt_top", "start": [308, 9], "end": [308, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Manifold/BumpFunction.lean", "full_name": "SmoothBumpFunction.nhds_basis_support", "start": [306, 1], "end": [309, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Subsemigroup/Center.lean", "full_name": "Set.center_eq_univ", "start": [156, 1], "end": [157, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "full_name": "AddLECancellable.tsub_lt_self_iff", "start": [449, 11], "end": [454, 16], "traced_tactics": [{"tactic": "refine'\n  \u27e8fun h => \u27e8(zero_le _).trans_lt h, (zero_le b).lt_of_ne _\u27e9, fun h => ha.tsub_lt_self h.1 h.2\u27e9", "annotated_tactic": ["refine'\n    \u27e8fun h => \u27e8(<a>zero_le</a> _).<a>trans_lt</a> h, (<a>zero_le</a> b).<a>lt_of_ne</a> _\u27e9, fun h => ha.tsub_lt_self h.1 h.2\u27e9", [{"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}, {"full_name": "LE.le.lt_of_ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [132, 7], "def_end_pos": [132, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nha : AddLECancellable a\n\u22a2 a - b < a \u2194 0 < a \u2227 0 < b", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nha : AddLECancellable a\nh : a - b < a\n\u22a2 0 \u2260 b"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nha : AddLECancellable a\nh : a - b < a\n\u22a2 0 \u2260 b", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na c d : \u03b1\nha : AddLECancellable a\nh : a - 0 < a\n\u22a2 False"}, {"tactic": "rw [tsub_zero] at h", "annotated_tactic": ["rw [<a>tsub_zero</a>] at h", [{"full_name": "tsub_zero", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [448, 9], "def_end_pos": [448, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na c d : \u03b1\nha : AddLECancellable a\nh : a - 0 < a\n\u22a2 False", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddCommMonoid \u03b1\ninst\u271d\u00b9 : Sub 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[656, 1], "end": [660, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Manifold/ContMDiff.lean", "full_name": "ContMDiffOn.congr", "start": [894, 1], "end": [896, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Matrix/Rank.lean", "full_name": "Matrix.rank_self_mul_conjTranspose", "start": [245, 1], "end": [247, 35], "traced_tactics": [{"tactic": "simpa only [rank_conjTranspose, conjTranspose_conjTranspose] using\n  rank_conjTranspose_mul_self A\u1d34", "annotated_tactic": ["simpa only [<a>rank_conjTranspose</a>, <a>conjTranspose_conjTranspose</a>] using\n    <a>rank_conjTranspose_mul_self</a> A\u1d34", [{"full_name": "Matrix.rank_conjTranspose", "def_path": "Mathlib/Data/Matrix/Rank.lean", "def_pos": [237, 9], "def_end_pos": [237, 27]}, {"full_name": "Matrix.conjTranspose_conjTranspose", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [2140, 9], "def_end_pos": [2140, 36]}, {"full_name": "Matrix.rank_conjTranspose_mul_self", "def_path": "Mathlib/Data/Matrix/Rank.lean", "def_pos": [224, 9], "def_end_pos": [224, 36]}]], "state_before": "l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Field R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\nA : Matrix m n R\n\u22a2 rank (A * A\u1d34) = rank A", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/BigOperators/Fin.lean", "full_name": "Fin.prod_univ_zero", "start": [60, 1], "end": [61, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/GroupAction/Quotient.lean", "full_name": "MulAction.card_eq_sum_card_group_div_card_stabilizer'", "start": [269, 1], "end": [280, 75], "traced_tactics": [{"tactic": "classical\n  have : \u2200 \u03c9 : \u03a9, Fintype.card \u03b1 / Fintype.card (stabilizer \u03b1 (\u03c6 \u03c9)) =\n      Fintype.card (\u03b1 \u29f8 stabilizer \u03b1 (\u03c6 \u03c9)) := by\n    intro \u03c9\n    rw [Fintype.card_congr (@Subgroup.groupEquivQuotientProdSubgroup \u03b1 _ (stabilizer \u03b1 <| \u03c6 \u03c9)),\n      Fintype.card_prod, Nat.mul_div_cancel]\n    exact Fintype.card_pos_iff.mpr (by infer_instance)\n  simp_rw [this, \u2190 Fintype.card_sigma,\n    Fintype.card_congr (selfEquivSigmaOrbitsQuotientStabilizer' \u03b1 \u03b2 h\u03c6)]", "annotated_tactic": ["classical\n    have : \u2200 \u03c9 : \u03a9, <a>Fintype.card</a> \u03b1 / <a>Fintype.card</a> (<a>stabilizer</a> \u03b1 (\u03c6 \u03c9)) =\n        <a>Fintype.card</a> (\u03b1 \u29f8 <a>stabilizer</a> \u03b1 (\u03c6 \u03c9)) := by\n      intro \u03c9\n      rw [<a>Fintype.card_congr</a> (@<a>Subgroup.groupEquivQuotientProdSubgroup</a> \u03b1 _ (<a>stabilizer</a> \u03b1 <| \u03c6 \u03c9)),\n        <a>Fintype.card_prod</a>, <a>Nat.mul_div_cancel</a>]\n      exact Fintype.card_pos_iff.mpr (by infer_instance)\n    simp_rw [this, \u2190 <a>Fintype.card_sigma</a>,\n      <a>Fintype.card_congr</a> (<a>selfEquivSigmaOrbitsQuotientStabilizer'</a> \u03b1 \u03b2 h\u03c6)]", [{"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "MulAction.stabilizer", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [206, 5], "def_end_pos": [206, 15]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "MulAction.stabilizer", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [206, 5], "def_end_pos": [206, 15]}, {"full_name": "Fintype.card_congr", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [149, 9], "def_end_pos": [149, 19]}, {"full_name": "Subgroup.groupEquivQuotientProdSubgroup", "def_path": "Mathlib/GroupTheory/Coset.lean", "def_pos": [607, 19], "def_end_pos": [607, 49]}, {"full_name": "MulAction.stabilizer", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [206, 5], "def_end_pos": [206, 15]}, {"full_name": "Fintype.card_prod", "def_path": "Mathlib/Data/Fintype/Prod.lean", "def_pos": [54, 9], "def_end_pos": [54, 26]}, {"full_name": "Nat.mul_div_cancel", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [641, 19], "def_end_pos": [641, 33]}, {"full_name": "Fintype.card_sigma", "def_path": "Mathlib/Data/Fintype/BigOperators.lean", "def_pos": [123, 16], "def_end_pos": [123, 34]}, {"full_name": "Fintype.card_congr", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [149, 9], "def_end_pos": [149, 19]}, {"full_name": "MulAction.selfEquivSigmaOrbitsQuotientStabilizer'", "def_path": "Mathlib/GroupTheory/GroupAction/Quotient.lean", "def_pos": [251, 19], "def_end_pos": [251, 58]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2076 inst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : MulAction \u03b1 \u03b2\nx : \u03b2\ninst\u271d\u00b3 : Fintype \u03b1\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : Fintype (Quotient (orbitRel \u03b1 \u03b2))\ninst\u271d : (b : \u03b2) \u2192 Fintype { x // x \u2208 stabilizer \u03b1 b }\n\u03c6 : Quotient (orbitRel \u03b1 \u03b2) \u2192 \u03b2\nh\u03c6 : LeftInverse Quotient.mk'' \u03c6\n\u22a2 Fintype.card \u03b2 = \u2211 \u03c9 : Quotient (orbitRel \u03b1 \u03b2), Fintype.card \u03b1 / Fintype.card { x // x \u2208 stabilizer \u03b1 (\u03c6 \u03c9) }", "state_after": "no goals"}, {"tactic": "have : \u2200 \u03c9 : \u03a9, Fintype.card \u03b1 / Fintype.card (stabilizer \u03b1 (\u03c6 \u03c9)) =\n    Fintype.card (\u03b1 \u29f8 stabilizer \u03b1 (\u03c6 \u03c9)) := by\n  intro \u03c9\n  rw [Fintype.card_congr (@Subgroup.groupEquivQuotientProdSubgroup \u03b1 _ (stabilizer \u03b1 <| \u03c6 \u03c9)),\n    Fintype.card_prod, Nat.mul_div_cancel]\n  exact Fintype.card_pos_iff.mpr (by infer_instance)", "annotated_tactic": ["have : \u2200 \u03c9 : \u03a9, <a>Fintype.card</a> \u03b1 / <a>Fintype.card</a> (<a>stabilizer</a> \u03b1 (\u03c6 \u03c9)) =\n        <a>Fintype.card</a> (\u03b1 \u29f8 <a>stabilizer</a> \u03b1 (\u03c6 \u03c9)) := by\n      intro \u03c9\n      rw [<a>Fintype.card_congr</a> (@<a>Subgroup.groupEquivQuotientProdSubgroup</a> \u03b1 _ (<a>stabilizer</a> \u03b1 <| \u03c6 \u03c9)),\n        <a>Fintype.card_prod</a>, <a>Nat.mul_div_cancel</a>]\n      exact Fintype.card_pos_iff.mpr (by infer_instance)", [{"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "MulAction.stabilizer", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [206, 5], "def_end_pos": [206, 15]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "MulAction.stabilizer", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [206, 5], "def_end_pos": [206, 15]}, {"full_name": "Fintype.card_congr", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [149, 9], "def_end_pos": [149, 19]}, {"full_name": "Subgroup.groupEquivQuotientProdSubgroup", "def_path": "Mathlib/GroupTheory/Coset.lean", "def_pos": [607, 19], "def_end_pos": [607, 49]}, {"full_name": "MulAction.stabilizer", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [206, 5], "def_end_pos": [206, 15]}, {"full_name": "Fintype.card_prod", "def_path": "Mathlib/Data/Fintype/Prod.lean", "def_pos": [54, 9], "def_end_pos": [54, 26]}, {"full_name": "Nat.mul_div_cancel", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [641, 19], "def_end_pos": [641, 33]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2076 inst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : MulAction \u03b1 \u03b2\nx : \u03b2\ninst\u271d\u00b3 : Fintype \u03b1\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : Fintype (Quotient (orbitRel \u03b1 \u03b2))\ninst\u271d : (b : \u03b2) \u2192 Fintype { x // x \u2208 stabilizer \u03b1 b }\n\u03c6 : Quotient (orbitRel \u03b1 \u03b2) \u2192 \u03b2\nh\u03c6 : LeftInverse Quotient.mk'' \u03c6\n\u22a2 Fintype.card \u03b2 = \u2211 \u03c9 : Quotient (orbitRel \u03b1 \u03b2), Fintype.card \u03b1 / Fintype.card { x // x \u2208 stabilizer \u03b1 (\u03c6 \u03c9) }", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2076 inst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : MulAction \u03b1 \u03b2\nx : \u03b2\ninst\u271d\u00b3 : Fintype \u03b1\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : Fintype (Quotient (orbitRel \u03b1 \u03b2))\ninst\u271d : (b : \u03b2) \u2192 Fintype { x // x \u2208 stabilizer \u03b1 b }\n\u03c6 : Quotient (orbitRel \u03b1 \u03b2) \u2192 \u03b2\nh\u03c6 : LeftInverse Quotient.mk'' \u03c6\nthis :\n  \u2200 (\u03c9 : Quotient (orbitRel \u03b1 \u03b2)),\n    Fintype.card \u03b1 / Fintype.card { x // x \u2208 stabilizer \u03b1 (\u03c6 \u03c9) } = Fintype.card (\u03b1 \u29f8 stabilizer \u03b1 (\u03c6 \u03c9))\n\u22a2 Fintype.card \u03b2 = \u2211 \u03c9 : Quotient (orbitRel \u03b1 \u03b2), Fintype.card \u03b1 / Fintype.card { x // x \u2208 stabilizer \u03b1 (\u03c6 \u03c9) }"}, {"tactic": "simp_rw [this, \u2190 Fintype.card_sigma,\n  Fintype.card_congr (selfEquivSigmaOrbitsQuotientStabilizer' \u03b1 \u03b2 h\u03c6)]", "annotated_tactic": ["simp_rw [this, \u2190 <a>Fintype.card_sigma</a>,\n      <a>Fintype.card_congr</a> (<a>selfEquivSigmaOrbitsQuotientStabilizer'</a> \u03b1 \u03b2 h\u03c6)]", [{"full_name": "Fintype.card_sigma", "def_path": "Mathlib/Data/Fintype/BigOperators.lean", "def_pos": [123, 16], "def_end_pos": [123, 34]}, {"full_name": "Fintype.card_congr", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [149, 9], "def_end_pos": [149, 19]}, {"full_name": "MulAction.selfEquivSigmaOrbitsQuotientStabilizer'", "def_path": "Mathlib/GroupTheory/GroupAction/Quotient.lean", "def_pos": [251, 19], "def_end_pos": [251, 58]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2076 inst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : MulAction \u03b1 \u03b2\nx : \u03b2\ninst\u271d\u00b3 : Fintype \u03b1\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : Fintype (Quotient (orbitRel \u03b1 \u03b2))\ninst\u271d : (b : \u03b2) \u2192 Fintype { x // x \u2208 stabilizer \u03b1 b }\n\u03c6 : Quotient (orbitRel \u03b1 \u03b2) \u2192 \u03b2\nh\u03c6 : LeftInverse Quotient.mk'' \u03c6\nthis :\n  \u2200 (\u03c9 : Quotient (orbitRel \u03b1 \u03b2)),\n    Fintype.card \u03b1 / Fintype.card { x // x \u2208 stabilizer \u03b1 (\u03c6 \u03c9) } = Fintype.card (\u03b1 \u29f8 stabilizer \u03b1 (\u03c6 \u03c9))\n\u22a2 Fintype.card \u03b2 = \u2211 \u03c9 : Quotient (orbitRel \u03b1 \u03b2), Fintype.card \u03b1 / Fintype.card { x // x \u2208 stabilizer \u03b1 (\u03c6 \u03c9) }", "state_after": "no goals"}, {"tactic": "intro \u03c9", "annotated_tactic": ["intro \u03c9", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2076 inst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : MulAction \u03b1 \u03b2\nx : \u03b2\ninst\u271d\u00b3 : Fintype \u03b1\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : Fintype (Quotient (orbitRel \u03b1 \u03b2))\ninst\u271d : (b : \u03b2) \u2192 Fintype { x // x \u2208 stabilizer \u03b1 b }\n\u03c6 : Quotient (orbitRel \u03b1 \u03b2) \u2192 \u03b2\nh\u03c6 : LeftInverse Quotient.mk'' \u03c6\n\u22a2 \u2200 (\u03c9 : Quotient (orbitRel \u03b1 \u03b2)),\n    Fintype.card \u03b1 / Fintype.card { x // x \u2208 stabilizer \u03b1 (\u03c6 \u03c9) } = Fintype.card (\u03b1 \u29f8 stabilizer \u03b1 (\u03c6 \u03c9))", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2076 inst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : MulAction \u03b1 \u03b2\nx : \u03b2\ninst\u271d\u00b3 : Fintype \u03b1\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : Fintype (Quotient (orbitRel \u03b1 \u03b2))\ninst\u271d : (b : \u03b2) \u2192 Fintype { x // x \u2208 stabilizer \u03b1 b }\n\u03c6 : Quotient (orbitRel \u03b1 \u03b2) \u2192 \u03b2\nh\u03c6 : LeftInverse Quotient.mk'' \u03c6\n\u03c9 : Quotient (orbitRel \u03b1 \u03b2)\n\u22a2 Fintype.card \u03b1 / Fintype.card { x // x \u2208 stabilizer \u03b1 (\u03c6 \u03c9) } = Fintype.card (\u03b1 \u29f8 stabilizer \u03b1 (\u03c6 \u03c9))"}, {"tactic": "rw [Fintype.card_congr (@Subgroup.groupEquivQuotientProdSubgroup \u03b1 _ (stabilizer \u03b1 <| \u03c6 \u03c9)),\n  Fintype.card_prod, Nat.mul_div_cancel]", "annotated_tactic": ["rw [<a>Fintype.card_congr</a> (@<a>Subgroup.groupEquivQuotientProdSubgroup</a> \u03b1 _ (<a>stabilizer</a> \u03b1 <| \u03c6 \u03c9)),\n        <a>Fintype.card_prod</a>, <a>Nat.mul_div_cancel</a>]", [{"full_name": "Fintype.card_congr", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [149, 9], "def_end_pos": [149, 19]}, {"full_name": "Subgroup.groupEquivQuotientProdSubgroup", "def_path": "Mathlib/GroupTheory/Coset.lean", "def_pos": [607, 19], "def_end_pos": [607, 49]}, {"full_name": "MulAction.stabilizer", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [206, 5], "def_end_pos": [206, 15]}, {"full_name": "Fintype.card_prod", "def_path": "Mathlib/Data/Fintype/Prod.lean", "def_pos": [54, 9], "def_end_pos": [54, 26]}, {"full_name": "Nat.mul_div_cancel", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [641, 19], "def_end_pos": [641, 33]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2076 inst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : MulAction \u03b1 \u03b2\nx : \u03b2\ninst\u271d\u00b3 : Fintype \u03b1\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : Fintype (Quotient (orbitRel \u03b1 \u03b2))\ninst\u271d : (b : \u03b2) \u2192 Fintype { x // x \u2208 stabilizer \u03b1 b }\n\u03c6 : Quotient (orbitRel \u03b1 \u03b2) \u2192 \u03b2\nh\u03c6 : LeftInverse Quotient.mk'' \u03c6\n\u03c9 : Quotient (orbitRel \u03b1 \u03b2)\n\u22a2 Fintype.card \u03b1 / Fintype.card { x // x \u2208 stabilizer \u03b1 (\u03c6 \u03c9) } = Fintype.card (\u03b1 \u29f8 stabilizer \u03b1 (\u03c6 \u03c9))", "state_after": "case H\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2076 inst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : MulAction \u03b1 \u03b2\nx : \u03b2\ninst\u271d\u00b3 : Fintype \u03b1\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : Fintype (Quotient (orbitRel \u03b1 \u03b2))\ninst\u271d : (b : \u03b2) \u2192 Fintype { x // x \u2208 stabilizer \u03b1 b }\n\u03c6 : Quotient (orbitRel \u03b1 \u03b2) \u2192 \u03b2\nh\u03c6 : LeftInverse Quotient.mk'' \u03c6\n\u03c9 : Quotient (orbitRel \u03b1 \u03b2)\n\u22a2 0 < Fintype.card { x // x \u2208 stabilizer \u03b1 (\u03c6 \u03c9) }"}, {"tactic": "exact Fintype.card_pos_iff.mpr (by infer_instance)", "annotated_tactic": ["exact Fintype.card_pos_iff.mpr (by infer_instance)", []], "state_before": "case H\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2076 inst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : MulAction \u03b1 \u03b2\nx : \u03b2\ninst\u271d\u00b3 : Fintype \u03b1\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : Fintype (Quotient (orbitRel \u03b1 \u03b2))\ninst\u271d : (b : \u03b2) \u2192 Fintype { x // x \u2208 stabilizer \u03b1 b }\n\u03c6 : Quotient (orbitRel \u03b1 \u03b2) \u2192 \u03b2\nh\u03c6 : LeftInverse Quotient.mk'' \u03c6\n\u03c9 : Quotient (orbitRel \u03b1 \u03b2)\n\u22a2 0 < Fintype.card { x // x \u2208 stabilizer \u03b1 (\u03c6 \u03c9) }", "state_after": "no goals"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2076 inst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : MulAction \u03b1 \u03b2\nx : \u03b2\ninst\u271d\u00b3 : Fintype \u03b1\ninst\u271d\u00b2 : Fintype \u03b2\ninst\u271d\u00b9 : Fintype (Quotient (orbitRel \u03b1 \u03b2))\ninst\u271d : (b : \u03b2) \u2192 Fintype { x // x \u2208 stabilizer \u03b1 b }\n\u03c6 : Quotient (orbitRel \u03b1 \u03b2) \u2192 \u03b2\nh\u03c6 : LeftInverse Quotient.mk'' \u03c6\n\u03c9 : Quotient (orbitRel \u03b1 \u03b2)\n\u22a2 Nonempty { x // x \u2208 stabilizer \u03b1 (\u03c6 \u03c9) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/WithZeroTopology.lean", "full_name": "WithZeroTopology.hasBasis_nhds_zero", "start": [62, 1], "end": [65, 86], "traced_tactics": [{"tactic": "rw [nhds_zero]", "annotated_tactic": ["rw [<a>nhds_zero</a>]", [{"full_name": "WithZeroTopology.nhds_zero", "def_path": "Mathlib/Topology/Algebra/WithZeroTopology.lean", "def_pos": [56, 9], "def_end_pos": [56, 18]}]], "state_before": "\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\n\u22a2 HasBasis (\ud835\udcdd 0) (fun \u03b3 => \u03b3 \u2260 0) Iio", "state_after": "\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\n\u22a2 HasBasis (\u2a05 \u03b3, \u2a05 (_ : \u03b3 \u2260 0), \ud835\udcdf (Iio \u03b3)) (fun \u03b3 => \u03b3 \u2260 0) Iio"}, {"tactic": "refine' hasBasis_biInf_principal _ \u27e81, one_ne_zero\u27e9", "annotated_tactic": ["refine' <a>hasBasis_biInf_principal</a> _ \u27e81, <a>one_ne_zero</a>\u27e9", [{"full_name": "Filter.hasBasis_biInf_principal", "def_path": "Mathlib/Order/Filter/Bases.lean", "def_pos": [774, 9], "def_end_pos": [774, 33]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}]], "state_before": "\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\n\u22a2 HasBasis (\u2a05 \u03b3, \u2a05 (_ : \u03b3 \u2260 0), \ud835\udcdf (Iio \u03b3)) (fun \u03b3 => \u03b3 \u2260 0) Iio", "state_after": "\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\n\u22a2 DirectedOn ((fun \u03b3 => Iio \u03b3) \u207b\u00b9'o fun x x_1 => x \u2265 x_1) fun \u03b3 => \u03b3 = 0 \u2192 False"}, {"tactic": "exact directedOn_iff_directed.2 (directed_of_inf fun a b hab => Iio_subset_Iio hab)", "annotated_tactic": ["exact <a>directedOn_iff_directed</a>.2 (<a>directed_of_inf</a> fun a b hab => <a>Iio_subset_Iio</a> hab)", [{"full_name": "directedOn_iff_directed", "def_path": "Mathlib/Order/Directed.lean", "def_pos": [55, 9], "def_end_pos": [55, 32]}, {"full_name": "directed_of_inf", "def_path": "Mathlib/Order/Directed.lean", "def_pos": [144, 9], "def_end_pos": [144, 24]}, {"full_name": "Set.Iio_subset_Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [609, 9], "def_end_pos": [609, 23]}]], "state_before": "\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\n\u22a2 DirectedOn ((fun \u03b3 => Iio \u03b3) \u207b\u00b9'o fun x x_1 => x \u2265 x_1) fun \u03b3 => \u03b3 = 0 \u2192 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/SuccPred/Basic.lean", "full_name": "Order.bot_lt_succ", "start": [554, 1], "end": [555, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Algebra/Tower.lean", "full_name": "IsScalarTower.coe_toAlgHom", "start": [148, 1], "end": [149, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "full_name": "parallelogram_law_with_nnnorm", "start": [1123, 1], "end": [1125, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.fdiv_self", "start": [225, 19], "end": [226, 61], "traced_tactics": [{"tactic": "have := Int.mul_fdiv_cancel 1 H", "annotated_tactic": ["have := <a>Int.mul_fdiv_cancel</a> 1 H", [{"full_name": "Int.mul_fdiv_cancel", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [189, 17], "def_end_pos": [189, 32]}]], "state_before": "a : Int\nH : a \u2260 0\n\u22a2 fdiv a a = 1", "state_after": "a : Int\nH : a \u2260 0\nthis : fdiv (1 * a) a = 1\n\u22a2 fdiv a a = 1"}, {"tactic": "rwa [Int.one_mul] at this", "annotated_tactic": ["rwa [<a>Int.one_mul</a>] at this", [{"full_name": "Int.one_mul", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [521, 27], "def_end_pos": [521, 34]}]], "state_before": "a : Int\nH : a \u2260 0\nthis : fdiv (1 * a) a = 1\n\u22a2 fdiv a a = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "full_name": "Rat.zero_den", "start": [9, 9], "end": [9, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.codisjoint_inf_right", "start": [546, 11], "end": [548, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finsupp/Basic.lean", "full_name": "Finsupp.comapSMul_single", "start": [1429, 1], "end": [1430, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Finsupp.lean", "full_name": "Finsupp.disjoint_supported_supported_iff", "start": [316, 1], "end": [322, 16], "traced_tactics": [{"tactic": "refine' \u27e8fun h => Set.disjoint_left.mpr fun x hx1 hx2 => _, disjoint_supported_supported\u27e9", "annotated_tactic": ["refine' \u27e8fun h => Set.disjoint_left.mpr fun x hx1 hx2 => _, <a>disjoint_supported_supported</a>\u27e9", [{"full_name": "Finsupp.disjoint_supported_supported", "def_path": "Mathlib/LinearAlgebra/Finsupp.lean", "def_pos": [311, 9], "def_end_pos": [311, 37]}]], "state_before": "\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nR : Type u_5\nS : Type u_6\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : Semiring S\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : AddCommMonoid P\ninst\u271d\u00b9 : Module R P\ninst\u271d : Nontrivial M\ns t : Set \u03b1\n\u22a2 Disjoint (supported M R s) (supported M R t) \u2194 Disjoint s t", "state_after": "\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nR : Type u_5\nS : Type u_6\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : Semiring S\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : AddCommMonoid P\ninst\u271d\u00b9 : Module R P\ninst\u271d : Nontrivial M\ns t : Set \u03b1\nh : Disjoint (supported M R s) (supported M R t)\nx : \u03b1\nhx1 : x \u2208 s\nhx2 : x \u2208 t\n\u22a2 False"}, {"tactic": "rcases exists_ne (0 : M) with \u27e8y, hy\u27e9", "annotated_tactic": ["rcases <a>exists_ne</a> (0 : M) with \u27e8y, hy\u27e9", [{"full_name": "exists_ne", "def_path": "Mathlib/Logic/Nontrivial/Defs.lean", "def_pos": [51, 9], "def_end_pos": [51, 18]}]], "state_before": "\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nR : Type u_5\nS : Type u_6\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : Semiring S\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : AddCommMonoid P\ninst\u271d\u00b9 : Module R P\ninst\u271d : Nontrivial M\ns t : Set \u03b1\nh : Disjoint (supported M R s) (supported M R t)\nx : \u03b1\nhx1 : x \u2208 s\nhx2 : x \u2208 t\n\u22a2 False", "state_after": "case intro\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nR : Type u_5\nS : Type u_6\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : Semiring S\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : AddCommMonoid P\ninst\u271d\u00b9 : Module R P\ninst\u271d : Nontrivial M\ns t : Set \u03b1\nh : Disjoint (supported M R s) (supported M R t)\nx : \u03b1\nhx1 : x \u2208 s\nhx2 : x \u2208 t\ny : M\nhy : y \u2260 0\n\u22a2 False"}, {"tactic": "have := h.le_bot \u27e8single_mem_supported R y hx1, single_mem_supported R y hx2\u27e9", "annotated_tactic": ["have := h.le_bot \u27e8<a>single_mem_supported</a> R y hx1, <a>single_mem_supported</a> R y hx2\u27e9", [{"full_name": "Finsupp.single_mem_supported", "def_path": "Mathlib/LinearAlgebra/Finsupp.lean", "def_pos": [211, 9], "def_end_pos": [211, 29]}, {"full_name": "Finsupp.single_mem_supported", "def_path": "Mathlib/LinearAlgebra/Finsupp.lean", "def_pos": [211, 9], "def_end_pos": [211, 29]}]], "state_before": "case intro\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nR : Type u_5\nS : Type u_6\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : Semiring S\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : AddCommMonoid P\ninst\u271d\u00b9 : Module R P\ninst\u271d : Nontrivial M\ns t : Set \u03b1\nh : Disjoint (supported M R s) (supported M R t)\nx : \u03b1\nhx1 : x \u2208 s\nhx2 : x \u2208 t\ny : M\nhy : y \u2260 0\n\u22a2 False", "state_after": "case intro\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nR : Type u_5\nS : Type u_6\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : Semiring S\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : AddCommMonoid P\ninst\u271d\u00b9 : Module R P\ninst\u271d : Nontrivial M\ns t : Set \u03b1\nh : Disjoint (supported M R s) (supported M R t)\nx : \u03b1\nhx1 : x \u2208 s\nhx2 : x \u2208 t\ny : M\nhy : y \u2260 0\nthis : (fun\u2080 | x => y) \u2208 \u22a5\n\u22a2 False"}, {"tactic": "rw [mem_bot, single_eq_zero] at this", "annotated_tactic": ["rw [<a>mem_bot</a>, <a>single_eq_zero</a>] at this", [{"full_name": "Submodule.mem_bot", "def_path": "Mathlib/Algebra/Module/Submodule/Lattice.lean", "def_pos": [76, 9], "def_end_pos": [76, 16]}, {"full_name": "Finsupp.single_eq_zero", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [438, 9], "def_end_pos": [438, 23]}]], "state_before": "case intro\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nR : Type u_5\nS : Type u_6\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : Semiring S\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : AddCommMonoid P\ninst\u271d\u00b9 : Module R P\ninst\u271d : Nontrivial M\ns t : Set \u03b1\nh : Disjoint (supported M R s) (supported M R t)\nx : \u03b1\nhx1 : x \u2208 s\nhx2 : x \u2208 t\ny : M\nhy : y \u2260 0\nthis : (fun\u2080 | x => y) \u2208 \u22a5\n\u22a2 False", "state_after": "case intro\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nR : Type u_5\nS : Type u_6\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : Semiring S\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : AddCommMonoid P\ninst\u271d\u00b9 : Module R P\ninst\u271d : Nontrivial M\ns t : Set \u03b1\nh : Disjoint (supported M R s) (supported M R t)\nx : \u03b1\nhx1 : x \u2208 s\nhx2 : x \u2208 t\ny : M\nhy : y \u2260 0\nthis : y = 0\n\u22a2 False"}, {"tactic": "exact hy this", "annotated_tactic": ["exact hy this", []], "state_before": "case intro\n\u03b1 : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nR : Type u_5\nS : Type u_6\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : Semiring S\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : AddCommMonoid P\ninst\u271d\u00b9 : Module R P\ninst\u271d : Nontrivial M\ns t : Set \u03b1\nh : Disjoint (supported M R s) (supported M R t)\nx : \u03b1\nhx1 : x \u2208 s\nhx2 : x \u2208 t\ny : M\nhy : y \u2260 0\nthis : y = 0\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Real/EReal.lean", "full_name": "EReal.bot_mul_of_neg", "start": [986, 1], "end": [988, 25], "traced_tactics": [{"tactic": "rw [EReal.mul_comm]", "annotated_tactic": ["rw [<a>EReal.mul_comm</a>]", [{"full_name": "EReal.mul_comm", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [221, 19], "def_end_pos": [221, 27]}]], "state_before": "x : EReal\nh : x < 0\n\u22a2 \u22a5 * x = \u22a4", "state_after": "x : EReal\nh : x < 0\n\u22a2 x * \u22a5 = \u22a4"}, {"tactic": "exact mul_bot_of_neg h", "annotated_tactic": ["exact <a>mul_bot_of_neg</a> h", [{"full_name": "EReal.mul_bot_of_neg", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [975, 9], "def_end_pos": [975, 23]}]], "state_before": "x : EReal\nh : x < 0\n\u22a2 x * \u22a5 = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "full_name": "MeasureTheory.condexpL2_comp_continuousLinearMap", "start": [281, 1], "end": [299, 92], "traced_tactics": [{"tactic": "refine' Lp.ae_eq_of_forall_set_integral_eq' \ud835\udd5c' hm _ _ two_ne_zero ENNReal.coe_ne_top\n  (fun s _ h\u03bcs => integrableOn_condexpL2_of_measure_ne_top hm h\u03bcs.ne _) (fun s _ h\u03bcs =>\n    integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim h\u03bcs.ne) _ _ _", "annotated_tactic": ["refine' <a>Lp.ae_eq_of_forall_set_integral_eq'</a> \ud835\udd5c' hm _ _ <a>two_ne_zero</a> <a>ENNReal.coe_ne_top</a>\n    (fun s _ h\u03bcs => <a>integrableOn_condexpL2_of_measure_ne_top</a> hm h\u03bcs.ne _) (fun s _ h\u03bcs =>\n      <a>integrableOn_Lp_of_measure_ne_top</a> _ fact_one_le_two_ennreal.elim h\u03bcs.ne) _ _ _", [{"full_name": "MeasureTheory.Lp.ae_eq_of_forall_set_integral_eq'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean", "def_pos": [93, 9], "def_end_pos": [93, 44]}, {"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [62, 7], "def_end_pos": [62, 18]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [302, 17], "def_end_pos": [302, 27]}, {"full_name": "MeasureTheory.integrableOn_condexpL2_of_measure_ne_top", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [84, 9], "def_end_pos": [84, 49]}, {"full_name": "MeasureTheory.integrableOn_Lp_of_measure_ne_top", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [363, 9], "def_end_pos": [363, 42]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 E'' \ud835\udd5c' hm) (compLp T f)) =\u1d50[\u03bc] \u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f))", "state_after": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\n\u22a2 \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s < \u22a4 \u2192\n        \u222b (x : \u03b1) in s, \u2191\u2191\u2191(\u2191(condexpL2 E'' \ud835\udd5c' hm) (compLp T f)) x \u2202\u03bc =\n          \u222b (x : \u03b1) in s, \u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f)) x \u2202\u03bc\n\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 E'' \ud835\udd5c' hm) (compLp T f))) \u03bc\n\ncase refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f))) \u03bc"}, {"tactic": "intro s hs h\u03bcs", "annotated_tactic": ["intro s hs h\u03bcs", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\n\u22a2 \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s < \u22a4 \u2192\n        \u222b (x : \u03b1) in s, \u2191\u2191\u2191(\u2191(condexpL2 E'' \ud835\udd5c' hm) (compLp T f)) x \u2202\u03bc =\n          \u222b (x : \u03b1) in s, \u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f)) x \u2202\u03bc", "state_after": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191\u2191(\u2191(condexpL2 E'' \ud835\udd5c' hm) (compLp T f)) x \u2202\u03bc =\n    \u222b (x : \u03b1) in s, \u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f)) x \u2202\u03bc"}, {"tactic": "rw [T.set_integral_compLp _ (hm s hs),\n  T.integral_comp_comm\n    (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim h\u03bcs.ne),\n  \u2190 lpMeas_coe, \u2190 lpMeas_coe, integral_condexpL2_eq hm f hs h\u03bcs.ne,\n  integral_condexpL2_eq hm (T.compLp f) hs h\u03bcs.ne, T.set_integral_compLp _ (hm s hs),\n  T.integral_comp_comm\n    (integrableOn_Lp_of_measure_ne_top f fact_one_le_two_ennreal.elim h\u03bcs.ne)]", "annotated_tactic": ["rw [T.set_integral_compLp _ (hm s hs),\n      T.integral_comp_comm\n        (<a>integrableOn_Lp_of_measure_ne_top</a> _ fact_one_le_two_ennreal.elim h\u03bcs.ne),\n      \u2190 <a>lpMeas_coe</a>, \u2190 <a>lpMeas_coe</a>, <a>integral_condexpL2_eq</a> hm f hs h\u03bcs.ne,\n      <a>integral_condexpL2_eq</a> hm (T.compLp f) hs h\u03bcs.ne, T.set_integral_compLp _ (hm s hs),\n      T.integral_comp_comm\n        (<a>integrableOn_Lp_of_measure_ne_top</a> f fact_one_le_two_ennreal.elim h\u03bcs.ne)]", [{"full_name": "MeasureTheory.integrableOn_Lp_of_measure_ne_top", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [363, 9], "def_end_pos": [363, 42]}, {"full_name": "MeasureTheory.lpMeas_coe", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [257, 9], "def_end_pos": [257, 19]}, {"full_name": "MeasureTheory.lpMeas_coe", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [257, 9], "def_end_pos": [257, 19]}, {"full_name": "MeasureTheory.integral_condexpL2_eq", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [256, 9], "def_end_pos": [256, 30]}, {"full_name": "MeasureTheory.integral_condexpL2_eq", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [256, 9], "def_end_pos": [256, 30]}, {"full_name": "MeasureTheory.integrableOn_Lp_of_measure_ne_top", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [363, 9], "def_end_pos": [363, 42]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191\u2191(\u2191(condexpL2 E'' \ud835\udd5c' hm) (compLp T f)) x \u2202\u03bc =\n    \u222b (x : \u03b1) in s, \u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f)) x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "exact lpMeas.aeStronglyMeasurable' _", "annotated_tactic": ["exact <a>lpMeas.aeStronglyMeasurable'</a> _", [{"full_name": "MeasureTheory.lpMeas.aeStronglyMeasurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [242, 9], "def_end_pos": [242, 37]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 E'' \ud835\udd5c' hm) (compLp T f))) \u03bc", "state_after": "no goals"}, {"tactic": "have h_coe := T.coeFn_compLp (condexpL2 E' \ud835\udd5c hm f : \u03b1 \u2192\u2082[\u03bc] E')", "annotated_tactic": ["have h_coe := T.coeFn_compLp (<a>condexpL2</a> E' \ud835\udd5c hm f : \u03b1 \u2192\u2082[\u03bc] E')", [{"full_name": "MeasureTheory.condexpL2", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [71, 19], "def_end_pos": [71, 28]}]], "state_before": "case refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f))) \u03bc", "state_after": "case refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\nh_coe : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f)) a = \u2191T (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) a)\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f))) \u03bc"}, {"tactic": "rw [\u2190 EventuallyEq] at h_coe", "annotated_tactic": ["rw [\u2190 <a>EventuallyEq</a>] at h_coe", [{"full_name": "Filter.EventuallyEq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1438, 5], "def_end_pos": [1438, 17]}]], "state_before": "case refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\nh_coe : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f)) a = \u2191T (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) a)\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f))) \u03bc", "state_after": "case refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\nh_coe : (fun a => \u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f)) a) =\u1d50[\u03bc] fun a => \u2191T (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) a)\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f))) \u03bc"}, {"tactic": "refine' AEStronglyMeasurable'.congr _ h_coe.symm", "annotated_tactic": ["refine' <a>AEStronglyMeasurable'.congr</a> _ h_coe.symm", [{"full_name": "MeasureTheory.AEStronglyMeasurable'.congr", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [59, 9], "def_end_pos": [59, 14]}]], "state_before": "case refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\nh_coe : (fun a => \u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f)) a) =\u1d50[\u03bc] fun a => \u2191T (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) a)\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f))) \u03bc", "state_after": "case refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\nh_coe : (fun a => \u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f)) a) =\u1d50[\u03bc] fun a => \u2191T (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) a)\n\u22a2 AEStronglyMeasurable' m (fun a => \u2191T (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) a)) \u03bc"}, {"tactic": "exact (lpMeas.aeStronglyMeasurable' (condexpL2 E' \ud835\udd5c hm f)).continuous_comp T.continuous", "annotated_tactic": ["exact (<a>lpMeas.aeStronglyMeasurable'</a> (<a>condexpL2</a> E' \ud835\udd5c hm f)).<a>continuous_comp</a> T.continuous", [{"full_name": "MeasureTheory.lpMeas.aeStronglyMeasurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [242, 9], "def_end_pos": [242, 37]}, {"full_name": "MeasureTheory.condexpL2", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [71, 19], "def_end_pos": [71, 28]}, {"full_name": "MeasureTheory.AEStronglyMeasurable'.continuous_comp", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [123, 9], "def_end_pos": [123, 24]}]], "state_before": "case refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\nh_coe : (fun a => \u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f)) a) =\u1d50[\u03bc] fun a => \u2191T (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) a)\n\u22a2 AEStronglyMeasurable' m (fun a => \u2191T (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) a)) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/ModelTheory/Basic.lean", "full_name": "FirstOrder.Language.Equiv.toEmbedding_toHom", "start": [816, 1], "end": [817, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/BigOperators/Multiset/Lemmas.lean", 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u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\ninst\u271d\u2078 : DecidableEq \u03b1'\ninst\u271d\u2077 : DecidableEq \u03b2'\ninst\u271d\u2076 : DecidableEq \u03b3\ninst\u271d\u2075 : DecidableEq \u03b3'\ninst\u271d\u2074 : DecidableEq \u03b4\ninst\u271d\u00b3 : DecidableEq \u03b4'\ninst\u271d\u00b2 : DecidableEq \u03b5\ninst\u271d\u00b9 : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na a' : \u03b1\nb b' : \u03b2\nc : \u03b3\ninst\u271d : DecidableEq \u03b2\n\u22a2 \u2191(image\u2082 f s (t \u222a t')) = \u2191(image\u2082 f s t \u222a image\u2082 f s t')", "state_after": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\n\u03b6 : Type u_11\n\u03b6' : Type u_12\n\u03bd : Type u_13\ninst\u271d\u2078 : DecidableEq \u03b1'\ninst\u271d\u2077 : DecidableEq \u03b2'\ninst\u271d\u2076 : DecidableEq \u03b3\ninst\u271d\u2075 : DecidableEq \u03b3'\ninst\u271d\u2074 : DecidableEq \u03b4\ninst\u271d\u00b3 : DecidableEq \u03b4'\ninst\u271d\u00b2 : DecidableEq \u03b5\ninst\u271d\u00b9 : DecidableEq \u03b5'\nf f' : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ng g' : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4\ns s' : Finset \u03b1\nt t' : Finset \u03b2\nu u' : Finset \u03b3\na a' : \u03b1\nb b' : \u03b2\nc : \u03b3\ninst\u271d : DecidableEq \u03b2\n\u22a2 image2 f (\u2191s) (\u2191t \u222a \u2191t') = image2 f \u2191s \u2191t \u222a image2 f \u2191s \u2191t'"}, {"tactic": "exact image2_union_right", "annotated_tactic": ["exact <a>image2_union_right</a>", [{"full_name": "Set.image2_union_right", "def_path": "Mathlib/Data/Set/NAry.lean", 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\u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn : \u2115\nh : \u2211 i : \u03b9, Finset.card (A i) = n\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d : \u2115\nM : Fin (Nat.succ n\u271d) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn : \u2115\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)"}, {"tactic": "induction' n using Nat.strong_induction_on with n IH generalizing A", "annotated_tactic": ["induction' n using <a>Nat.strong_induction_on</a> with n IH generalizing A", [{"full_name": "Nat.strong_induction_on", "def_path": "Mathlib/Init/Data/Nat/Lemmas.lean", "def_pos": [500, 19], "def_end_pos": [500, 38]}]], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d : \u2115\nM : Fin (Nat.succ n\u271d) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn : \u2115\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "case h\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)"}, {"tactic": "by_cases Ai_empty : \u2203 i, A i = \u2205", "annotated_tactic": ["by_cases Ai_empty : \u2203 i, A i = \u2205", []], "state_before": "case h\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2203 i, A i = \u2205\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\n\ncase neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u00ac\u2203 i, A i = \u2205\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)"}, {"tactic": "push_neg at Ai_empty", "annotated_tactic": ["push_neg at Ai_empty", []], "state_before": "case neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u00ac\u2203 i, A i = \u2205\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "case neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)"}, {"tactic": "by_cases Ai_singleton : \u2200 i, (A i).card \u2264 1", "annotated_tactic": ["by_cases Ai_singleton : \u2200 i, (A i).<a>card</a> \u2264 1", [{"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}]], "state_before": "case neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\n\ncase neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u00ac\u2200 (i : \u03b9), Finset.card (A i) \u2264 1\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)"}, {"tactic": "push_neg at Ai_singleton", "annotated_tactic": ["push_neg at Ai_singleton", []], "state_before": "case neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u00ac\u2200 (i : \u03b9), Finset.card (A i) \u2264 1\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "case neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2203 i, 1 < Finset.card (A i)\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)"}, {"tactic": "obtain \u27e8i\u2080, hi\u2080\u27e9 : \u2203 i, 1 < (A i).card := Ai_singleton", "annotated_tactic": ["obtain \u27e8i\u2080, hi\u2080\u27e9 : \u2203 i, 1 < (A i).<a>card</a> := Ai_singleton", [{"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}]], "state_before": "case neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2203 i, 1 < Finset.card (A i)\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "case neg.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)"}, {"tactic": "obtain \u27e8j\u2081, j\u2082, _, hj\u2082, _\u27e9 : \u2203 j\u2081 j\u2082, j\u2081 \u2208 A i\u2080 \u2227 j\u2082 \u2208 A i\u2080 \u2227 j\u2081 \u2260 j\u2082 :=\n  Finset.one_lt_card_iff.1 hi\u2080", "annotated_tactic": ["obtain \u27e8j\u2081, j\u2082, _, hj\u2082, _\u27e9 : \u2203 j\u2081 j\u2082, j\u2081 \u2208 A i\u2080 \u2227 j\u2082 \u2208 A i\u2080 \u2227 j\u2081 \u2260 j\u2082 :=\n    <a>Finset.one_lt_card_iff</a>.1 hi\u2080", [{"full_name": "Finset.one_lt_card_iff", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [580, 9], "def_end_pos": [580, 24]}]], "state_before": "case neg.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "case neg.intro.intro.intro.intro.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)"}, {"tactic": "let B := Function.update A i\u2080 (A i\u2080 \\ {j\u2082})", "annotated_tactic": ["let B := <a>Function.update</a> A i\u2080 (A i\u2080 \\ {j\u2082})", [{"full_name": "Function.update", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [550, 5], "def_end_pos": [550, 11]}]], "state_before": "case neg.intro.intro.intro.intro.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "case neg.intro.intro.intro.intro.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)"}, {"tactic": "let C := Function.update A i\u2080 {j\u2082}", "annotated_tactic": ["let C := <a>Function.update</a> A i\u2080 {j\u2082}", [{"full_name": "Function.update", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [550, 5], "def_end_pos": [550, 11]}]], "state_before": "case neg.intro.intro.intro.intro.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "case neg.intro.intro.intro.intro.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)"}, {"tactic": "have Brec : (f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, f fun i => g i (r i) := by\n  have : (\u2211 i, Finset.card (B i)) < \u2211 i, Finset.card (A i) := by\n    refine'\n      Finset.sum_lt_sum (fun i _ => Finset.card_le_of_subset (B_subset_A i))\n        \u27e8i\u2080, Finset.mem_univ _, _\u27e9\n    have : {j\u2082} \u2286 A i\u2080 := by simp [hj\u2082]\n    simp only [Finset.card_sdiff this, Function.update_same, Finset.card_singleton]\n    exact Nat.pred_lt (ne_of_gt (lt_trans Nat.zero_lt_one hi\u2080))\n  rw [h] at this\n  exact IH _ this B rfl", "annotated_tactic": ["have Brec : (f fun i => \u2211 j in B i, g i j) = \u2211 r in <a>piFinset</a> B, f fun i => g i (r i) := by\n    have : (\u2211 i, <a>Finset.card</a> (B i)) < \u2211 i, <a>Finset.card</a> (A i) := by\n      refine'\n        <a>Finset.sum_lt_sum</a> (fun i _ => <a>Finset.card_le_of_subset</a> (B_subset_A i))\n          \u27e8i\u2080, <a>Finset.mem_univ</a> _, _\u27e9\n      have : {j\u2082} \u2286 A i\u2080 := by simp [hj\u2082]\n      simp only [<a>Finset.card_sdiff</a> this, <a>Function.update_same</a>, <a>Finset.card_singleton</a>]\n      exact <a>Nat.pred_lt</a> (<a>ne_of_gt</a> (<a>lt_trans</a> <a>Nat.zero_lt_one</a> hi\u2080))\n    rw [h] at this\n    exact IH _ this B <a>rfl</a>", [{"full_name": "Fintype.piFinset", "def_path": "Mathlib/Data/Fintype/Pi.lean", "def_pos": [28, 5], "def_end_pos": [28, 13]}, {"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}, {"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}, {"full_name": "Finset.sum_lt_sum", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [436, 15], "def_end_pos": [436, 25]}, {"full_name": "Finset.card_le_of_subset", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [61, 9], "def_end_pos": [61, 26]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Finset.card_sdiff", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [440, 9], "def_end_pos": [440, 19]}, {"full_name": "Function.update_same", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}, {"full_name": "Finset.card_singleton", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [86, 9], "def_end_pos": [86, 23]}, {"full_name": "Nat.pred_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [227, 9], "def_end_pos": [227, 16]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}, {"full_name": "lt_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [90, 9], "def_end_pos": [90, 17]}, {"full_name": "Nat.zero_lt_one", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [401, 19], "def_end_pos": [401, 30]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case neg.intro.intro.intro.intro.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "case neg.intro.intro.intro.intro.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)"}, {"tactic": "have Crec : (f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, f fun i => g i (r i) := by\n  have : (\u2211 i, Finset.card (C i)) < \u2211 i, Finset.card (A i) :=\n    Finset.sum_lt_sum (fun i _ => Finset.card_le_of_subset (C_subset_A i))\n      \u27e8i\u2080, Finset.mem_univ _, by simp [hi\u2080]\u27e9\n  rw [h] at this\n  exact IH _ this C rfl", "annotated_tactic": ["have Crec : (f fun i => \u2211 j in C i, g i j) = \u2211 r in <a>piFinset</a> C, f fun i => g i (r i) := by\n    have : (\u2211 i, <a>Finset.card</a> (C i)) < \u2211 i, <a>Finset.card</a> (A i) :=\n      <a>Finset.sum_lt_sum</a> (fun i _ => <a>Finset.card_le_of_subset</a> (C_subset_A i))\n        \u27e8i\u2080, <a>Finset.mem_univ</a> _, by simp [hi\u2080]\u27e9\n    rw [h] at this\n    exact IH _ this C <a>rfl</a>", [{"full_name": "Fintype.piFinset", "def_path": "Mathlib/Data/Fintype/Pi.lean", "def_pos": [28, 5], "def_end_pos": [28, 13]}, {"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}, {"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}, {"full_name": "Finset.sum_lt_sum", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [436, 15], "def_end_pos": [436, 25]}, {"full_name": "Finset.card_le_of_subset", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [61, 9], "def_end_pos": [61, 26]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case neg.intro.intro.intro.intro.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "case neg.intro.intro.intro.intro.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)"}, {"tactic": "have D : Disjoint (piFinset B) (piFinset C) :=\n  haveI : Disjoint (B i\u2080) (C i\u2080) := by simp\n  piFinset_disjoint_of_disjoint B C this", "annotated_tactic": ["have D : <a>Disjoint</a> (<a>piFinset</a> B) (<a>piFinset</a> C) :=\n    haveI : <a>Disjoint</a> (B i\u2080) (C i\u2080) := by simp\n    <a>piFinset_disjoint_of_disjoint</a> B C this", [{"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}, {"full_name": "Fintype.piFinset", "def_path": "Mathlib/Data/Fintype/Pi.lean", "def_pos": [28, 5], "def_end_pos": [28, 13]}, {"full_name": "Fintype.piFinset", "def_path": "Mathlib/Data/Fintype/Pi.lean", "def_pos": [28, 5], "def_end_pos": [28, 13]}, {"full_name": "Disjoint", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [41, 5], "def_end_pos": [41, 13]}, {"full_name": "Fintype.piFinset_disjoint_of_disjoint", "def_path": "Mathlib/Data/Fintype/Pi.lean", "def_pos": [72, 9], "def_end_pos": [72, 38]}]], "state_before": "case neg.intro.intro.intro.intro.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "case neg.intro.intro.intro.intro.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)"}, {"tactic": "rw [A_eq_BC]", "annotated_tactic": ["rw [A_eq_BC]", []], "state_before": "case neg.intro.intro.intro.intro.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\npi_BC : piFinset A = piFinset B \u222a piFinset C\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "case neg.intro.intro.intro.intro.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\npi_BC : piFinset A = piFinset B \u222a piFinset C\n\u22a2 \u2191f (update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)) =\n    \u2211 r in piFinset A, \u2191f fun i => g i (r i)"}, {"tactic": "simp only [MultilinearMap.map_add, Beq, Ceq, Brec, Crec, pi_BC]", "annotated_tactic": ["simp only [<a>MultilinearMap.map_add</a>, Beq, Ceq, Brec, Crec, pi_BC]", [{"full_name": "MultilinearMap.map_add", "def_path": "Mathlib/LinearAlgebra/Multilinear/Basic.lean", "def_pos": [161, 19], "def_end_pos": [161, 26]}]], "state_before": "case neg.intro.intro.intro.intro.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\npi_BC : piFinset A = piFinset B \u222a piFinset C\n\u22a2 \u2191f (update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)) =\n    \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "case neg.intro.intro.intro.intro.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\npi_BC : piFinset A = piFinset B \u222a piFinset C\n\u22a2 ((\u2211 x in piFinset (update A i\u2080 (A i\u2080 \\ {j\u2082})), \u2191f fun i => g i (x i)) +\n      \u2211 x in piFinset (update A i\u2080 {j\u2082}), \u2191f fun i => g i (x i)) =\n    \u2211 x in piFinset (update A i\u2080 (A i\u2080 \\ {j\u2082})) \u222a piFinset (update A i\u2080 {j\u2082}), \u2191f fun i => g i (x i)"}, {"tactic": "rw [\u2190 Finset.sum_union D]", "annotated_tactic": ["rw [\u2190 <a>Finset.sum_union</a> D]", [{"full_name": "Finset.sum_union", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [408, 3], "def_end_pos": [408, 14]}]], "state_before": "case neg.intro.intro.intro.intro.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\npi_BC : piFinset A = piFinset B \u222a piFinset C\n\u22a2 ((\u2211 x in piFinset (update A i\u2080 (A i\u2080 \\ {j\u2082})), \u2191f fun i => g i (x i)) +\n      \u2211 x in piFinset (update A i\u2080 {j\u2082}), \u2191f fun i => g i (x i)) =\n    \u2211 x in piFinset (update A i\u2080 (A i\u2080 \\ {j\u2082})) \u222a piFinset (update A i\u2080 {j\u2082}), \u2191f fun i => g i (x i)", "state_after": "no goals"}, {"tactic": "rcases Ai_empty with \u27e8i, hi\u27e9", "annotated_tactic": ["rcases Ai_empty with \u27e8i, hi\u27e9", []], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2203 i, A i = \u2205\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "case pos.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\ni : \u03b9\nhi : A i = \u2205\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)"}, {"tactic": "have : \u2211 j in A i, g i j = 0 := by rw [hi, Finset.sum_empty]", "annotated_tactic": ["have : \u2211 j in A i, g i j = 0 := by rw [hi, <a>Finset.sum_empty</a>]", [{"full_name": "Finset.sum_empty", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [298, 3], "def_end_pos": [298, 14]}]], "state_before": "case pos.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\ni : \u03b9\nhi : A i = \u2205\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "case pos.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\ni : \u03b9\nhi : A i = \u2205\nthis : \u2211 j in A i, g i j = 0\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)"}, {"tactic": "rw [f.map_coord_zero i this]", "annotated_tactic": ["rw [f.map_coord_zero i this]", []], "state_before": "case pos.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\ni : \u03b9\nhi : A i = \u2205\nthis : \u2211 j in A i, g i j = 0\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "case pos.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\ni : \u03b9\nhi : A i = \u2205\nthis : \u2211 j in A i, g i j = 0\n\u22a2 0 = \u2211 r in piFinset A, \u2191f fun i => g i (r i)"}, {"tactic": "have : piFinset A = \u2205 := by\n  refine Finset.eq_empty_of_forall_not_mem fun r hr => ?_\n  have : r i \u2208 A i := mem_piFinset.mp hr i\n  simp [hi] at this", "annotated_tactic": ["have : <a>piFinset</a> A = \u2205 := by\n      refine <a>Finset.eq_empty_of_forall_not_mem</a> fun r hr => ?_\n      have : r i \u2208 A i := mem_piFinset.mp hr i\n      simp [hi] at this", [{"full_name": "Fintype.piFinset", "def_path": "Mathlib/Data/Fintype/Pi.lean", "def_pos": [28, 5], "def_end_pos": [28, 13]}, {"full_name": "Finset.eq_empty_of_forall_not_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [575, 9], "def_end_pos": [575, 35]}]], "state_before": "case pos.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\ni : \u03b9\nhi : A i = \u2205\nthis : \u2211 j in A i, g i j = 0\n\u22a2 0 = \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "case pos.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\ni : \u03b9\nhi : A i = \u2205\nthis\u271d : \u2211 j in A i, g i j = 0\nthis : piFinset A = \u2205\n\u22a2 0 = \u2211 r in piFinset A, \u2191f fun i => g i (r i)"}, {"tactic": "rw [this, Finset.sum_empty]", "annotated_tactic": ["rw [this, <a>Finset.sum_empty</a>]", [{"full_name": "Finset.sum_empty", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [298, 3], "def_end_pos": [298, 14]}]], "state_before": "case pos.intro\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\ni : \u03b9\nhi : A i = \u2205\nthis\u271d : \u2211 j in A i, g i j = 0\nthis : piFinset A = \u2205\n\u22a2 0 = \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "no goals"}, {"tactic": "rw [hi, Finset.sum_empty]", "annotated_tactic": ["rw [hi, <a>Finset.sum_empty</a>]", [{"full_name": "Finset.sum_empty", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [298, 3], "def_end_pos": [298, 14]}]], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\ni : \u03b9\nhi : A i = \u2205\n\u22a2 \u2211 j in A i, g i j = 0", "state_after": "no goals"}, {"tactic": "refine Finset.eq_empty_of_forall_not_mem fun r hr => ?_", "annotated_tactic": ["refine <a>Finset.eq_empty_of_forall_not_mem</a> fun r hr => ?_", [{"full_name": "Finset.eq_empty_of_forall_not_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [575, 9], "def_end_pos": [575, 35]}]], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\ni : \u03b9\nhi : A i = \u2205\nthis : \u2211 j in A i, g i j = 0\n\u22a2 piFinset A = \u2205", "state_after": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\ni : \u03b9\nhi : A i = \u2205\nthis : \u2211 j in A i, g i j = 0\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\n\u22a2 False"}, {"tactic": "have : r i \u2208 A i := mem_piFinset.mp hr i", "annotated_tactic": ["have : r i \u2208 A i := mem_piFinset.mp hr i", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\ni : \u03b9\nhi : A i = \u2205\nthis : \u2211 j in A i, g i j = 0\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\n\u22a2 False", "state_after": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\ni : \u03b9\nhi : A i = \u2205\nthis\u271d : \u2211 j in A i, g i j = 0\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nthis : r i \u2208 A i\n\u22a2 False"}, {"tactic": "simp [hi] at this", "annotated_tactic": ["simp [hi] at this", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\ni : \u03b9\nhi : A i = \u2205\nthis\u271d : \u2211 j in A i, g i j = 0\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nthis : r i \u2208 A i\n\u22a2 False", "state_after": "no goals"}, {"tactic": "have Ai_card : \u2200 i, (A i).card = 1 := by\n  intro i\n  have pos : Finset.card (A i) \u2260 0 := by simp [Finset.card_eq_zero, Ai_empty i]\n  have : Finset.card (A i) \u2264 1 := Ai_singleton i\n  exact le_antisymm this (Nat.succ_le_of_lt (_root_.pos_iff_ne_zero.mpr pos))", "annotated_tactic": ["have Ai_card : \u2200 i, (A i).<a>card</a> = 1 := by\n      intro i\n      have pos : <a>Finset.card</a> (A i) \u2260 0 := by simp [<a>Finset.card_eq_zero</a>, Ai_empty i]\n      have : <a>Finset.card</a> (A i) \u2264 1 := Ai_singleton i\n      exact <a>le_antisymm</a> this (<a>Nat.succ_le_of_lt</a> (_root_.pos_iff_ne_zero.mpr pos))", [{"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}, {"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}, {"full_name": "Finset.card_eq_zero", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [70, 9], "def_end_pos": [70, 21]}, {"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}, {"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "Nat.succ_le_of_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [313, 9], "def_end_pos": [313, 22]}]], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\nAi_card : \u2200 (i : \u03b9), Finset.card (A i) = 1\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)"}, {"tactic": "have :\n  \u2200 r : \u2200 i, \u03b1 i, r \u2208 piFinset A \u2192 (f fun i => g i (r i)) = f fun i => \u2211 j in A i, g i j := by\n  intro r hr\n  congr with i\n  have : \u2200 j \u2208 A i, g i j = g i (r i) := by\n    intro j hj\n    congr\n    apply Finset.card_le_one_iff.1 (Ai_singleton i) hj\n    exact mem_piFinset.mp hr i\n  simp only [Finset.sum_congr rfl this, Finset.mem_univ, Finset.sum_const, Ai_card i, one_nsmul]", "annotated_tactic": ["have :\n      \u2200 r : \u2200 i, \u03b1 i, r \u2208 <a>piFinset</a> A \u2192 (f fun i => g i (r i)) = f fun i => \u2211 j in A i, g i j := by\n      intro r hr\n      congr with i\n      have : \u2200 j \u2208 A i, g i j = g i (r i) := by\n        intro j hj\n        congr\n        apply <a>Finset.card_le_one_iff</a>.1 (Ai_singleton i) hj\n        exact mem_piFinset.mp hr i\n      simp only [<a>Finset.sum_congr</a> <a>rfl</a> this, <a>Finset.mem_univ</a>, <a>Finset.sum_const</a>, Ai_card i, <a>one_nsmul</a>]", [{"full_name": "Fintype.piFinset", "def_path": "Mathlib/Data/Fintype/Pi.lean", "def_pos": [28, 5], "def_end_pos": [28, 13]}, {"full_name": "Finset.card_le_one_iff", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [551, 9], "def_end_pos": [551, 24]}, {"full_name": "Finset.sum_congr", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [376, 3], "def_end_pos": [376, 14]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Finset.sum_const", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1440, 3], "def_end_pos": [1440, 14]}, {"full_name": "one_nsmul", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}]], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\nAi_card : \u2200 (i : \u03b9), Finset.card (A i) = 1\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\nAi_card : \u2200 (i : \u03b9), Finset.card (A i) = 1\nthis : \u2200 (r : (i : \u03b9) \u2192 \u03b1 i), r \u2208 piFinset A \u2192 (\u2191f fun i => g i (r i)) = \u2191f fun i => \u2211 j in A i, g i j\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)"}, {"tactic": "simp only [Finset.sum_congr rfl this, Ai_card, card_piFinset, prod_const_one, one_nsmul,\n  Finset.sum_const]", "annotated_tactic": ["simp only [<a>Finset.sum_congr</a> <a>rfl</a> this, Ai_card, <a>card_piFinset</a>, <a>prod_const_one</a>, <a>one_nsmul</a>,\n      <a>Finset.sum_const</a>]", [{"full_name": "Finset.sum_congr", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [376, 3], "def_end_pos": [376, 14]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Fintype.card_piFinset", "def_path": "Mathlib/Data/Fintype/BigOperators.lean", "def_pos": [135, 9], "def_end_pos": [135, 30]}, {"full_name": "Finset.prod_const_one", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 23]}, {"full_name": "one_nsmul", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}, {"full_name": "Finset.sum_const", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1440, 3], "def_end_pos": [1440, 14]}]], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\nAi_card : \u2200 (i : \u03b9), Finset.card (A i) = 1\nthis : \u2200 (r : (i : \u03b9) \u2192 \u03b1 i), r \u2208 piFinset A \u2192 (\u2191f fun i => g i (r i)) = \u2191f fun i => \u2211 j in A i, g i j\n\u22a2 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)", "state_after": "no goals"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\n\u22a2 \u2200 (i : \u03b9), Finset.card (A i) = 1", "state_after": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\ni : \u03b9\n\u22a2 Finset.card (A i) = 1"}, {"tactic": "have pos : Finset.card (A i) \u2260 0 := by simp [Finset.card_eq_zero, Ai_empty i]", "annotated_tactic": ["have pos : <a>Finset.card</a> (A i) \u2260 0 := by simp [<a>Finset.card_eq_zero</a>, Ai_empty i]", [{"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}, {"full_name": "Finset.card_eq_zero", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [70, 9], "def_end_pos": [70, 21]}]], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\ni : \u03b9\n\u22a2 Finset.card (A i) = 1", "state_after": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\ni : \u03b9\npos : Finset.card (A i) \u2260 0\n\u22a2 Finset.card (A i) = 1"}, {"tactic": "have : Finset.card (A i) \u2264 1 := Ai_singleton i", "annotated_tactic": ["have : <a>Finset.card</a> (A i) \u2264 1 := Ai_singleton i", [{"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}]], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\ni : \u03b9\npos : Finset.card (A i) \u2260 0\n\u22a2 Finset.card (A i) = 1", "state_after": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\ni : \u03b9\npos : Finset.card (A i) \u2260 0\nthis : Finset.card (A i) \u2264 1\n\u22a2 Finset.card (A i) = 1"}, {"tactic": "exact le_antisymm this (Nat.succ_le_of_lt (_root_.pos_iff_ne_zero.mpr pos))", "annotated_tactic": ["exact <a>le_antisymm</a> this (<a>Nat.succ_le_of_lt</a> (_root_.pos_iff_ne_zero.mpr pos))", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "Nat.succ_le_of_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [313, 9], "def_end_pos": [313, 22]}]], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\ni : \u03b9\npos : Finset.card (A i) \u2260 0\nthis : Finset.card (A i) \u2264 1\n\u22a2 Finset.card (A i) = 1", "state_after": "no goals"}, {"tactic": "simp [Finset.card_eq_zero, Ai_empty i]", "annotated_tactic": ["simp [<a>Finset.card_eq_zero</a>, Ai_empty i]", [{"full_name": "Finset.card_eq_zero", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [70, 9], "def_end_pos": [70, 21]}]], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\ni : \u03b9\n\u22a2 Finset.card (A i) \u2260 0", "state_after": "no goals"}, {"tactic": "intro r hr", "annotated_tactic": ["intro r hr", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\nAi_card : \u2200 (i : \u03b9), Finset.card (A i) = 1\n\u22a2 \u2200 (r : (i : \u03b9) \u2192 \u03b1 i), r \u2208 piFinset A \u2192 (\u2191f fun i => g i (r i)) = \u2191f fun i => \u2211 j in A i, g i j", "state_after": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\nAi_card : \u2200 (i : \u03b9), Finset.card (A i) = 1\nr : (i : \u03b9) \u2192 \u03b1 i\nhr : r \u2208 piFinset A\n\u22a2 (\u2191f fun i => g i (r i)) = \u2191f fun i => \u2211 j in A i, g i j"}, {"tactic": "congr with i", "annotated_tactic": ["congr with i", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\nAi_card : \u2200 (i : \u03b9), Finset.card (A i) = 1\nr : (i : \u03b9) \u2192 \u03b1 i\nhr : r \u2208 piFinset A\n\u22a2 (\u2191f fun i => g i (r i)) = \u2191f fun i => \u2211 j in A i, g i j", "state_after": "case h.e_6.h.h\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\nAi_card : \u2200 (i : \u03b9), Finset.card (A i) = 1\nr : (i : \u03b9) \u2192 \u03b1 i\nhr : r \u2208 piFinset A\ni : \u03b9\n\u22a2 g i (r i) = \u2211 j in A i, g i j"}, {"tactic": "have : \u2200 j \u2208 A i, g i j = g i (r i) := by\n  intro j hj\n  congr\n  apply Finset.card_le_one_iff.1 (Ai_singleton i) hj\n  exact mem_piFinset.mp hr i", "annotated_tactic": ["have : \u2200 j \u2208 A i, g i j = g i (r i) := by\n        intro j hj\n        congr\n        apply <a>Finset.card_le_one_iff</a>.1 (Ai_singleton i) hj\n        exact mem_piFinset.mp hr i", [{"full_name": "Finset.card_le_one_iff", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [551, 9], "def_end_pos": [551, 24]}]], "state_before": "case h.e_6.h.h\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\nAi_card : \u2200 (i : \u03b9), Finset.card (A i) = 1\nr : (i : \u03b9) \u2192 \u03b1 i\nhr : r \u2208 piFinset A\ni : \u03b9\n\u22a2 g i (r i) = \u2211 j in A i, g i j", "state_after": "case h.e_6.h.h\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\nAi_card : \u2200 (i : \u03b9), Finset.card (A i) = 1\nr : (i : \u03b9) \u2192 \u03b1 i\nhr : r \u2208 piFinset A\ni : \u03b9\nthis : \u2200 (j : \u03b1 i), j \u2208 A i \u2192 g i j = g i (r i)\n\u22a2 g i (r i) = \u2211 j in A i, g i j"}, {"tactic": "simp only [Finset.sum_congr rfl this, Finset.mem_univ, Finset.sum_const, Ai_card i, one_nsmul]", "annotated_tactic": ["simp only [<a>Finset.sum_congr</a> <a>rfl</a> this, <a>Finset.mem_univ</a>, <a>Finset.sum_const</a>, Ai_card i, <a>one_nsmul</a>]", [{"full_name": "Finset.sum_congr", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [376, 3], "def_end_pos": [376, 14]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Finset.sum_const", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1440, 3], "def_end_pos": [1440, 14]}, {"full_name": "one_nsmul", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}]], "state_before": "case h.e_6.h.h\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\nAi_card : \u2200 (i : \u03b9), Finset.card (A i) = 1\nr : (i : \u03b9) \u2192 \u03b1 i\nhr : r \u2208 piFinset A\ni : \u03b9\nthis : \u2200 (j : \u03b1 i), j \u2208 A i \u2192 g i j = g i (r i)\n\u22a2 g i (r i) = \u2211 j in A i, g i j", "state_after": "no goals"}, {"tactic": "intro j hj", "annotated_tactic": ["intro j hj", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\nAi_card : \u2200 (i : \u03b9), Finset.card (A i) = 1\nr : (i : \u03b9) \u2192 \u03b1 i\nhr : r \u2208 piFinset A\ni : \u03b9\n\u22a2 \u2200 (j : \u03b1 i), j \u2208 A i \u2192 g i j = g i (r i)", "state_after": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\nAi_card : \u2200 (i : \u03b9), Finset.card (A i) = 1\nr : (i : \u03b9) \u2192 \u03b1 i\nhr : r \u2208 piFinset A\ni : \u03b9\nj : \u03b1 i\nhj : j \u2208 A i\n\u22a2 g i j = g i (r i)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\nAi_card : \u2200 (i : \u03b9), Finset.card (A i) = 1\nr : (i : \u03b9) \u2192 \u03b1 i\nhr : r \u2208 piFinset A\ni : \u03b9\nj : \u03b1 i\nhj : j \u2208 A i\n\u22a2 g i j = g i (r i)", "state_after": "case e_a\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\nAi_card : \u2200 (i : \u03b9), Finset.card (A i) = 1\nr : (i : \u03b9) \u2192 \u03b1 i\nhr : r \u2208 piFinset A\ni : \u03b9\nj : \u03b1 i\nhj : j \u2208 A i\n\u22a2 j = r i"}, {"tactic": "apply Finset.card_le_one_iff.1 (Ai_singleton i) hj", "annotated_tactic": ["apply <a>Finset.card_le_one_iff</a>.1 (Ai_singleton i) hj", [{"full_name": "Finset.card_le_one_iff", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [551, 9], "def_end_pos": [551, 24]}]], "state_before": "case e_a\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\nAi_card : \u2200 (i : \u03b9), Finset.card (A i) = 1\nr : (i : \u03b9) \u2192 \u03b1 i\nhr : r \u2208 piFinset A\ni : \u03b9\nj : \u03b1 i\nhj : j \u2208 A i\n\u22a2 j = r i", "state_after": "case e_a\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\nAi_card : \u2200 (i : \u03b9), Finset.card (A i) = 1\nr : (i : \u03b9) \u2192 \u03b1 i\nhr : r \u2208 piFinset A\ni : \u03b9\nj : \u03b1 i\nhj : j \u2208 A i\n\u22a2 r i \u2208 A i"}, {"tactic": "exact mem_piFinset.mp hr i", "annotated_tactic": ["exact mem_piFinset.mp hr i", []], "state_before": "case e_a\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\nAi_singleton : \u2200 (i : \u03b9), Finset.card (A i) \u2264 1\nAi_card : \u2200 (i : \u03b9), Finset.card (A i) = 1\nr : (i : \u03b9) \u2192 \u03b1 i\nhr : r \u2208 piFinset A\ni : \u03b9\nj : \u03b1 i\nhj : j \u2208 A i\n\u22a2 r i \u2208 A i", "state_after": "no goals"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\n\u22a2 \u2200 (i : \u03b9), B i \u2286 A i", "state_after": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\ni : \u03b9\n\u22a2 B i \u2286 A i"}, {"tactic": "by_cases hi : i = i\u2080", "annotated_tactic": ["by_cases hi : i = i\u2080", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\ni : \u03b9\n\u22a2 B i \u2286 A i", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\ni : \u03b9\nhi : i = i\u2080\n\u22a2 B i \u2286 A i\n\ncase neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\ni : \u03b9\nhi : \u00aci = i\u2080\n\u22a2 B i \u2286 A i"}, {"tactic": "rw [hi]", "annotated_tactic": ["rw [hi]", []], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\ni : \u03b9\nhi : i = i\u2080\n\u22a2 B i \u2286 A i", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\ni : \u03b9\nhi : i = i\u2080\n\u22a2 B i\u2080 \u2286 A i\u2080"}, {"tactic": "simp only [sdiff_subset, update_same]", "annotated_tactic": ["simp only [<a>sdiff_subset</a>, <a>update_same</a>]", [{"full_name": "Finset.sdiff_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2236, 9], "def_end_pos": [2236, 21]}, {"full_name": "Function.update_same", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}]], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\ni : \u03b9\nhi : i = i\u2080\n\u22a2 B i\u2080 \u2286 A i\u2080", "state_after": "no goals"}, {"tactic": "simp only [hi, update_noteq, Ne.def, not_false_iff, Finset.Subset.refl]", "annotated_tactic": ["simp only [hi, <a>update_noteq</a>, <a>Ne.def</a>, <a>not_false_iff</a>, <a>Finset.Subset.refl</a>]", [{"full_name": "Function.update_noteq", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [560, 9], "def_end_pos": [560, 21]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "Finset.Subset.refl", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [339, 9], "def_end_pos": [339, 20]}]], "state_before": "case neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\ni : \u03b9\nhi : \u00aci = i\u2080\n\u22a2 B i \u2286 A i", "state_after": "no goals"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\n\u22a2 \u2200 (i : \u03b9), C i \u2286 A i", "state_after": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\ni : \u03b9\n\u22a2 C i \u2286 A i"}, {"tactic": "by_cases hi : i = i\u2080", "annotated_tactic": ["by_cases hi : i = i\u2080", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\ni : \u03b9\n\u22a2 C i \u2286 A i", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\n\u22a2 C i \u2286 A i\n\ncase neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\ni : \u03b9\nhi : \u00aci = i\u2080\n\u22a2 C i \u2286 A i"}, {"tactic": "rw [hi]", "annotated_tactic": ["rw [hi]", []], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\n\u22a2 C i \u2286 A i", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\n\u22a2 C i\u2080 \u2286 A i\u2080"}, {"tactic": "simp only [hj\u2082, Finset.singleton_subset_iff, update_same]", "annotated_tactic": ["simp only [hj\u2082, <a>Finset.singleton_subset_iff</a>, <a>update_same</a>]", [{"full_name": "Finset.singleton_subset_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [772, 9], "def_end_pos": [772, 29]}, {"full_name": "Function.update_same", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}]], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\n\u22a2 C i\u2080 \u2286 A i\u2080", "state_after": "no goals"}, {"tactic": "simp only [hi, update_noteq, Ne.def, not_false_iff, Finset.Subset.refl]", "annotated_tactic": ["simp only [hi, <a>update_noteq</a>, <a>Ne.def</a>, <a>not_false_iff</a>, <a>Finset.Subset.refl</a>]", [{"full_name": "Function.update_noteq", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [560, 9], "def_end_pos": [560, 21]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "Finset.Subset.refl", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [339, 9], "def_end_pos": [339, 20]}]], "state_before": "case neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\ni : \u03b9\nhi : \u00aci = i\u2080\n\u22a2 C i \u2286 A i", "state_after": "no goals"}, {"tactic": "ext i", "annotated_tactic": ["ext i", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\n\u22a2 (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)", "state_after": "case h\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\n\u22a2 \u2211 j in A i, g i j = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j) i"}, {"tactic": "by_cases hi : i = i\u2080", "annotated_tactic": ["by_cases hi : i = i\u2080", []], "state_before": "case h\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\n\u22a2 \u2211 j in A i, g i j = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j) i", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\n\u22a2 \u2211 j in A i, g i j = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j) i\n\ncase neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\nhi : \u00aci = i\u2080\n\u22a2 \u2211 j in A i, g i j = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j) i"}, {"tactic": "rw [hi, update_same]", "annotated_tactic": ["rw [hi, <a>update_same</a>]", [{"full_name": "Function.update_same", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}]], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\n\u22a2 \u2211 j in A i, g i j = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j) i", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\n\u22a2 \u2211 j in A i\u2080, g i\u2080 j = \u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j"}, {"tactic": "have : A i\u2080 = B i\u2080 \u222a C i\u2080 := by\n  simp only [Function.update_same, Finset.sdiff_union_self_eq_union]\n  symm\n  simp only [hj\u2082, Finset.singleton_subset_iff, Finset.union_eq_left]", "annotated_tactic": ["have : A i\u2080 = B i\u2080 \u222a C i\u2080 := by\n        simp only [<a>Function.update_same</a>, <a>Finset.sdiff_union_self_eq_union</a>]\n        symm\n        simp only [hj\u2082, <a>Finset.singleton_subset_iff</a>, <a>Finset.union_eq_left</a>]", [{"full_name": "Function.update_same", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}, {"full_name": "Finset.sdiff_union_self_eq_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2160, 9], "def_end_pos": [2160, 34]}, {"full_name": "Finset.singleton_subset_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [772, 9], "def_end_pos": [772, 29]}, {"full_name": "Finset.union_eq_left", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1503, 15], "def_end_pos": [1503, 28]}]], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\n\u22a2 \u2211 j in A i\u2080, g i\u2080 j = \u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\nthis : A i\u2080 = B i\u2080 \u222a C i\u2080\n\u22a2 \u2211 j in A i\u2080, g i\u2080 j = \u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\nthis : A i\u2080 = B i\u2080 \u222a C i\u2080\n\u22a2 \u2211 j in A i\u2080, g i\u2080 j = \u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\nthis : A i\u2080 = B i\u2080 \u222a C i\u2080\n\u22a2 \u2211 j in B i\u2080 \u222a C i\u2080, g i\u2080 j = \u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j"}, {"tactic": "refine Finset.sum_union <| Finset.disjoint_right.2 fun j hj => ?_", "annotated_tactic": ["refine <a>Finset.sum_union</a> <| <a>Finset.disjoint_right</a>.2 fun j hj => ?_", [{"full_name": "Finset.sum_union", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [408, 3], "def_end_pos": [408, 14]}, {"full_name": "Finset.disjoint_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [945, 9], "def_end_pos": [945, 23]}]], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\nthis : A i\u2080 = B i\u2080 \u222a C i\u2080\n\u22a2 \u2211 j in B i\u2080 \u222a C i\u2080, g i\u2080 j = \u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\nthis : A i\u2080 = B i\u2080 \u222a C i\u2080\nj : \u03b1 i\u2080\nhj : j \u2208 C i\u2080\n\u22a2 \u00acj \u2208 B i\u2080"}, {"tactic": "have : j = j\u2082 := by\n  simpa using hj", "annotated_tactic": ["have : j = j\u2082 := by\n        simpa using hj", []], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\nthis : A i\u2080 = B i\u2080 \u222a C i\u2080\nj : \u03b1 i\u2080\nhj : j \u2208 C i\u2080\n\u22a2 \u00acj \u2208 B i\u2080", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\nthis\u271d : A i\u2080 = B i\u2080 \u222a C i\u2080\nj : \u03b1 i\u2080\nhj : j \u2208 C i\u2080\nthis : j = j\u2082\n\u22a2 \u00acj \u2208 B i\u2080"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\nthis\u271d : A i\u2080 = B i\u2080 \u222a C i\u2080\nj : \u03b1 i\u2080\nhj : j \u2208 C i\u2080\nthis : j = j\u2082\n\u22a2 \u00acj \u2208 B i\u2080", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\nthis\u271d : A i\u2080 = B i\u2080 \u222a C i\u2080\nj : \u03b1 i\u2080\nhj : j \u2208 C i\u2080\nthis : j = j\u2082\n\u22a2 \u00acj\u2082 \u2208 B i\u2080"}, {"tactic": "simp only [mem_sdiff, eq_self_iff_true, not_true, not_false_iff, Finset.mem_singleton,\n  update_same, and_false_iff]", "annotated_tactic": ["simp only [<a>mem_sdiff</a>, <a>eq_self_iff_true</a>, <a>not_true</a>, <a>not_false_iff</a>, <a>Finset.mem_singleton</a>,\n        <a>update_same</a>, <a>and_false_iff</a>]", [{"full_name": "Finset.mem_sdiff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2077, 9], "def_end_pos": [2077, 18]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "not_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [80, 17], "def_end_pos": [80, 25]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}, {"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 22]}, {"full_name": "Function.update_same", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}, {"full_name": "and_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [149, 9], "def_end_pos": [149, 22]}]], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d\u00b9 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\nthis\u271d : A i\u2080 = B i\u2080 \u222a C i\u2080\nj : \u03b1 i\u2080\nhj : j \u2208 C i\u2080\nthis : j = j\u2082\n\u22a2 \u00acj\u2082 \u2208 B i\u2080", "state_after": "no goals"}, {"tactic": "simp only [Function.update_same, Finset.sdiff_union_self_eq_union]", "annotated_tactic": ["simp only [<a>Function.update_same</a>, <a>Finset.sdiff_union_self_eq_union</a>]", [{"full_name": "Function.update_same", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}, {"full_name": "Finset.sdiff_union_self_eq_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2160, 9], "def_end_pos": [2160, 34]}]], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\n\u22a2 A i\u2080 = B i\u2080 \u222a C i\u2080", "state_after": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\n\u22a2 A i\u2080 = A i\u2080 \u222a {j\u2082}"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\n\u22a2 A i\u2080 = A i\u2080 \u222a {j\u2082}", "state_after": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\n\u22a2 A i\u2080 \u222a {j\u2082} = A i\u2080"}, {"tactic": "simp only [hj\u2082, Finset.singleton_subset_iff, Finset.union_eq_left]", "annotated_tactic": ["simp only [hj\u2082, <a>Finset.singleton_subset_iff</a>, <a>Finset.union_eq_left</a>]", [{"full_name": "Finset.singleton_subset_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [772, 9], "def_end_pos": [772, 29]}, {"full_name": "Finset.union_eq_left", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1503, 15], "def_end_pos": [1503, 28]}]], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\n\u22a2 A i\u2080 \u222a {j\u2082} = A i\u2080", "state_after": "no goals"}, {"tactic": "simpa using hj", "annotated_tactic": ["simpa using hj", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\nhi : i = i\u2080\nthis : A i\u2080 = B i\u2080 \u222a C i\u2080\nj : \u03b1 i\u2080\nhj : j \u2208 C i\u2080\n\u22a2 j = j\u2082", "state_after": "no goals"}, {"tactic": "simp [hi]", "annotated_tactic": ["simp [hi]", []], "state_before": "case neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\ni : \u03b9\nhi : \u00aci = i\u2080\n\u22a2 \u2211 j in A i, g i j = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j) i", "state_after": "no goals"}, {"tactic": "ext i", "annotated_tactic": ["ext i", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\n\u22a2 update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j", "state_after": "case h\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\ni : \u03b9\n\u22a2 update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) i = \u2211 j in B i, g i j"}, {"tactic": "by_cases hi : i = i\u2080", "annotated_tactic": ["by_cases hi : i = i\u2080", []], "state_before": "case h\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\ni : \u03b9\n\u22a2 update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) i = \u2211 j in B i, g i j", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\ni : \u03b9\nhi : i = i\u2080\n\u22a2 update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) i = \u2211 j in B i, g i j\n\ncase neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\ni : \u03b9\nhi : \u00aci = i\u2080\n\u22a2 update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) i = \u2211 j in B i, g i j"}, {"tactic": "rw [hi]", "annotated_tactic": ["rw [hi]", []], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\ni : \u03b9\nhi : i = i\u2080\n\u22a2 update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) i = \u2211 j in B i, g i j", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\ni : \u03b9\nhi : i = i\u2080\n\u22a2 update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) i\u2080 = \u2211 j in B i\u2080, g i\u2080 j"}, {"tactic": "simp only [update_same]", "annotated_tactic": ["simp only [<a>update_same</a>]", [{"full_name": "Function.update_same", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}]], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\ni : \u03b9\nhi : i = i\u2080\n\u22a2 update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) i\u2080 = \u2211 j in B i\u2080, g i\u2080 j", "state_after": "no goals"}, {"tactic": "simp only [hi, update_noteq, Ne.def, not_false_iff]", "annotated_tactic": ["simp only [hi, <a>update_noteq</a>, <a>Ne.def</a>, <a>not_false_iff</a>]", [{"full_name": "Function.update_noteq", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [560, 9], "def_end_pos": [560, 21]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "case neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\ni : \u03b9\nhi : \u00aci = i\u2080\n\u22a2 update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) i = \u2211 j in B i, g i j", "state_after": "no goals"}, {"tactic": "ext i", "annotated_tactic": ["ext i", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\n\u22a2 update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j", "state_after": "case h\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\ni : \u03b9\n\u22a2 update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) i = \u2211 j in C i, g i j"}, {"tactic": "by_cases hi : i = i\u2080", "annotated_tactic": ["by_cases hi : i = i\u2080", []], "state_before": "case h\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\ni : \u03b9\n\u22a2 update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) i = \u2211 j in C i, g i j", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\ni : \u03b9\nhi : i = i\u2080\n\u22a2 update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) i = \u2211 j in C i, g i j\n\ncase neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\ni : \u03b9\nhi : \u00aci = i\u2080\n\u22a2 update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) i = \u2211 j in C i, g i j"}, {"tactic": "rw [hi]", "annotated_tactic": ["rw [hi]", []], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\ni : \u03b9\nhi : i = i\u2080\n\u22a2 update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) i = \u2211 j in C i, g i j", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\ni : \u03b9\nhi : i = i\u2080\n\u22a2 update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) i\u2080 = \u2211 j in C i\u2080, g i\u2080 j"}, {"tactic": "simp only [update_same]", "annotated_tactic": ["simp only [<a>update_same</a>]", [{"full_name": "Function.update_same", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}]], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\ni : \u03b9\nhi : i = i\u2080\n\u22a2 update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) i\u2080 = \u2211 j in C i\u2080, g i\u2080 j", "state_after": "no goals"}, {"tactic": "simp only [hi, update_noteq, Ne.def, not_false_iff]", "annotated_tactic": ["simp only [hi, <a>update_noteq</a>, <a>Ne.def</a>, <a>not_false_iff</a>]", [{"full_name": "Function.update_noteq", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [560, 9], "def_end_pos": [560, 21]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "not_false_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "case neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\ni : \u03b9\nhi : \u00aci = i\u2080\n\u22a2 update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) i = \u2211 j in C i, g i j", "state_after": "no goals"}, {"tactic": "have : (\u2211 i, Finset.card (B i)) < \u2211 i, Finset.card (A i) := by\n  refine'\n    Finset.sum_lt_sum (fun i _ => Finset.card_le_of_subset (B_subset_A i))\n      \u27e8i\u2080, Finset.mem_univ _, _\u27e9\n  have : {j\u2082} \u2286 A i\u2080 := by simp [hj\u2082]\n  simp only [Finset.card_sdiff this, Function.update_same, Finset.card_singleton]\n  exact Nat.pred_lt (ne_of_gt (lt_trans Nat.zero_lt_one hi\u2080))", "annotated_tactic": ["have : (\u2211 i, <a>Finset.card</a> (B i)) < \u2211 i, <a>Finset.card</a> (A i) := by\n      refine'\n        <a>Finset.sum_lt_sum</a> (fun i _ => <a>Finset.card_le_of_subset</a> (B_subset_A i))\n          \u27e8i\u2080, <a>Finset.mem_univ</a> _, _\u27e9\n      have : {j\u2082} \u2286 A i\u2080 := by simp [hj\u2082]\n      simp only [<a>Finset.card_sdiff</a> this, <a>Function.update_same</a>, <a>Finset.card_singleton</a>]\n      exact <a>Nat.pred_lt</a> (<a>ne_of_gt</a> (<a>lt_trans</a> <a>Nat.zero_lt_one</a> hi\u2080))", [{"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}, {"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}, {"full_name": "Finset.sum_lt_sum", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [436, 15], "def_end_pos": [436, 25]}, {"full_name": "Finset.card_le_of_subset", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [61, 9], "def_end_pos": [61, 26]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Finset.card_sdiff", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [440, 9], "def_end_pos": [440, 19]}, {"full_name": "Function.update_same", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}, {"full_name": "Finset.card_singleton", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [86, 9], "def_end_pos": [86, 23]}, {"full_name": "Nat.pred_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [227, 9], "def_end_pos": [227, 16]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}, {"full_name": "lt_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [90, 9], "def_end_pos": [90, 17]}, {"full_name": "Nat.zero_lt_one", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [401, 19], "def_end_pos": [401, 30]}]], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\n\u22a2 (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)", "state_after": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nthis : \u2211 i : \u03b9, Finset.card (B i) < \u2211 i : \u03b9, Finset.card (A i)\n\u22a2 (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)"}, {"tactic": "rw [h] at this", "annotated_tactic": ["rw [h] at this", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nthis : \u2211 i : \u03b9, Finset.card (B i) < \u2211 i : \u03b9, Finset.card (A i)\n\u22a2 (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)", "state_after": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nthis : \u2211 i : \u03b9, Finset.card (B i) < n\n\u22a2 (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)"}, {"tactic": "exact IH _ this B rfl", "annotated_tactic": ["exact IH _ this B <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nthis : \u2211 i : \u03b9, Finset.card (B i) < n\n\u22a2 (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)", "state_after": "no goals"}, {"tactic": "refine'\n  Finset.sum_lt_sum (fun i _ => Finset.card_le_of_subset (B_subset_A i))\n    \u27e8i\u2080, Finset.mem_univ _, _\u27e9", "annotated_tactic": ["refine'\n        <a>Finset.sum_lt_sum</a> (fun i _ => <a>Finset.card_le_of_subset</a> (B_subset_A i))\n          \u27e8i\u2080, <a>Finset.mem_univ</a> _, _\u27e9", [{"full_name": "Finset.sum_lt_sum", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [436, 15], "def_end_pos": [436, 25]}, {"full_name": "Finset.card_le_of_subset", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [61, 9], "def_end_pos": [61, 26]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\n\u22a2 \u2211 i : \u03b9, Finset.card (B i) < \u2211 i : \u03b9, Finset.card (A i)", "state_after": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\n\u22a2 Finset.card (B i\u2080) < Finset.card (A i\u2080)"}, {"tactic": "have : {j\u2082} \u2286 A i\u2080 := by simp [hj\u2082]", "annotated_tactic": ["have : {j\u2082} \u2286 A i\u2080 := by simp [hj\u2082]", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\n\u22a2 Finset.card (B i\u2080) < Finset.card (A i\u2080)", "state_after": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nthis : {j\u2082} \u2286 A i\u2080\n\u22a2 Finset.card (B i\u2080) < Finset.card (A i\u2080)"}, {"tactic": "simp only [Finset.card_sdiff this, Function.update_same, Finset.card_singleton]", "annotated_tactic": ["simp only [<a>Finset.card_sdiff</a> this, <a>Function.update_same</a>, <a>Finset.card_singleton</a>]", [{"full_name": "Finset.card_sdiff", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [440, 9], "def_end_pos": [440, 19]}, {"full_name": "Function.update_same", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}, {"full_name": "Finset.card_singleton", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [86, 9], "def_end_pos": [86, 23]}]], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nthis : {j\u2082} \u2286 A i\u2080\n\u22a2 Finset.card (B i\u2080) < Finset.card (A i\u2080)", "state_after": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nthis : {j\u2082} \u2286 A i\u2080\n\u22a2 Finset.card (A i\u2080) - 1 < Finset.card (A i\u2080)"}, {"tactic": "exact Nat.pred_lt (ne_of_gt (lt_trans Nat.zero_lt_one hi\u2080))", "annotated_tactic": ["exact <a>Nat.pred_lt</a> (<a>ne_of_gt</a> (<a>lt_trans</a> <a>Nat.zero_lt_one</a> hi\u2080))", [{"full_name": "Nat.pred_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [227, 9], "def_end_pos": [227, 16]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}, {"full_name": "lt_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [90, 9], "def_end_pos": [90, 17]}, {"full_name": "Nat.zero_lt_one", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [401, 19], "def_end_pos": [401, 30]}]], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nthis : {j\u2082} \u2286 A i\u2080\n\u22a2 Finset.card (A i\u2080) - 1 < Finset.card (A i\u2080)", "state_after": "no goals"}, {"tactic": "simp [hj\u2082]", "annotated_tactic": ["simp [hj\u2082]", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\n\u22a2 {j\u2082} \u2286 A i\u2080", "state_after": "no goals"}, {"tactic": "have : (\u2211 i, Finset.card (C i)) < \u2211 i, Finset.card (A i) :=\n  Finset.sum_lt_sum (fun i _ => Finset.card_le_of_subset (C_subset_A i))\n    \u27e8i\u2080, Finset.mem_univ _, by simp [hi\u2080]\u27e9", "annotated_tactic": ["have : (\u2211 i, <a>Finset.card</a> (C i)) < \u2211 i, <a>Finset.card</a> (A i) :=\n      <a>Finset.sum_lt_sum</a> (fun i _ => <a>Finset.card_le_of_subset</a> (C_subset_A i))\n        \u27e8i\u2080, <a>Finset.mem_univ</a> _, by simp [hi\u2080]\u27e9", [{"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}, {"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [43, 5], "def_end_pos": [43, 9]}, {"full_name": "Finset.sum_lt_sum", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [436, 15], "def_end_pos": [436, 25]}, {"full_name": "Finset.card_le_of_subset", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [61, 9], "def_end_pos": [61, 26]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\n\u22a2 (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)", "state_after": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nthis : \u2211 i : \u03b9, Finset.card (C i) < \u2211 i : \u03b9, Finset.card (A i)\n\u22a2 (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)"}, {"tactic": "rw [h] at this", "annotated_tactic": ["rw [h] at this", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nthis : \u2211 i : \u03b9, Finset.card (C i) < \u2211 i : \u03b9, Finset.card (A i)\n\u22a2 (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)", "state_after": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nthis : \u2211 i : \u03b9, Finset.card (C i) < n\n\u22a2 (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)"}, {"tactic": "exact IH _ this C rfl", "annotated_tactic": ["exact IH _ this C <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nthis : \u2211 i : \u03b9, Finset.card (C i) < n\n\u22a2 (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)", "state_after": "no goals"}, {"tactic": "simp [hi\u2080]", "annotated_tactic": ["simp [hi\u2080]", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\n\u22a2 Finset.card (C i\u2080) < Finset.card (A i\u2080)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\n\u22a2 Disjoint (B i\u2080) (C i\u2080)", "state_after": "no goals"}, {"tactic": "apply Finset.Subset.antisymm", "annotated_tactic": ["apply <a>Finset.Subset.antisymm</a>", [{"full_name": "Finset.Subset.antisymm", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [367, 9], "def_end_pos": [367, 24]}]], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\n\u22a2 piFinset A = piFinset B \u222a piFinset C", "state_after": "case H\u2081\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\n\u22a2 piFinset A \u2286 piFinset B \u222a piFinset C\n\ncase H\u2082\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\n\u22a2 piFinset B \u222a piFinset C \u2286 piFinset A"}, {"tactic": "intro r hr", "annotated_tactic": ["intro r hr", []], "state_before": "case H\u2081\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\n\u22a2 piFinset A \u2286 piFinset B \u222a piFinset C", "state_after": "case H\u2081\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\n\u22a2 r \u2208 piFinset B \u222a piFinset C"}, {"tactic": "by_cases hri\u2080 : r i\u2080 = j\u2082", "annotated_tactic": ["by_cases hri\u2080 : r i\u2080 = j\u2082", []], "state_before": "case H\u2081\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\n\u22a2 r \u2208 piFinset B \u222a piFinset C", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : r i\u2080 = j\u2082\n\u22a2 r \u2208 piFinset B \u222a piFinset C\n\ncase neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : \u00acr i\u2080 = j\u2082\n\u22a2 r \u2208 piFinset B \u222a piFinset C"}, {"tactic": "apply Finset.mem_union_right", "annotated_tactic": ["apply <a>Finset.mem_union_right</a>", [{"full_name": "Finset.mem_union_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1386, 9], "def_end_pos": [1386, 24]}]], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : r i\u2080 = j\u2082\n\u22a2 r \u2208 piFinset B \u222a piFinset C", "state_after": "case pos.h\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : r i\u2080 = j\u2082\n\u22a2 r \u2208 piFinset C"}, {"tactic": "refine mem_piFinset.2 fun i => ?_", "annotated_tactic": ["refine <a>mem_piFinset</a>.2 fun i => ?_", [{"full_name": "Fintype.mem_piFinset", "def_path": "Mathlib/Data/Fintype/Pi.lean", "def_pos": [34, 9], "def_end_pos": [34, 21]}]], "state_before": "case pos.h\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : r i\u2080 = j\u2082\n\u22a2 r \u2208 piFinset C", "state_after": "case pos.h\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : r i\u2080 = j\u2082\ni : \u03b9\n\u22a2 r i \u2208 C i"}, {"tactic": "by_cases hi : i = i\u2080", "annotated_tactic": ["by_cases hi : i = i\u2080", []], "state_before": "case pos.h\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : r i\u2080 = j\u2082\ni : \u03b9\n\u22a2 r i \u2208 C i", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : r i\u2080 = j\u2082\ni : \u03b9\nhi : i = i\u2080\n\u22a2 r i \u2208 C i\n\ncase neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : r i\u2080 = j\u2082\ni : \u03b9\nhi : \u00aci = i\u2080\n\u22a2 r i \u2208 C i"}, {"tactic": "have : r i\u2080 \u2208 C i\u2080 := by simp [hri\u2080]", "annotated_tactic": ["have : r i\u2080 \u2208 C i\u2080 := by simp [hri\u2080]", []], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : r i\u2080 = j\u2082\ni : \u03b9\nhi : i = i\u2080\n\u22a2 r i \u2208 C i", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : r i\u2080 = j\u2082\ni : \u03b9\nhi : i = i\u2080\nthis : r i\u2080 \u2208 C i\u2080\n\u22a2 r i \u2208 C i"}, {"tactic": "rwa [hi]", "annotated_tactic": ["rwa [hi]", []], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : r i\u2080 = j\u2082\ni : \u03b9\nhi : i = i\u2080\nthis : r i\u2080 \u2208 C i\u2080\n\u22a2 r i \u2208 C i", "state_after": "no goals"}, {"tactic": "simp [hri\u2080]", "annotated_tactic": ["simp [hri\u2080]", []], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : r i\u2080 = j\u2082\ni : \u03b9\nhi : i = i\u2080\n\u22a2 r i\u2080 \u2208 C i\u2080", "state_after": "no goals"}, {"tactic": "simp [hi, mem_piFinset.1 hr i]", "annotated_tactic": ["simp [hi, <a>mem_piFinset</a>.1 hr i]", [{"full_name": "Fintype.mem_piFinset", "def_path": "Mathlib/Data/Fintype/Pi.lean", "def_pos": [34, 9], "def_end_pos": [34, 21]}]], "state_before": "case neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : r i\u2080 = j\u2082\ni : \u03b9\nhi : \u00aci = i\u2080\n\u22a2 r i \u2208 C i", "state_after": "no goals"}, {"tactic": "apply Finset.mem_union_left", "annotated_tactic": ["apply <a>Finset.mem_union_left</a>", [{"full_name": "Finset.mem_union_left", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1382, 9], "def_end_pos": [1382, 23]}]], "state_before": "case neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : \u00acr i\u2080 = j\u2082\n\u22a2 r \u2208 piFinset B \u222a piFinset C", "state_after": "case neg.h\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : \u00acr i\u2080 = j\u2082\n\u22a2 r \u2208 piFinset B"}, {"tactic": "refine mem_piFinset.2 fun i => ?_", "annotated_tactic": ["refine <a>mem_piFinset</a>.2 fun i => ?_", [{"full_name": "Fintype.mem_piFinset", "def_path": "Mathlib/Data/Fintype/Pi.lean", "def_pos": [34, 9], "def_end_pos": [34, 21]}]], "state_before": "case neg.h\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : \u00acr i\u2080 = j\u2082\n\u22a2 r \u2208 piFinset B", "state_after": "case neg.h\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : \u00acr i\u2080 = j\u2082\ni : \u03b9\n\u22a2 r i \u2208 B i"}, {"tactic": "by_cases hi : i = i\u2080", "annotated_tactic": ["by_cases hi : i = i\u2080", []], "state_before": "case neg.h\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : \u00acr i\u2080 = j\u2082\ni : \u03b9\n\u22a2 r i \u2208 B i", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : \u00acr i\u2080 = j\u2082\ni : \u03b9\nhi : i = i\u2080\n\u22a2 r i \u2208 B i\n\ncase neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : \u00acr i\u2080 = j\u2082\ni : \u03b9\nhi : \u00aci = i\u2080\n\u22a2 r i \u2208 B i"}, {"tactic": "have : r i\u2080 \u2208 B i\u2080 := by simp [hri\u2080, mem_piFinset.1 hr i\u2080]", "annotated_tactic": ["have : r i\u2080 \u2208 B i\u2080 := by simp [hri\u2080, <a>mem_piFinset</a>.1 hr i\u2080]", [{"full_name": "Fintype.mem_piFinset", "def_path": "Mathlib/Data/Fintype/Pi.lean", "def_pos": [34, 9], "def_end_pos": [34, 21]}]], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : \u00acr i\u2080 = j\u2082\ni : \u03b9\nhi : i = i\u2080\n\u22a2 r i \u2208 B i", "state_after": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : \u00acr i\u2080 = j\u2082\ni : \u03b9\nhi : i = i\u2080\nthis : r i\u2080 \u2208 B i\u2080\n\u22a2 r i \u2208 B i"}, {"tactic": "rwa [hi]", "annotated_tactic": ["rwa [hi]", []], "state_before": "case pos\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : \u00acr i\u2080 = j\u2082\ni : \u03b9\nhi : i = i\u2080\nthis : r i\u2080 \u2208 B i\u2080\n\u22a2 r i \u2208 B i", "state_after": "no goals"}, {"tactic": "simp [hri\u2080, mem_piFinset.1 hr i\u2080]", "annotated_tactic": ["simp [hri\u2080, <a>mem_piFinset</a>.1 hr i\u2080]", [{"full_name": "Fintype.mem_piFinset", "def_path": "Mathlib/Data/Fintype/Pi.lean", "def_pos": [34, 9], "def_end_pos": [34, 21]}]], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : \u00acr i\u2080 = j\u2082\ni : \u03b9\nhi : i = i\u2080\n\u22a2 r i\u2080 \u2208 B i\u2080", "state_after": "no goals"}, {"tactic": "simp [hi, mem_piFinset.1 hr i]", "annotated_tactic": ["simp [hi, <a>mem_piFinset</a>.1 hr i]", [{"full_name": "Fintype.mem_piFinset", "def_path": "Mathlib/Data/Fintype/Pi.lean", "def_pos": [34, 9], "def_end_pos": [34, 21]}]], "state_before": "case neg\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\nr : (a : \u03b9) \u2192 \u03b1 a\nhr : r \u2208 piFinset A\nhri\u2080 : \u00acr i\u2080 = j\u2082\ni : \u03b9\nhi : \u00aci = i\u2080\n\u22a2 r i \u2208 B i", "state_after": "no goals"}, {"tactic": "exact\n  Finset.union_subset (piFinset_subset _ _ fun i => B_subset_A i)\n    (piFinset_subset _ _ fun i => C_subset_A i)", "annotated_tactic": ["exact\n        <a>Finset.union_subset</a> (<a>piFinset_subset</a> _ _ fun i => B_subset_A i)\n          (<a>piFinset_subset</a> _ _ fun i => C_subset_A i)", [{"full_name": "Finset.union_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1403, 9], "def_end_pos": [1403, 21]}, {"full_name": "Fintype.piFinset_subset", "def_path": "Mathlib/Data/Fintype/Pi.lean", "def_pos": [53, 9], "def_end_pos": [53, 24]}, {"full_name": "Fintype.piFinset_subset", "def_path": "Mathlib/Data/Fintype/Pi.lean", "def_pos": [53, 9], "def_end_pos": [53, 24]}]], "state_before": "case H\u2082\nR : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn\u271d\u00b9 : \u2115\nM : Fin (Nat.succ n\u271d\u00b9) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : (i : Fin (Nat.succ n\u271d\u00b9)) \u2192 Module R (M i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Module R M\u2083\ninst\u271d\u00b2 : Module R M'\nf f' : MultilinearMap R M\u2081 M\u2082\n\u03b1 : \u03b9 \u2192 Type u_1\ng : (i : \u03b9) \u2192 \u03b1 i \u2192 M\u2081 i\nA\u271d : (i : \u03b9) \u2192 Finset (\u03b1 i)\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nn\u271d : \u2115\nh\u271d : \u2211 i : \u03b9, Finset.card (A\u271d i) = n\u271d\nthis : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) := fun i => Classical.decEq (\u03b1 i)\nn : \u2115\nIH :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 (A : (i : \u03b9) \u2192 Finset (\u03b1 i)),\n        \u2211 i : \u03b9, Finset.card (A i) = m \u2192 (\u2191f fun i => \u2211 j in A i, g i j) = \u2211 r in piFinset A, \u2191f fun i => g i (r i)\nA : (i : \u03b9) \u2192 Finset (\u03b1 i)\nh : \u2211 i : \u03b9, Finset.card (A i) = n\nAi_empty : \u2200 (i : \u03b9), A i \u2260 \u2205\ni\u2080 : \u03b9\nhi\u2080 : 1 < Finset.card (A i\u2080)\nj\u2081 j\u2082 : \u03b1 i\u2080\nleft\u271d : j\u2081 \u2208 A i\u2080\nhj\u2082 : j\u2082 \u2208 A i\u2080\nright\u271d : j\u2081 \u2260 j\u2082\nB : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 (A i\u2080 \\ {j\u2082})\nC : (a : \u03b9) \u2192 Finset (\u03b1 a) := update A i\u2080 {j\u2082}\nB_subset_A : \u2200 (i : \u03b9), B i \u2286 A i\nC_subset_A : \u2200 (i : \u03b9), C i \u2286 A i\nA_eq_BC :\n  (fun i => \u2211 j in A i, g i j) = update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j + \u2211 j in C i\u2080, g i\u2080 j)\nBeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in B i\u2080, g i\u2080 j) = fun i => \u2211 j in B i, g i j\nCeq : update (fun i => \u2211 j in A i, g i j) i\u2080 (\u2211 j in C i\u2080, g i\u2080 j) = fun i => \u2211 j in C i, g i j\nBrec : (\u2191f fun i => \u2211 j in B i, g i j) = \u2211 r in piFinset B, \u2191f fun i => g i (r i)\nCrec : (\u2191f fun i => \u2211 j in C i, g i j) = \u2211 r in piFinset C, \u2191f fun i => g i (r i)\nD : Disjoint (piFinset B) (piFinset C)\n\u22a2 piFinset B \u222a piFinset C \u2286 piFinset A", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/GradedMonoid.lean", "full_name": "SetLike.homogeneous_coe", "start": [711, 1], "end": [712, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Surreal/Dyadic.lean", "full_name": "Surreal.nsmul_pow_two_powHalf'", "start": [192, 1], "end": [198, 44], "traced_tactics": [{"tactic": "induction' k with k hk", "annotated_tactic": ["induction' k with k hk", []], "state_before": "n k : \u2115\n\u22a2 2 ^ n \u2022 powHalf (n + k) = powHalf k", "state_after": "case zero\nn : \u2115\n\u22a2 2 ^ n \u2022 powHalf (n + Nat.zero) = powHalf Nat.zero\n\ncase succ\nn k : \u2115\nhk : 2 ^ n \u2022 powHalf (n + k) = powHalf k\n\u22a2 2 ^ n \u2022 powHalf (n + Nat.succ k) = powHalf (Nat.succ k)"}, {"tactic": "simp only [add_zero, Surreal.nsmul_pow_two_powHalf, Nat.zero_eq, eq_self_iff_true,\n  Surreal.powHalf_zero]", "annotated_tactic": ["simp only [<a>add_zero</a>, <a>Surreal.nsmul_pow_two_powHalf</a>, <a>Nat.zero_eq</a>, <a>eq_self_iff_true</a>,\n      <a>Surreal.powHalf_zero</a>]", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "Surreal.nsmul_pow_two_powHalf", "def_path": "Mathlib/SetTheory/Surreal/Dyadic.lean", "def_pos": [184, 9], "def_end_pos": [184, 30]}, {"full_name": "Nat.zero_eq", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [83, 17], "def_end_pos": [83, 24]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "Surreal.powHalf_zero", "def_path": "Mathlib/SetTheory/Surreal/Dyadic.lean", "def_pos": [174, 9], "def_end_pos": [174, 21]}]], "state_before": "case zero\nn : \u2115\n\u22a2 2 ^ n \u2022 powHalf (n + Nat.zero) = powHalf Nat.zero", "state_after": "no goals"}, {"tactic": "rw [\u2190 double_powHalf_succ_eq_powHalf (n + k), \u2190 double_powHalf_succ_eq_powHalf k,\n  smul_algebra_smul_comm] at hk", "annotated_tactic": ["rw [\u2190 <a>double_powHalf_succ_eq_powHalf</a> (n + k), \u2190 <a>double_powHalf_succ_eq_powHalf</a> k,\n      <a>smul_algebra_smul_comm</a>] at hk", [{"full_name": "Surreal.double_powHalf_succ_eq_powHalf", "def_path": "Mathlib/SetTheory/Surreal/Dyadic.lean", "def_pos": [179, 9], "def_end_pos": [179, 39]}, {"full_name": "Surreal.double_powHalf_succ_eq_powHalf", "def_path": "Mathlib/SetTheory/Surreal/Dyadic.lean", "def_pos": [179, 9], "def_end_pos": [179, 39]}, {"full_name": "smul_algebra_smul_comm", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [878, 9], "def_end_pos": [878, 31]}]], "state_before": "case succ\nn k : \u2115\nhk : 2 ^ n \u2022 powHalf (n + k) = powHalf k\n\u22a2 2 ^ n \u2022 powHalf (n + Nat.succ k) = powHalf (Nat.succ k)", "state_after": "case succ\nn k : \u2115\nhk : 2 \u2022 2 ^ n \u2022 powHalf (Nat.succ (n + k)) = 2 \u2022 powHalf (Nat.succ k)\n\u22a2 2 ^ n \u2022 powHalf (n + Nat.succ k) = powHalf (Nat.succ k)"}, {"tactic": "rwa [\u2190 zsmul_eq_zsmul_iff' two_ne_zero]", "annotated_tactic": ["rwa [\u2190 <a>zsmul_eq_zsmul_iff'</a> <a>two_ne_zero</a>]", [{"full_name": "zsmul_eq_zsmul_iff'", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [437, 3], "def_end_pos": [437, 14]}, {"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [62, 7], "def_end_pos": [62, 18]}]], "state_before": "case succ\nn k : \u2115\nhk : 2 \u2022 2 ^ n \u2022 powHalf (Nat.succ (n + k)) = 2 \u2022 powHalf (Nat.succ k)\n\u22a2 2 ^ n \u2022 powHalf (n + Nat.succ k) = powHalf (Nat.succ k)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Deprecated/Subfield.lean", "full_name": "Field.subset_closure", "start": [129, 1], "end": [130, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Semicontinuous.lean", "full_name": "lowerSemicontinuous_iff_isOpen_preimage", "start": [254, 1], "end": [257, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Compare.lean", "full_name": "eq_iff_eq_of_cmp_eq_cmp", "start": [262, 1], "end": [264, 53], "traced_tactics": [{"tactic": "rw [le_antisymm_iff, le_antisymm_iff, le_iff_le_of_cmp_eq_cmp h,\n    le_iff_le_of_cmp_eq_cmp (cmp_eq_cmp_symm.1 h)]", "annotated_tactic": ["rw [<a>le_antisymm_iff</a>, <a>le_antisymm_iff</a>, <a>le_iff_le_of_cmp_eq_cmp</a> h,\n      <a>le_iff_le_of_cmp_eq_cmp</a> (<a>cmp_eq_cmp_symm</a>.1 h)]", [{"full_name": "le_antisymm_iff", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [192, 9], "def_end_pos": [192, 24]}, {"full_name": "le_antisymm_iff", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [192, 9], "def_end_pos": [192, 24]}, {"full_name": "le_iff_le_of_cmp_eq_cmp", "def_path": "Mathlib/Order/Compare.lean", "def_pos": [255, 9], "def_end_pos": [255, 32]}, {"full_name": "le_iff_le_of_cmp_eq_cmp", "def_path": "Mathlib/Order/Compare.lean", "def_pos": [255, 9], "def_end_pos": [255, 32]}, {"full_name": "cmp_eq_cmp_symm", "def_path": "Mathlib/Order/Compare.lean", "def_pos": [246, 9], "def_end_pos": [246, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\nx y : \u03b1\n\u03b2 : Type u_3\ninst\u271d : LinearOrder \u03b2\nx' y' : \u03b2\nh : cmp x y = cmp x' y'\n\u22a2 x = y \u2194 x' = y'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/Deriv/Inverse.lean", "full_name": "HasDerivAt.tendsto_punctured_nhds", "start": [112, 1], "end": [115, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean", "full_name": "CategoryTheory.IsPushout.of_map", "start": [1046, 1], "end": [1053, 53], "traced_tactics": [{"tactic": "refine' \u27e8\u27e8e\u27e9, \u27e8isColimitOfReflects F <| _\u27e9\u27e9", "annotated_tactic": ["refine' \u27e8\u27e8e\u27e9, \u27e8<a>isColimitOfReflects</a> F <| _\u27e9\u27e9", [{"full_name": "CategoryTheory.Limits.isColimitOfReflects", "def_path": "Mathlib/CategoryTheory/Limits/Preserves/Basic.lean", "def_pos": [444, 5], "def_end_pos": [444, 24]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : ReflectsColimit (span f g) F\ne : f \u226b h = g \u226b i\nH : IsPushout (F.map f) (F.map g) (F.map h) (F.map i)\n\u22a2 IsPushout f g h i", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : ReflectsColimit (span f g) F\ne : f \u226b h = g \u226b i\nH : IsPushout (F.map f) (F.map g) (F.map h) (F.map i)\n\u22a2 IsColimit (F.mapCocone (PushoutCocone.mk h i (_ : f \u226b h = g \u226b i)))"}, {"tactic": "refine'\n  (IsColimit.equivOfNatIsoOfIso (spanCompIso F f g) _ _ (WalkingSpan.ext _ _ _)).symm\n    H.isColimit", "annotated_tactic": ["refine'\n    (<a>IsColimit.equivOfNatIsoOfIso</a> (<a>spanCompIso</a> F f g) _ _ (<a>WalkingSpan.ext</a> _ _ _)).<a>symm</a>\n      H.isColimit", [{"full_name": "CategoryTheory.Limits.IsColimit.equivOfNatIsoOfIso", "def_path": "Mathlib/CategoryTheory/Limits/IsLimit.lean", "def_pos": [804, 5], "def_end_pos": [804, 23]}, {"full_name": "CategoryTheory.Limits.spanCompIso", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [330, 5], "def_end_pos": [330, 16]}, {"full_name": "CategoryTheory.Limits.WalkingSpan.ext", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [169, 5], "def_end_pos": [169, 20]}, {"full_name": "Equiv.symm", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [157, 15], "def_end_pos": [157, 19]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : ReflectsColimit (span f g) F\ne : f \u226b h = g \u226b i\nH : IsPushout (F.map f) (F.map g) (F.map h) (F.map i)\n\u22a2 IsColimit (F.mapCocone (PushoutCocone.mk h i (_ : f \u226b h = g \u226b i)))", "state_after": "case refine'_1\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : ReflectsColimit (span f g) F\ne : f \u226b h = g \u226b i\nH : IsPushout (F.map f) (F.map g) (F.map h) (F.map i)\n\u22a2 ((Cocones.precompose (spanCompIso F f g).inv).obj (F.mapCocone (PushoutCocone.mk h i (_ : f \u226b h = g \u226b i)))).pt \u2245\n    (cocone H).pt\n\ncase refine'_2\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : ReflectsColimit (span f g) F\ne : f \u226b h = g \u226b i\nH : IsPushout (F.map f) (F.map g) (F.map h) (F.map i)\n\u22a2 ((Cocones.precompose (spanCompIso F f g).inv).obj (F.mapCocone (PushoutCocone.mk h i (_ : f \u226b h = g \u226b i)))).\u03b9.app\n        WalkingCospan.left \u226b\n      ?refine'_1.hom =\n    (cocone H).\u03b9.app WalkingCospan.left\n\ncase refine'_3\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : ReflectsColimit (span f g) F\ne : f \u226b h = g \u226b i\nH : IsPushout (F.map f) (F.map g) (F.map h) (F.map i)\n\u22a2 ((Cocones.precompose (spanCompIso F f g).inv).obj (F.mapCocone (PushoutCocone.mk h i (_ : f \u226b h = g \u226b i)))).\u03b9.app\n        WalkingCospan.right \u226b\n      ?refine'_1.hom =\n    (cocone H).\u03b9.app WalkingCospan.right"}, {"tactic": "exacts [Iso.refl _, (Category.comp_id _).trans (Category.id_comp _),\n  (Category.comp_id _).trans (Category.id_comp _)]", "annotated_tactic": ["exacts [<a>Iso.refl</a> _, (<a>Category.comp_id</a> _).<a>trans</a> (<a>Category.id_comp</a> _),\n    (<a>Category.comp_id</a> _).<a>trans</a> (<a>Category.id_comp</a> _)]", [{"full_name": "CategoryTheory.Iso.refl", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [134, 5], "def_end_pos": [134, 9]}, {"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [157, 3], "def_end_pos": [157, 10]}, {"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}, {"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [155, 3], "def_end_pos": [155, 10]}, {"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [157, 3], "def_end_pos": [157, 10]}, {"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}, {"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [155, 3], "def_end_pos": [155, 10]}]], "state_before": "case refine'_1\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : ReflectsColimit (span f g) F\ne : f \u226b h = g \u226b i\nH : IsPushout (F.map f) (F.map g) (F.map h) (F.map i)\n\u22a2 ((Cocones.precompose (spanCompIso F f g).inv).obj (F.mapCocone (PushoutCocone.mk h i (_ : f \u226b h = g \u226b i)))).pt \u2245\n    (cocone H).pt\n\ncase refine'_2\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : ReflectsColimit (span f g) F\ne : f \u226b h = g \u226b i\nH : IsPushout (F.map f) (F.map g) (F.map h) (F.map i)\n\u22a2 ((Cocones.precompose (spanCompIso F f g).inv).obj (F.mapCocone (PushoutCocone.mk h i (_ : f \u226b h = g \u226b i)))).\u03b9.app\n        WalkingCospan.left \u226b\n      ?refine'_1.hom =\n    (cocone H).\u03b9.app WalkingCospan.left\n\ncase refine'_3\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nW X Y Z : C\nf : W \u27f6 X\ng : W \u27f6 Y\nh : X \u27f6 Z\ni : Y \u27f6 Z\ninst\u271d : ReflectsColimit (span f g) F\ne : f \u226b h = g \u226b i\nH : IsPushout (F.map f) (F.map g) (F.map h) (F.map i)\n\u22a2 ((Cocones.precompose (spanCompIso F f g).inv).obj (F.mapCocone (PushoutCocone.mk h i (_ : f \u226b h = g \u226b i)))).\u03b9.app\n        WalkingCospan.right \u226b\n      ?refine'_1.hom =\n    (cocone H).\u03b9.app WalkingCospan.right", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Jacobson.lean", "full_name": "Ideal.isJacobson_of_isIntegral", "start": [132, 1], "end": [148, 66], "traced_tactics": [{"tactic": "rw [isJacobson_iff_prime_eq]", "annotated_tactic": ["rw [<a>isJacobson_iff_prime_eq</a>]", [{"full_name": "Ideal.isJacobson_iff_prime_eq", "def_path": "Mathlib/RingTheory/Jacobson.lean", "def_pos": [70, 9], "def_end_pos": [70, 32]}]], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\n\u22a2 IsJacobson S", "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\n\u22a2 \u2200 (P : Ideal S), IsPrime P \u2192 jacobson P = P"}, {"tactic": "intro P hP", "annotated_tactic": ["intro P hP", []], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\n\u22a2 \u2200 (P : Ideal S), IsPrime P \u2192 jacobson P = P", "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\n\u22a2 jacobson P = P"}, {"tactic": "by_cases hP_top : comap (algebraMap R S) P = \u22a4", "annotated_tactic": ["by_cases hP_top : <a>comap</a> (<a>algebraMap</a> R S) P = \u22a4", [{"full_name": "Ideal.comap", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [1372, 5], "def_end_pos": [1372, 10]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}]], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\n\u22a2 jacobson P = P", "state_after": "case pos\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : comap (algebraMap R S) P = \u22a4\n\u22a2 jacobson P = P\n\ncase neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\n\u22a2 jacobson P = P"}, {"tactic": "simp [comap_eq_top_iff.1 hP_top]", "annotated_tactic": ["simp [<a>comap_eq_top_iff</a>.1 hP_top]", [{"full_name": "Ideal.comap_eq_top_iff", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [1512, 9], "def_end_pos": [1512, 25]}]], "state_before": "case pos\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : comap (algebraMap R S) P = \u22a4\n\u22a2 jacobson P = P", "state_after": "no goals"}, {"tactic": "haveI : Nontrivial (R \u29f8 comap (algebraMap R S) P) := Quotient.nontrivial hP_top", "annotated_tactic": ["haveI : <a>Nontrivial</a> (R \u29f8 <a>comap</a> (<a>algebraMap</a> R S) P) := <a>Quotient.nontrivial</a> hP_top", [{"full_name": "Nontrivial", "def_path": "Mathlib/Logic/Nontrivial/Defs.lean", "def_pos": [29, 7], "def_end_pos": [29, 17]}, {"full_name": "Ideal.comap", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [1372, 5], "def_end_pos": [1372, 10]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "Ideal.Quotient.nontrivial", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [148, 19], "def_end_pos": [148, 29]}]], "state_before": "case neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\n\u22a2 jacobson P = P", "state_after": "case neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis : Nontrivial (R \u29f8 comap (algebraMap R S) P)\n\u22a2 jacobson P = P"}, {"tactic": "rw [jacobson_eq_iff_jacobson_quotient_eq_bot]", "annotated_tactic": ["rw [<a>jacobson_eq_iff_jacobson_quotient_eq_bot</a>]", [{"full_name": "Ideal.jacobson_eq_iff_jacobson_quotient_eq_bot", "def_path": "Mathlib/RingTheory/JacobsonIdeal.lean", "def_pos": [272, 9], "def_end_pos": [272, 49]}]], "state_before": "case neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis : Nontrivial (R \u29f8 comap (algebraMap R S) P)\n\u22a2 jacobson P = P", "state_after": "case neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis : Nontrivial (R \u29f8 comap (algebraMap R S) P)\n\u22a2 jacobson \u22a5 = \u22a5"}, {"tactic": "refine' eq_bot_of_comap_eq_bot (isIntegral_quotient_of_isIntegral hRS) _", "annotated_tactic": ["refine' <a>eq_bot_of_comap_eq_bot</a> (<a>isIntegral_quotient_of_isIntegral</a> hRS) _", [{"full_name": "Ideal.eq_bot_of_comap_eq_bot", "def_path": "Mathlib/RingTheory/Ideal/Over.lean", "def_pos": [275, 9], "def_end_pos": [275, 31]}, {"full_name": "isIntegral_quotient_of_isIntegral", "def_path": "Mathlib/RingTheory/IntegralClosure.lean", "def_pos": [1108, 9], "def_end_pos": [1108, 42]}]], "state_before": "case neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis : Nontrivial (R \u29f8 comap (algebraMap R S) P)\n\u22a2 jacobson \u22a5 = \u22a5", "state_after": "case neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis : Nontrivial (R \u29f8 comap (algebraMap R S) P)\n\u22a2 comap (algebraMap (R \u29f8 comap (algebraMap R S) P) (S \u29f8 P)) (jacobson \u22a5) = \u22a5"}, {"tactic": "rw [eq_bot_iff, \u2190 jacobson_eq_iff_jacobson_quotient_eq_bot.1\n  ((isJacobson_iff_prime_eq.1 hR) (comap (algebraMap R S) P) (comap_isPrime _ _)),\n  comap_jacobson]", "annotated_tactic": ["rw [<a>eq_bot_iff</a>, \u2190 <a>jacobson_eq_iff_jacobson_quotient_eq_bot</a>.1\n      ((<a>isJacobson_iff_prime_eq</a>.1 hR) (<a>comap</a> (<a>algebraMap</a> R S) P) (<a>comap_isPrime</a> _ _)),\n      <a>comap_jacobson</a>]", [{"full_name": "eq_bot_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [363, 9], "def_end_pos": [363, 19]}, {"full_name": "Ideal.jacobson_eq_iff_jacobson_quotient_eq_bot", "def_path": "Mathlib/RingTheory/JacobsonIdeal.lean", "def_pos": [272, 9], "def_end_pos": [272, 49]}, {"full_name": "Ideal.isJacobson_iff_prime_eq", "def_path": "Mathlib/RingTheory/Jacobson.lean", "def_pos": [70, 9], "def_end_pos": [70, 32]}, {"full_name": "Ideal.comap", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [1372, 5], "def_end_pos": [1372, 10]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "Ideal.comap_isPrime", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [1564, 9], "def_end_pos": [1564, 22]}, {"full_name": "Ideal.comap_jacobson", "def_path": "Mathlib/RingTheory/JacobsonIdeal.lean", "def_pos": [204, 9], "def_end_pos": [204, 23]}]], "state_before": "case neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis : Nontrivial (R \u29f8 comap (algebraMap R S) P)\n\u22a2 comap (algebraMap (R \u29f8 comap (algebraMap R S) P) (S \u29f8 P)) (jacobson \u22a5) = \u22a5", "state_after": "case neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis : Nontrivial (R \u29f8 comap (algebraMap R S) P)\n\u22a2 sInf (comap (algebraMap (R \u29f8 comap (algebraMap R S) P) (S \u29f8 P)) '' {J | \u22a5 \u2264 J \u2227 IsMaximal J}) \u2264 jacobson \u22a5"}, {"tactic": "refine' sInf_le_sInf fun J hJ => _", "annotated_tactic": ["refine' <a>sInf_le_sInf</a> fun J hJ => _", [{"full_name": "sInf_le_sInf", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [286, 9], "def_end_pos": [286, 21]}]], "state_before": "case neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis : Nontrivial (R \u29f8 comap (algebraMap R S) P)\n\u22a2 sInf (comap (algebraMap (R \u29f8 comap (algebraMap R S) P) (S \u29f8 P)) '' {J | \u22a5 \u2264 J \u2227 IsMaximal J}) \u2264 jacobson \u22a5", "state_after": "case neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis : Nontrivial (R \u29f8 comap (algebraMap R S) P)\nJ : Ideal (R \u29f8 comap (algebraMap R S) P)\nhJ : J \u2208 {J | \u22a5 \u2264 J \u2227 IsMaximal J}\n\u22a2 J \u2208 comap (algebraMap (R \u29f8 comap (algebraMap R S) P) (S \u29f8 P)) '' {J | \u22a5 \u2264 J \u2227 IsMaximal J}"}, {"tactic": "simp only [true_and_iff, Set.mem_image, bot_le, Set.mem_setOf_eq]", "annotated_tactic": ["simp only [<a>true_and_iff</a>, <a>Set.mem_image</a>, <a>bot_le</a>, <a>Set.mem_setOf_eq</a>]", [{"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [147, 9], "def_end_pos": [147, 21]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}, {"full_name": "bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [256, 9], "def_end_pos": [256, 15]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "case neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis : Nontrivial (R \u29f8 comap (algebraMap R S) P)\nJ : Ideal (R \u29f8 comap (algebraMap R S) P)\nhJ : J \u2208 {J | \u22a5 \u2264 J \u2227 IsMaximal J}\n\u22a2 J \u2208 comap (algebraMap (R \u29f8 comap (algebraMap R S) P) (S \u29f8 P)) '' {J | \u22a5 \u2264 J \u2227 IsMaximal J}", "state_after": "case neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis : Nontrivial (R \u29f8 comap (algebraMap R S) P)\nJ : Ideal (R \u29f8 comap (algebraMap R S) P)\nhJ : J \u2208 {J | \u22a5 \u2264 J \u2227 IsMaximal J}\n\u22a2 \u2203 x, IsMaximal x \u2227 comap (algebraMap (R \u29f8 comap (algebraMap R S) P) (S \u29f8 P)) x = J"}, {"tactic": "have : J.IsMaximal := by simpa using hJ", "annotated_tactic": ["have : J.IsMaximal := by simpa using hJ", []], "state_before": "case neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis : Nontrivial (R \u29f8 comap (algebraMap R S) P)\nJ : Ideal (R \u29f8 comap (algebraMap R S) P)\nhJ : J \u2208 {J | \u22a5 \u2264 J \u2227 IsMaximal J}\n\u22a2 \u2203 x, IsMaximal x \u2227 comap (algebraMap (R \u29f8 comap (algebraMap R S) P) (S \u29f8 P)) x = J", "state_after": "case neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis\u271d : Nontrivial (R \u29f8 comap (algebraMap R S) P)\nJ : Ideal (R \u29f8 comap (algebraMap R S) P)\nhJ : J \u2208 {J | \u22a5 \u2264 J \u2227 IsMaximal J}\nthis : IsMaximal J\n\u22a2 \u2203 x, IsMaximal x \u2227 comap (algebraMap (R \u29f8 comap (algebraMap R S) P) (S \u29f8 P)) x = J"}, {"tactic": "exact exists_ideal_over_maximal_of_isIntegral (isIntegral_quotient_of_isIntegral hRS) J\n  (comap_bot_le_of_injective _ algebraMap_quotient_injective)", "annotated_tactic": ["exact <a>exists_ideal_over_maximal_of_isIntegral</a> (<a>isIntegral_quotient_of_isIntegral</a> hRS) J\n      (<a>comap_bot_le_of_injective</a> _ <a>algebraMap_quotient_injective</a>)", [{"full_name": "Ideal.exists_ideal_over_maximal_of_isIntegral", "def_path": "Mathlib/RingTheory/Ideal/Over.lean", "def_pos": [407, 9], "def_end_pos": [407, 48]}, {"full_name": "isIntegral_quotient_of_isIntegral", "def_path": "Mathlib/RingTheory/IntegralClosure.lean", "def_pos": [1108, 9], "def_end_pos": [1108, 42]}, {"full_name": "Ideal.comap_bot_le_of_injective", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [1692, 9], "def_end_pos": [1692, 34]}, {"full_name": "Ideal.algebraMap_quotient_injective", "def_path": "Mathlib/RingTheory/Ideal/QuotientOperations.lean", "def_pos": [588, 9], "def_end_pos": [588, 38]}]], "state_before": "case neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis\u271d : Nontrivial (R \u29f8 comap (algebraMap R S) P)\nJ : Ideal (R \u29f8 comap (algebraMap R S) P)\nhJ : J \u2208 {J | \u22a5 \u2264 J \u2227 IsMaximal J}\nthis : IsMaximal J\n\u22a2 \u2203 x, IsMaximal x \u2227 comap (algebraMap (R \u29f8 comap (algebraMap R S) P) (S \u29f8 P)) x = J", "state_after": "no goals"}, {"tactic": "simpa using hJ", "annotated_tactic": ["simpa using hJ", []], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI : Ideal R\ninst\u271d : Algebra R S\nhRS : Algebra.IsIntegral R S\nhR : IsJacobson R\nP : Ideal S\nhP : IsPrime P\nhP_top : \u00accomap (algebraMap R S) P = \u22a4\nthis : Nontrivial (R \u29f8 comap (algebraMap R S) P)\nJ : Ideal (R \u29f8 comap (algebraMap R S) P)\nhJ : J \u2208 {J | \u22a5 \u2264 J \u2227 IsMaximal J}\n\u22a2 IsMaximal J", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Torsion.lean", "full_name": "ExponentExists.isTorsion", "start": [132, 1], "end": [134, 60], "traced_tactics": [{"tactic": "obtain \u27e8n, npos, hn\u27e9 := h", "annotated_tactic": ["obtain \u27e8n, npos, hn\u27e9 := h", []], "state_before": "G : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Group H\nh : ExponentExists G\ng : G\n\u22a2 IsOfFinOrder g", "state_after": "case intro.intro\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Group H\ng : G\nn : \u2115\nnpos : 0 < n\nhn : \u2200 (g : G), g ^ n = 1\n\u22a2 IsOfFinOrder g"}, {"tactic": "exact (isOfFinOrder_iff_pow_eq_one g).mpr \u27e8n, npos, hn g\u27e9", "annotated_tactic": ["exact (<a>isOfFinOrder_iff_pow_eq_one</a> g).<a>mpr</a> \u27e8n, npos, hn g\u27e9", [{"full_name": "isOfFinOrder_iff_pow_eq_one", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [70, 9], "def_end_pos": [70, 36]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "case intro.intro\nG : Type u_1\nH : Type u_2\ninst\u271d\u00b9 : Group G\nN : Subgroup G\ninst\u271d : Group H\ng : G\nn : \u2115\nnpos : 0 < n\nhn : \u2200 (g : G), g ^ n = 1\n\u22a2 IsOfFinOrder g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Game/PGame.lean", "full_name": "SetTheory.PGame.zero_lf_le", "start": [661, 1], "end": [663, 7], "traced_tactics": [{"tactic": "rw [lf_iff_exists_le]", "annotated_tactic": ["rw [<a>lf_iff_exists_le</a>]", [{"full_name": "SetTheory.PGame.lf_iff_exists_le", "def_path": "Mathlib/SetTheory/Game/PGame.lean", "def_pos": [455, 9], "def_end_pos": [455, 25]}]], "state_before": "x : PGame\n\u22a2 0 \u29cf x \u2194 \u2203 i, 0 \u2264 moveLeft x i", "state_after": "x : PGame\n\u22a2 ((\u2203 i, 0 \u2264 moveLeft x i) \u2228 \u2203 j, moveRight 0 j \u2264 x) \u2194 \u2203 i, 0 \u2264 moveLeft x i"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "x : PGame\n\u22a2 ((\u2203 i, 0 \u2264 moveLeft x i) \u2228 \u2203 j, moveRight 0 j \u2264 x) \u2194 \u2203 i, 0 \u2264 moveLeft x i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Monad.lean", "full_name": "MvPolynomial.mem_vars_bind\u2081", "start": [391, 1], "end": [394, 95], "traced_tactics": [{"tactic": "classical\nsimpa only [exists_prop, Finset.mem_biUnion, mem_support_iff, Ne.def] using vars_bind\u2081 f \u03c6 h", "annotated_tactic": ["classical\n  simpa only [<a>exists_prop</a>, <a>Finset.mem_biUnion</a>, <a>mem_support_iff</a>, <a>Ne.def</a>] using <a>vars_bind\u2081</a> f \u03c6 h", [{"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Finset.mem_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3613, 9], "def_end_pos": [3613, 20]}, {"full_name": "MvPolynomial.mem_support_iff", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [587, 9], "def_end_pos": [587, 24]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "MvPolynomial.vars_bind\u2081", "def_path": "Mathlib/Data/MvPolynomial/Monad.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}]], "state_before": "\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d f : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nj : \u03c4\nh : j \u2208 vars (\u2191(bind\u2081 f) \u03c6)\n\u22a2 \u2203 i, i \u2208 vars \u03c6 \u2227 j \u2208 vars (f i)", "state_after": "no goals"}, {"tactic": "simpa only [exists_prop, Finset.mem_biUnion, mem_support_iff, Ne.def] using vars_bind\u2081 f \u03c6 h", "annotated_tactic": ["simpa only [<a>exists_prop</a>, <a>Finset.mem_biUnion</a>, <a>mem_support_iff</a>, <a>Ne.def</a>] using <a>vars_bind\u2081</a> f \u03c6 h", [{"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Finset.mem_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3613, 9], "def_end_pos": [3613, 20]}, {"full_name": "MvPolynomial.mem_support_iff", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [587, 9], "def_end_pos": [587, 24]}, {"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "MvPolynomial.vars_bind\u2081", "def_path": "Mathlib/Data/MvPolynomial/Monad.lean", "def_pos": [357, 9], "def_end_pos": [357, 19]}]], "state_before": "\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : CommSemiring T\nf\u271d f : \u03c3 \u2192 MvPolynomial \u03c4 R\n\u03c6 : MvPolynomial \u03c3 R\nj : \u03c4\nh : j \u2208 vars (\u2191(bind\u2081 f) \u03c6)\n\u22a2 \u2203 i, i \u2208 vars \u03c6 \u2227 j \u2208 vars (f i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Ordinal/FixedPoint.lean", "full_name": "Ordinal.fp_iff_derivBFamily", "start": [384, 1], "end": [389, 15], "traced_tactics": [{"tactic": "rw [\u2190 le_iff_derivBFamily H]", "annotated_tactic": ["rw [\u2190 <a>le_iff_derivBFamily</a> H]", [{"full_name": "Ordinal.le_iff_derivBFamily", "def_path": "Mathlib/SetTheory/Ordinal/FixedPoint.lean", "def_pos": [374, 9], "def_end_pos": [374, 28]}]], "state_before": "o : Ordinal.{u}\nf : (b : Ordinal.{u}) \u2192 b < o \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v}\nH : \u2200 (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi)\na : Ordinal.{max u v}\n\u22a2 (\u2200 (i : Ordinal.{u}) (hi : i < o), f i hi a = a) \u2194 \u2203 b, derivBFamily o f b = a", "state_after": "o : Ordinal.{u}\nf : (b : Ordinal.{u}) \u2192 b < o \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v}\nH : \u2200 (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi)\na : Ordinal.{max u v}\n\u22a2 (\u2200 (i : Ordinal.{u}) (hi : i < o), f i hi a = a) \u2194 \u2200 (i : Ordinal.{u}) (hi : i < o), f i hi a \u2264 a"}, {"tactic": "refine' \u27e8fun h i hi => le_of_eq (h i hi), fun h i hi => _\u27e9", "annotated_tactic": ["refine' 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: Ordinal.{max u v}\nh : \u2200 (i : Ordinal.{u}) (hi : i < o), f i hi a \u2264 a\ni : Ordinal.{u}\nhi : i < o\n\u22a2 f i hi a \u2264 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean", "full_name": "CategoryTheory.Subgroupoid.map_comap_le", "start": [523, 1], "end": [525, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Monoid.lean", "full_name": "le_nhds_mul", "start": [140, 1], "end": [142, 33], "traced_tactics": [{"tactic": "rw [\u2190 map\u2082_mul, \u2190 map_uncurry_prod, \u2190 nhds_prod_eq]", "annotated_tactic": ["rw [\u2190 <a>map\u2082_mul</a>, \u2190 <a>map_uncurry_prod</a>, \u2190 <a>nhds_prod_eq</a>]", [{"full_name": "Filter.map\u2082_mul", "def_path": "Mathlib/Order/Filter/Pointwise.lean", "def_pos": [290, 9], "def_end_pos": [290, 17]}, {"full_name": "Filter.map_uncurry_prod", "def_path": "Mathlib/Order/Filter/NAry.lean", "def_pos": [283, 9], "def_end_pos": [283, 25]}, {"full_name": "nhds_prod_eq", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [513, 9], "def_end_pos": [513, 21]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Mul M\ninst\u271d : ContinuousMul M\na b : M\n\u22a2 \ud835\udcdd a * \ud835\udcdd b \u2264 \ud835\udcdd (a * b)", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Mul M\ninst\u271d : ContinuousMul M\na b : M\n\u22a2 map (Function.uncurry fun x x_1 => x * x_1) (\ud835\udcdd (a, b)) \u2264 \ud835\udcdd (a * b)"}, {"tactic": "exact continuous_mul.tendsto _", "annotated_tactic": ["exact continuous_mul.tendsto _", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\nX : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : Mul M\ninst\u271d : ContinuousMul M\na b : M\n\u22a2 map (Function.uncurry fun x x_1 => x * x_1) (\ud835\udcdd (a, b)) \u2264 \ud835\udcdd (a * b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.of_mem_filter", "start": [1986, 1], "end": [1987, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "full_name": "MeasureTheory.condexp_mono", "start": [347, 1], "end": [356, 78], "traced_tactics": [{"tactic": "by_cases hm : m \u2264 m0", "annotated_tactic": ["by_cases hm : m \u2264 m0", []], "state_before": "\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]\n\ncase pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]"}, {"tactic": "by_cases h\u03bcm : SigmaFinite (\u03bc.trim hm)", "annotated_tactic": ["by_cases h\u03bcm : <a>SigmaFinite</a> (\u03bc.trim hm)", [{"full_name": "MeasureTheory.SigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3289, 7], "def_end_pos": [3289, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]\n\ncase neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]\n\ncase pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]"}, {"tactic": "haveI : SigmaFinite (\u03bc.trim hm) := h\u03bcm", "annotated_tactic": ["haveI : <a>SigmaFinite</a> (\u03bc.trim hm) := h\u03bcm", [{"full_name": "MeasureTheory.SigmaFinite", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3289, 7], "def_end_pos": [3289, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]"}, {"tactic": "exact (condexp_ae_eq_condexpL1 hm _).trans_le\n  ((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_condexpL1 hm _).symm)", "annotated_tactic": ["exact (<a>condexp_ae_eq_condexpL1</a> hm _).<a>trans_le</a>\n    ((<a>condexpL1_mono</a> hf hg hfg).<a>trans_eq</a> (<a>condexp_ae_eq_condexpL1</a> hm _).<a>symm</a>)", [{"full_name": "MeasureTheory.condexp_ae_eq_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 32]}, {"full_name": "Filter.EventuallyEq.trans_le", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1684, 9], "def_end_pos": [1684, 30]}, {"full_name": "MeasureTheory.condexpL1_mono", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [592, 9], "def_end_pos": [592, 23]}, {"full_name": "Filter.EventuallyLE.trans_eq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1693, 9], "def_end_pos": [1693, 30]}, {"full_name": "MeasureTheory.condexp_ae_eq_condexpL1", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [136, 9], "def_end_pos": [136, 32]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1498, 9], "def_end_pos": [1498, 26]}]], "state_before": "case pos\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm this : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]", "state_after": "no goals"}, {"tactic": "simp_rw [condexp_of_not_le hm]", "annotated_tactic": ["simp_rw [<a>condexp_of_not_le</a> hm]", [{"full_name": "MeasureTheory.condexp_of_not_le", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [106, 9], "def_end_pos": [106, 26]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : \u00acm \u2264 m0\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : \u00acm \u2264 m0\n\u22a2 0 \u2264\u1d50[\u03bc] 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : \u00acm \u2264 m0\n\u22a2 0 \u2264\u1d50[\u03bc] 0", "state_after": "no goals"}, {"tactic": "simp_rw [condexp_of_not_sigmaFinite hm h\u03bcm]", "annotated_tactic": ["simp_rw [<a>condexp_of_not_sigmaFinite</a> hm h\u03bcm]", [{"full_name": "MeasureTheory.condexp_of_not_sigmaFinite", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [109, 9], "def_end_pos": [109, 35]}]], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u03bc[f|m] \u2264\u1d50[\u03bc] \u03bc[g|m]", "state_after": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 0 \u2264\u1d50[\u03bc] 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\n\u03b1 : Type u_1\nF : Type u_2\nF' : Type u_3\n\ud835\udd5c : Type u_4\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup F'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : CompleteSpace F'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nE : Type u_5\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : OrderedSMul \u211d E\nf g : \u03b1 \u2192 E\nhf : Integrable f\nhg : Integrable g\nhfg : f \u2264\u1d50[\u03bc] g\nhm : m \u2264 m0\nh\u03bcm : \u00acSigmaFinite (Measure.trim \u03bc hm)\n\u22a2 0 \u2264\u1d50[\u03bc] 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/AEDisjoint.lean", "full_name": "MeasureTheory.AEDisjoint.of_null_right", "start": [147, 1], "end": [148, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "full_name": "extChartAt_preimage_mem_nhdsWithin", "start": [1437, 1], "end": [1439, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "full_name": "le_one_of_mul_le_left", "start": [450, 1], "end": [453, 53], "traced_tactics": [{"tactic": "simpa only [one_mul]", "annotated_tactic": ["simpa only [<a>one_mul</a>]", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nh : a * b \u2264 b\n\u22a2 a * ?m.22212 h \u2264 1 * ?m.22212 h", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Control/Traversable/Basic.lean", "full_name": "ApplicativeTransformation.preserves_pure", "start": [140, 1], "end": [141, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Order/LeftRightLim.lean", "full_name": "leftLim_eq_of_tendsto", "start": [65, 1], "end": [72, 18], "traced_tactics": [{"tactic": "have h'' : \u2203 y, Tendsto f (\ud835\udcdd[<] a) (\ud835\udcdd y) := \u27e8y, h'\u27e9", "annotated_tactic": ["have h'' : \u2203 y, <a>Tendsto</a> f (\ud835\udcdd[<] a) (\ud835\udcdd y) := \u27e8y, h'\u27e9", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nh\u03b1 : TopologicalSpace \u03b1\nh'\u03b1 : OrderTopology \u03b1\ninst\u271d : T2Space \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ny : \u03b2\nh : \ud835\udcdd[Iio a] a \u2260 \u22a5\nh' : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd y)\n\u22a2 leftLim f a = y", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nh\u03b1 : TopologicalSpace \u03b1\nh'\u03b1 : OrderTopology \u03b1\ninst\u271d : T2Space \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ny : \u03b2\nh : \ud835\udcdd[Iio a] a \u2260 \u22a5\nh' : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd y)\nh'' : \u2203 y, Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd y)\n\u22a2 leftLim f a = y"}, {"tactic": "simp only [leftLim, h, h'', not_true, or_self_iff, if_false]", "annotated_tactic": ["simp only [<a>leftLim</a>, h, h'', <a>not_true</a>, <a>or_self_iff</a>, <a>if_false</a>]", [{"full_name": "Function.leftLim", "def_path": "Mathlib/Topology/Algebra/Order/LeftRightLim.lean", "def_pos": [48, 19], "def_end_pos": [48, 35]}, {"full_name": "not_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [80, 17], "def_end_pos": [80, 25]}, {"full_name": "or_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [190, 9], "def_end_pos": [190, 20]}, {"full_name": "if_false", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [729, 17], "def_end_pos": [729, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nh\u03b1 : TopologicalSpace \u03b1\nh'\u03b1 : OrderTopology \u03b1\ninst\u271d : T2Space \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ny : \u03b2\nh : \ud835\udcdd[Iio a] a \u2260 \u22a5\nh' : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd y)\nh'' : \u2203 y, Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd y)\n\u22a2 leftLim f a = y", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nh\u03b1 : TopologicalSpace \u03b1\nh'\u03b1 : OrderTopology \u03b1\ninst\u271d : T2Space \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ny : \u03b2\nh : \ud835\udcdd[Iio a] a \u2260 \u22a5\nh' : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd y)\nh'' : \u2203 y, Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd y)\n\u22a2 limUnder (\ud835\udcdd[Iio a] a) f = y"}, {"tactic": "haveI := neBot_iff.2 h", "annotated_tactic": ["haveI := <a>neBot_iff</a>.2 h", [{"full_name": "Filter.neBot_iff", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [500, 9], "def_end_pos": [500, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nh\u03b1 : TopologicalSpace \u03b1\nh'\u03b1 : OrderTopology \u03b1\ninst\u271d : T2Space \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ny : \u03b2\nh : \ud835\udcdd[Iio a] a \u2260 \u22a5\nh' : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd y)\nh'' : \u2203 y, Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd y)\n\u22a2 limUnder (\ud835\udcdd[Iio a] a) f = y", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nh\u03b1 : TopologicalSpace \u03b1\nh'\u03b1 : OrderTopology \u03b1\ninst\u271d : T2Space \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ny : \u03b2\nh : \ud835\udcdd[Iio a] a \u2260 \u22a5\nh' : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd y)\nh'' : \u2203 y, Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd y)\nthis : NeBot (\ud835\udcdd[Iio a] a)\n\u22a2 limUnder (\ud835\udcdd[Iio a] a) f = y"}, {"tactic": "exact lim_eq h'", "annotated_tactic": ["exact <a>lim_eq</a> h'", [{"full_name": "lim_eq", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [1062, 9], "def_end_pos": [1062, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nh\u03b1 : TopologicalSpace \u03b1\nh'\u03b1 : OrderTopology \u03b1\ninst\u271d : T2Space \u03b2\nf : \u03b1 \u2192 \u03b2\na : \u03b1\ny : \u03b2\nh : \ud835\udcdd[Iio a] a \u2260 \u22a5\nh' : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd y)\nh'' : \u2203 y, Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd y)\nthis : NeBot (\ud835\udcdd[Iio a] a)\n\u22a2 limUnder (\ud835\udcdd[Iio a] a) f = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Perm/Support.lean", "full_name": "Equiv.Perm.support_swap", "start": [430, 1], "end": [436, 12], "traced_tactics": [{"tactic": "ext z", "annotated_tactic": ["ext z", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : x \u2260 y\n\u22a2 support (swap x y) = {x, y}", "state_after": "case a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : x \u2260 y\nz : \u03b1\n\u22a2 z \u2208 support (swap x y) \u2194 z \u2208 {x, y}"}, {"tactic": "by_cases hx : z = x", "annotated_tactic": ["by_cases hx : z = x", []], "state_before": "case a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : x \u2260 y\nz : \u03b1\n\u22a2 z \u2208 support (swap x y) \u2194 z \u2208 {x, y}", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : x \u2260 y\nz : \u03b1\nhx : z = x\n\u22a2 z \u2208 support (swap x y) \u2194 z \u2208 {x, y}\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : x \u2260 y\nz : \u03b1\nhx : \u00acz = x\n\u22a2 z \u2208 support (swap x y) \u2194 z \u2208 {x, y}"}, {"tactic": "any_goals simpa [hx] using h.symm", "annotated_tactic": ["any_goals simpa [hx] using h.symm", []], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : x \u2260 y\nz : \u03b1\nhx : z = x\n\u22a2 z \u2208 support (swap x y) \u2194 z \u2208 {x, y}\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : x \u2260 y\nz : \u03b1\nhx : \u00acz = x\n\u22a2 z \u2208 support (swap x y) \u2194 z \u2208 {x, y}", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : x \u2260 y\nz : \u03b1\nhx : \u00acz = x\n\u22a2 z \u2208 support (swap x y) \u2194 z \u2208 {x, y}"}, {"tactic": "simpa [hx] using h.symm", "annotated_tactic": ["simpa [hx] using h.symm", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : x \u2260 y\nz : \u03b1\nhx : \u00acz = x\n\u22a2 z \u2208 support (swap x y) \u2194 z \u2208 {x, y}", "state_after": "no goals"}, {"tactic": "simp [swap_apply_of_ne_of_ne, hx, hy] <;>\nexact h", "annotated_tactic": ["simp [<a>swap_apply_of_ne_of_ne</a>, hx, hy] <;>\n    exact h", [{"full_name": "Equiv.swap_apply_of_ne_of_ne", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1650, 9], "def_end_pos": [1650, 31]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nh : x \u2260 y\nz : \u03b1\nhx : \u00acz = x\nhy : \u00acz = y\n\u22a2 z \u2208 support (swap x y) \u2194 z \u2208 {x, y}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "full_name": "MeasureTheory.JordanDecomposition.toSignedMeasure_smul", "start": [187, 1], "end": [193, 101], "traced_tactics": [{"tactic": "ext1 i hi", "annotated_tactic": ["ext1 i hi", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\nr : \u211d\u22650\n\u22a2 toSignedMeasure (r \u2022 j) = r \u2022 toSignedMeasure j", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\nr : \u211d\u22650\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(toSignedMeasure (r \u2022 j)) i = \u2191(r \u2022 toSignedMeasure j) i"}, {"tactic": "rw [VectorMeasure.smul_apply, toSignedMeasure, toSignedMeasure,\n  toSignedMeasure_sub_apply hi, toSignedMeasure_sub_apply hi, smul_sub, smul_posPart,\n  smul_negPart, \u2190 ENNReal.toReal_smul, \u2190 ENNReal.toReal_smul, smul_toOuterMeasure,\n  OuterMeasure.coe_smul, Pi.smul_apply, smul_toOuterMeasure, OuterMeasure.coe_smul, Pi.smul_apply]", "annotated_tactic": ["rw [<a>VectorMeasure.smul_apply</a>, <a>toSignedMeasure</a>, <a>toSignedMeasure</a>,\n    <a>toSignedMeasure_sub_apply</a> hi, <a>toSignedMeasure_sub_apply</a> hi, <a>smul_sub</a>, <a>smul_posPart</a>,\n    <a>smul_negPart</a>, \u2190 <a>ENNReal.toReal_smul</a>, \u2190 <a>ENNReal.toReal_smul</a>, <a>smul_toOuterMeasure</a>,\n    <a>OuterMeasure.coe_smul</a>, <a>Pi.smul_apply</a>, <a>smul_toOuterMeasure</a>, <a>OuterMeasure.coe_smul</a>, <a>Pi.smul_apply</a>]", [{"full_name": "MeasureTheory.VectorMeasure.smul_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [272, 9], "def_end_pos": [272, 19]}, {"full_name": "MeasureTheory.JordanDecomposition.toSignedMeasure", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [169, 5], "def_end_pos": [169, 20]}, {"full_name": "MeasureTheory.JordanDecomposition.toSignedMeasure", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [169, 5], "def_end_pos": [169, 20]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure_sub_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [514, 9], "def_end_pos": [514, 34]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure_sub_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [514, 9], "def_end_pos": [514, 34]}, {"full_name": "smul_sub", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [988, 9], "def_end_pos": [988, 17]}, {"full_name": "MeasureTheory.JordanDecomposition.smul_posPart", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [120, 9], "def_end_pos": [120, 21]}, {"full_name": "MeasureTheory.JordanDecomposition.smul_negPart", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [125, 9], "def_end_pos": [125, 21]}, {"full_name": "ENNReal.toReal_smul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2333, 9], "def_end_pos": [2333, 20]}, {"full_name": "ENNReal.toReal_smul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2333, 9], "def_end_pos": [2333, 20]}, {"full_name": "MeasureTheory.Measure.smul_toOuterMeasure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [830, 9], "def_end_pos": [830, 28]}, {"full_name": "MeasureTheory.OuterMeasure.coe_smul", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [306, 9], "def_end_pos": [306, 17]}, {"full_name": "Pi.smul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [116, 60], "def_end_pos": [116, 70]}, {"full_name": "MeasureTheory.Measure.smul_toOuterMeasure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [830, 9], "def_end_pos": [830, 28]}, {"full_name": "MeasureTheory.OuterMeasure.coe_smul", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [306, 9], "def_end_pos": [306, 17]}, {"full_name": "Pi.smul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [116, 60], "def_end_pos": [116, 70]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj : JordanDecomposition \u03b1\nr : \u211d\u22650\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 \u2191(toSignedMeasure (r \u2022 j)) i = \u2191(r \u2022 toSignedMeasure j) i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "full_name": "Set.Subsingleton.measurableSet", "start": [301, 1], "end": [302, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Bounded.lean", "full_name": "Set.Bounded.mono", "start": [26, 1], "end": [27, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Ordinal/FixedPoint.lean", "full_name": "Ordinal.nfp_le_iff", "start": [449, 1], "end": [451, 19], "traced_tactics": [{"tactic": "rw [\u2190 sup_iterate_eq_nfp]", "annotated_tactic": ["rw [\u2190 <a>sup_iterate_eq_nfp</a>]", [{"full_name": "Ordinal.sup_iterate_eq_nfp", "def_path": "Mathlib/SetTheory/Ordinal/FixedPoint.lean", "def_pos": [424, 9], "def_end_pos": [424, 27]}]], "state_before": "f : Ordinal.{u} \u2192 Ordinal.{u}\na b : Ordinal.{u}\n\u22a2 nfp f a \u2264 b \u2194 \u2200 (n : \u2115), f^[n] a \u2264 b", "state_after": "f : Ordinal.{u} \u2192 Ordinal.{u}\na b : Ordinal.{u}\n\u22a2 (fun a => sup fun n => f^[n] a) a \u2264 b \u2194 \u2200 (n : \u2115), f^[n] a \u2264 b"}, {"tactic": "exact sup_le_iff", "annotated_tactic": ["exact <a>sup_le_iff</a>", [{"full_name": "Ordinal.sup_le_iff", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [1265, 9], "def_end_pos": [1265, 19]}]], "state_before": "f : Ordinal.{u} \u2192 Ordinal.{u}\na b : Ordinal.{u}\n\u22a2 (fun a => sup fun n => f^[n] a) a \u2264 b \u2194 \u2200 (n : \u2115), f^[n] a \u2264 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "full_name": "Complex.continuous_sinh", "start": [72, 1], "end": [74, 13], "traced_tactics": [{"tactic": "change Continuous fun z => (exp z - exp (-z)) / 2", "annotated_tactic": ["change <a>Continuous</a> fun z => (<a>exp</a> z - <a>exp</a> (-z)) / 2", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1591, 11], "def_end_pos": [1591, 21]}, {"full_name": "Complex.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [379, 5], "def_end_pos": [379, 8]}, {"full_name": "Complex.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [379, 5], "def_end_pos": [379, 8]}]], "state_before": "\u22a2 Continuous sinh", "state_after": "\u22a2 Continuous fun z => (cexp z - cexp (-z)) / 2"}, {"tactic": "continuity", "annotated_tactic": ["continuity", []], "state_before": "\u22a2 Continuous fun z => (cexp z - cexp (-z)) / 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "IsClosed.mul_right_of_isCompact", "start": [1449, 1], "end": [1452, 68], "traced_tactics": [{"tactic": "rw [\u2190 image_op_smul]", "annotated_tactic": ["rw [\u2190 <a>image_op_smul</a>]", [{"full_name": "Set.image_op_smul", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [422, 9], "def_end_pos": [422, 22]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nG : Type w\nH : Type x\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : TopologicalGroup \u03b1\ns t : Set \u03b1\nht : IsClosed t\nhs : IsCompact s\n\u22a2 IsClosed (t * s)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nG : Type w\nH : Type x\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : TopologicalGroup \u03b1\ns t : Set \u03b1\nht : IsClosed t\nhs : IsCompact s\n\u22a2 IsClosed (op '' s \u2022 t)"}, {"tactic": "exact IsClosed.smul_left_of_isCompact ht (hs.image continuous_op)", "annotated_tactic": ["exact <a>IsClosed.smul_left_of_isCompact</a> ht (hs.image <a>continuous_op</a>)", [{"full_name": "IsClosed.smul_left_of_isCompact", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1258, 9], "def_end_pos": [1258, 40]}, {"full_name": "MulOpposite.continuous_op", "def_path": "Mathlib/Topology/Algebra/Constructions.lean", "def_pos": [44, 9], "def_end_pos": [44, 22]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nG : Type w\nH : Type x\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : TopologicalGroup \u03b1\ns t : Set \u03b1\nht : IsClosed t\nhs : IsCompact s\n\u22a2 IsClosed (op '' s \u2022 t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "Complex.cos_neg", "start": [860, 1], "end": [860, 87], "traced_tactics": [{"tactic": "simp [cos, sub_eq_add_neg, exp_neg, add_comm]", "annotated_tactic": ["simp [<a>cos</a>, <a>sub_eq_add_neg</a>, <a>exp_neg</a>, <a>add_comm</a>]", [{"full_name": "Complex.cos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [391, 5], "def_end_pos": [391, 8]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "Complex.exp_neg", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [552, 9], "def_end_pos": [552, 16]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "x y : \u2102\n\u22a2 cos (-x) = cos x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "full_name": "MulPosMono.toMulPosStrictMono", "start": [991, 1], "end": [993, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Ideal/Operations.lean", "full_name": "Ideal.pow_sup_eq_top", "start": [738, 1], "end": [740, 37], "traced_tactics": [{"tactic": "rw [\u2190 Finset.card_range n, \u2190 Finset.prod_const]", "annotated_tactic": ["rw [\u2190 <a>Finset.card_range</a> n, \u2190 <a>Finset.prod_const</a>]", [{"full_name": "Finset.card_range", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [177, 9], "def_end_pos": [177, 19]}, {"full_name": "Finset.prod_const", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 19]}]], "state_before": "R : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nn : \u2115\nh : I \u2294 J = \u22a4\n\u22a2 I ^ n \u2294 J = \u22a4", "state_after": "R : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nn : \u2115\nh : I \u2294 J = \u22a4\n\u22a2 (\u220f _x in Finset.range n, I) \u2294 J = \u22a4"}, {"tactic": "exact prod_sup_eq_top fun _ _ => h", "annotated_tactic": ["exact <a>prod_sup_eq_top</a> fun _ _ => h", [{"full_name": "Ideal.prod_sup_eq_top", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [723, 9], "def_end_pos": [723, 24]}]], "state_before": "R : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nn : \u2115\nh : I \u2294 J = \u22a4\n\u22a2 (\u220f _x in Finset.range n, I) \u2294 J = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Matrix/Rank.lean", "full_name": "Matrix.rank_mul_le_left", "start": [75, 1], "end": [78, 100], "traced_tactics": [{"tactic": "rw [rank, rank, mulVecLin_mul]", "annotated_tactic": ["rw [<a>rank</a>, <a>rank</a>, <a>mulVecLin_mul</a>]", [{"full_name": "Matrix.rank", "def_path": "Mathlib/Data/Matrix/Rank.lean", "def_pos": [48, 19], "def_end_pos": [48, 23]}, {"full_name": "Matrix.rank", "def_path": "Mathlib/Data/Matrix/Rank.lean", "def_pos": [48, 19], "def_end_pos": [48, 23]}, {"full_name": "Matrix.mulVecLin_mul", "def_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "def_pos": [252, 9], "def_end_pos": [252, 29]}]], "state_before": "l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : StrongRankCondition R\nA : Matrix m n R\nB : Matrix n o R\n\u22a2 rank (A * B) \u2264 rank A", "state_after": "l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : StrongRankCondition R\nA : Matrix m n R\nB : Matrix n o R\n\u22a2 finrank R { x // x \u2208 LinearMap.range (LinearMap.comp (mulVecLin A) (mulVecLin B)) } \u2264\n    finrank R { x // x \u2208 LinearMap.range (mulVecLin A) }"}, {"tactic": "exact Cardinal.toNat_le_of_le_of_lt_aleph0 (rank_lt_aleph0 _ _) (LinearMap.rank_comp_le_left _ _)", "annotated_tactic": ["exact <a>Cardinal.toNat_le_of_le_of_lt_aleph0</a> (<a>rank_lt_aleph0</a> _ _) (<a>LinearMap.rank_comp_le_left</a> _ _)", [{"full_name": "Cardinal.toNat_le_of_le_of_lt_aleph0", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1738, 9], "def_end_pos": [1738, 36]}, {"full_name": "FiniteDimensional.rank_lt_aleph0", "def_path": "Mathlib/LinearAlgebra/FreeModule/Finite/Rank.lean", "def_pos": [63, 9], "def_end_pos": [63, 23]}, {"full_name": "LinearMap.rank_comp_le_left", "def_path": "Mathlib/LinearAlgebra/Dimension.lean", "def_pos": [1391, 9], "def_end_pos": [1391, 26]}]], "state_before": "l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : StrongRankCondition R\nA : Matrix m n R\nB : Matrix n o R\n\u22a2 finrank R { x // x \u2208 LinearMap.range (LinearMap.comp (mulVecLin A) (mulVecLin B)) } \u2264\n    finrank R { x // x \u2208 LinearMap.range (mulVecLin A) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/SymmDiff.lean", "full_name": "inf_le_bihimp", "start": [275, 1], "end": [276, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Logic/Equiv/Basic.lean", "full_name": "Equiv.Perm.sumCongr_swap_refl", "start": [1739, 1], "end": [1746, 43], "traced_tactics": [{"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "\u03b1\u271d : Sort ?u.118581\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ni j : \u03b1\n\u22a2 sumCongr (swap i j) (Equiv.refl \u03b2) = swap (Sum.inl i) (Sum.inl j)", "state_after": "case H\n\u03b1\u271d : Sort ?u.118581\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ni j : \u03b1\nx : \u03b1 \u2295 \u03b2\n\u22a2 \u2191(sumCongr (swap i j) (Equiv.refl \u03b2)) x = \u2191(swap (Sum.inl i) (Sum.inl j)) x"}, {"tactic": "cases x", "annotated_tactic": ["cases x", []], "state_before": "case H\n\u03b1\u271d : Sort ?u.118581\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ni j : \u03b1\nx : \u03b1 \u2295 \u03b2\n\u22a2 \u2191(sumCongr (swap i j) (Equiv.refl \u03b2)) x = \u2191(swap (Sum.inl i) (Sum.inl j)) x", "state_after": "case H.inl\n\u03b1\u271d : Sort ?u.118581\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ni j val\u271d : \u03b1\n\u22a2 \u2191(sumCongr (swap i j) (Equiv.refl \u03b2)) (Sum.inl val\u271d) = \u2191(swap (Sum.inl i) (Sum.inl j)) (Sum.inl val\u271d)\n\ncase H.inr\n\u03b1\u271d : Sort ?u.118581\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ni j : \u03b1\nval\u271d : \u03b2\n\u22a2 \u2191(sumCongr (swap i j) (Equiv.refl \u03b2)) (Sum.inr val\u271d) = \u2191(swap (Sum.inl i) (Sum.inl j)) (Sum.inr val\u271d)"}, {"tactic": "simp only [Equiv.sumCongr_apply, Sum.map, coe_refl, comp.right_id, Sum.elim_inl, comp_apply,\n  swap_apply_def, Sum.inl.injEq]", "annotated_tactic": ["simp only [<a>Equiv.sumCongr_apply</a>, <a>Sum.map</a>, <a>coe_refl</a>, <a>comp.right_id</a>, <a>Sum.elim_inl</a>, <a>comp_apply</a>,\n      <a>swap_apply_def</a>, Sum.inl.injEq]", [{"full_name": "Equiv.sumCongr_apply", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [296, 9], "def_end_pos": [296, 14]}, {"full_name": "Sum.map", "def_path": "lake-packages/std/Std/Data/Sum/Basic.lean", "def_pos": [104, 15], "def_end_pos": [104, 18]}, {"full_name": "Equiv.coe_refl", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [258, 17], "def_end_pos": [258, 25]}, {"full_name": "Function.comp.right_id", "def_path": "Mathlib/Init/Function.lean", "def_pos": [100, 9], "def_end_pos": [100, 22]}, {"full_name": "Sum.elim_inl", "def_path": "lake-packages/std/Std/Data/Sum/Basic.lean", "def_pos": [97, 17], "def_end_pos": [97, 25]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "Equiv.swap_apply_def", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1636, 9], "def_end_pos": [1636, 23]}]], "state_before": "case H.inl\n\u03b1\u271d : Sort ?u.118581\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ni j val\u271d : \u03b1\n\u22a2 \u2191(sumCongr (swap i j) (Equiv.refl \u03b2)) (Sum.inl val\u271d) = \u2191(swap (Sum.inl i) (Sum.inl j)) (Sum.inl val\u271d)", "state_after": "case H.inl\n\u03b1\u271d : Sort ?u.118581\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ni j val\u271d : \u03b1\n\u22a2 Sum.inl (if val\u271d = i then j else if val\u271d = j then i else val\u271d) =\n    if val\u271d = i then Sum.inl j else if val\u271d = j then Sum.inl i else Sum.inl val\u271d"}, {"tactic": "split_ifs <;> rfl", "annotated_tactic": ["split_ifs <;> rfl", []], "state_before": "case H.inl\n\u03b1\u271d : Sort ?u.118581\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ni j val\u271d : \u03b1\n\u22a2 Sum.inl (if val\u271d = i then j else if val\u271d = j then i else val\u271d) =\n    if val\u271d = i then Sum.inl j else if val\u271d = j then Sum.inl i else Sum.inl val\u271d", "state_after": "no goals"}, {"tactic": "simp [Sum.map, swap_apply_of_ne_of_ne]", "annotated_tactic": ["simp [<a>Sum.map</a>, <a>swap_apply_of_ne_of_ne</a>]", [{"full_name": "Sum.map", "def_path": "lake-packages/std/Std/Data/Sum/Basic.lean", "def_pos": [104, 15], "def_end_pos": [104, 18]}, {"full_name": "Equiv.swap_apply_of_ne_of_ne", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1650, 9], "def_end_pos": [1650, 31]}]], "state_before": "case H.inr\n\u03b1\u271d : Sort ?u.118581\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ni j : \u03b1\nval\u271d : \u03b2\n\u22a2 \u2191(sumCongr (swap i j) (Equiv.refl \u03b2)) (Sum.inr val\u271d) = \u2191(swap (Sum.inl i) (Sum.inl j)) (Sum.inr val\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Isometry.lean", "full_name": "IsometryEquiv.symm_trans_apply", "start": [461, 1], "end": [463, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Divisibility/Basic.lean", "full_name": "dvd_of_eq", "start": [135, 1], "end": [135, 51], "traced_tactics": [{"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : Monoid \u03b1\na b : \u03b1\nh : a = b\n\u22a2 a \u2223 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean", "full_name": "Real.pi_div_two_eq_arcsin", "start": [246, 1], "end": [247, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/ZFC/Basic.lean", "full_name": "ZFSet.singleton_injective", "start": [1069, 1], "end": [1071, 51], "traced_tactics": [{"tactic": "let this := congr_arg sUnion H", "annotated_tactic": ["let this := <a>congr_arg</a> <a>sUnion</a> H", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "ZFSet.sUnion", "def_path": "Mathlib/SetTheory/ZFC/Basic.lean", "def_pos": [990, 5], "def_end_pos": [990, 11]}]], "state_before": "x y : ZFSet\nH : {x} = {y}\n\u22a2 x = y", "state_after": "x y : ZFSet\nH : {x} = {y}\nthis : \u22c3\u2080 {x} = \u22c3\u2080 {y} := congr_arg sUnion H\n\u22a2 x = y"}, {"tactic": "rwa [sUnion_singleton, sUnion_singleton] at this", "annotated_tactic": ["rwa [<a>sUnion_singleton</a>, <a>sUnion_singleton</a>] at this", [{"full_name": "ZFSet.sUnion_singleton", "def_path": "Mathlib/SetTheory/ZFC/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 25]}, {"full_name": "ZFSet.sUnion_singleton", "def_path": "Mathlib/SetTheory/ZFC/Basic.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 25]}]], "state_before": "x y : ZFSet\nH : {x} = {y}\nthis : \u22c3\u2080 {x} = \u22c3\u2080 {y} := congr_arg sUnion H\n\u22a2 x = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Abelian/Exact.lean", "full_name": "CategoryTheory.Abelian.exact_epi_comp_iff", "start": [116, 1], "end": [121, 61], "traced_tactics": [{"tactic": "refine' \u27e8fun hfg => _, fun h => exact_epi_comp h\u27e9", "annotated_tactic": ["refine' \u27e8fun hfg => _, fun h => <a>exact_epi_comp</a> h\u27e9", [{"full_name": "CategoryTheory.exact_epi_comp", "def_path": "Mathlib/Algebra/Homology/Exact.lean", "def_pos": [187, 9], "def_end_pos": [187, 23]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nW : C\nh : W \u27f6 X\ninst\u271d : Epi h\n\u22a2 Exact (h \u226b f) g \u2194 Exact f g", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nW : C\nh : W \u27f6 X\ninst\u271d : Epi h\nhfg : Exact (h \u226b f) g\n\u22a2 Exact f g"}, {"tactic": "let hc := isCokernelOfComp _ _ (colimit.isColimit (parallelPair (h \u226b f) 0))\n  (by rw [\u2190 cancel_epi h, \u2190 Category.assoc, CokernelCofork.condition, comp_zero]) rfl", "annotated_tactic": ["let hc := <a>isCokernelOfComp</a> _ _ (<a>colimit.isColimit</a> (<a>parallelPair</a> (h \u226b f) 0))\n    (by rw [\u2190 <a>cancel_epi</a> h, \u2190 <a>Category.assoc</a>, <a>CokernelCofork.condition</a>, <a>comp_zero</a>]) <a>rfl</a>", [{"full_name": "CategoryTheory.Limits.isCokernelOfComp", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean", "def_pos": [634, 5], "def_end_pos": [634, 21]}, {"full_name": "CategoryTheory.Limits.colimit.isColimit", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [740, 5], "def_end_pos": [740, 22]}, {"full_name": "CategoryTheory.Limits.parallelPair", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [210, 5], "def_end_pos": [210, 17]}, {"full_name": "CategoryTheory.cancel_epi", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [288, 9], "def_end_pos": [288, 19]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [159, 3], "def_end_pos": [159, 8]}, {"full_name": "CategoryTheory.Limits.CokernelCofork.condition", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean", "def_pos": [540, 9], "def_end_pos": [540, 33]}, {"full_name": "CategoryTheory.Limits.comp_zero", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean", "def_pos": [67, 9], "def_end_pos": [67, 18]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nW : C\nh : W \u27f6 X\ninst\u271d : Epi h\nhfg : Exact (h \u226b f) g\n\u22a2 Exact f g", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nW : C\nh : W \u27f6 X\ninst\u271d : Epi h\nhfg : Exact (h \u226b f) g\nhc : IsColimit\n  (CokernelCofork.of\u03c0 (Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)))\n    (_ : f \u226b Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)) = 0)) :=\n  isCokernelOfComp h (h \u226b f) (colimit.isColimit (parallelPair (h \u226b f) 0))\n    (_ : f \u226b Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)) = 0) (_ : h \u226b f = h \u226b f)\n\u22a2 Exact f g"}, {"tactic": "refine' (exact_iff' _ _ (limit.isLimit _) hc).2 \u27e8_, ((exact_iff _ _).1 hfg).2\u27e9", "annotated_tactic": ["refine' (<a>exact_iff'</a> _ _ (<a>limit.isLimit</a> _) hc).2 \u27e8_, ((<a>exact_iff</a> _ _).1 hfg).2\u27e9", [{"full_name": "CategoryTheory.Abelian.exact_iff'", "def_path": "Mathlib/CategoryTheory/Abelian/Exact.lean", "def_pos": [85, 9], "def_end_pos": [85, 19]}, {"full_name": "CategoryTheory.Limits.limit.isLimit", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [176, 5], "def_end_pos": [176, 18]}, {"full_name": "CategoryTheory.Abelian.exact_iff", "def_path": "Mathlib/CategoryTheory/Abelian/Exact.lean", "def_pos": [66, 9], "def_end_pos": [66, 18]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nW : C\nh : W \u27f6 X\ninst\u271d : Epi h\nhfg : Exact (h \u226b f) g\nhc : IsColimit\n  (CokernelCofork.of\u03c0 (Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)))\n    (_ : f \u226b Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)) = 0)) :=\n  isCokernelOfComp h (h \u226b f) (colimit.isColimit (parallelPair (h \u226b f) 0))\n    (_ : f \u226b Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)) = 0) (_ : h \u226b f = h \u226b f)\n\u22a2 Exact f g", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nW : C\nh : W \u27f6 X\ninst\u271d : Epi h\nhfg : Exact (h \u226b f) g\nhc : IsColimit\n  (CokernelCofork.of\u03c0 (Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)))\n    (_ : f \u226b Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)) = 0)) :=\n  isCokernelOfComp h (h \u226b f) (colimit.isColimit (parallelPair (h \u226b f) 0))\n    (_ : f \u226b Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)) = 0) (_ : h \u226b f = h \u226b f)\n\u22a2 f \u226b g = 0"}, {"tactic": "exact zero_of_epi_comp h (by rw [\u2190 hfg.1, Category.assoc])", "annotated_tactic": ["exact <a>zero_of_epi_comp</a> h (by rw [\u2190 hfg.1, <a>Category.assoc</a>])", [{"full_name": "CategoryTheory.Limits.zero_of_epi_comp", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean", "def_pos": [146, 9], "def_end_pos": [146, 25]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [159, 3], "def_end_pos": [159, 8]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nW : C\nh : W \u27f6 X\ninst\u271d : Epi h\nhfg : Exact (h \u226b f) g\nhc : IsColimit\n  (CokernelCofork.of\u03c0 (Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)))\n    (_ : f \u226b Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)) = 0)) :=\n  isCokernelOfComp h (h \u226b f) (colimit.isColimit (parallelPair (h \u226b f) 0))\n    (_ : f \u226b Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)) = 0) (_ : h \u226b f = h \u226b f)\n\u22a2 f \u226b g = 0", "state_after": "no goals"}, {"tactic": "rw [\u2190 cancel_epi h, \u2190 Category.assoc, CokernelCofork.condition, comp_zero]", "annotated_tactic": ["rw [\u2190 <a>cancel_epi</a> h, \u2190 <a>Category.assoc</a>, <a>CokernelCofork.condition</a>, <a>comp_zero</a>]", [{"full_name": "CategoryTheory.cancel_epi", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [288, 9], "def_end_pos": [288, 19]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [159, 3], "def_end_pos": [159, 8]}, {"full_name": "CategoryTheory.Limits.CokernelCofork.condition", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean", "def_pos": [540, 9], "def_end_pos": [540, 33]}, {"full_name": "CategoryTheory.Limits.comp_zero", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean", "def_pos": [67, 9], "def_end_pos": [67, 18]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nW : C\nh : W \u27f6 X\ninst\u271d : Epi h\nhfg : Exact (h \u226b f) g\n\u22a2 f \u226b Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)) = 0", "state_after": "no goals"}, {"tactic": "rw [\u2190 hfg.1, Category.assoc]", "annotated_tactic": ["rw [\u2190 hfg.1, <a>Category.assoc</a>]", [{"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [159, 3], "def_end_pos": [159, 8]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Abelian C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\nW : C\nh : W \u27f6 X\ninst\u271d : Epi h\nhfg : Exact (h \u226b f) g\nhc : IsColimit\n  (CokernelCofork.of\u03c0 (Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)))\n    (_ : f \u226b Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)) = 0)) :=\n  isCokernelOfComp h (h \u226b f) (colimit.isColimit (parallelPair (h \u226b f) 0))\n    (_ : f \u226b Cofork.\u03c0 (colimit.cocone (parallelPair (h \u226b f) 0)) = 0) (_ : h \u226b f = h \u226b f)\n\u22a2 h \u226b f \u226b g = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/Padics/Hensel.lean", "full_name": "deriv_sq_norm_ne_zero", "start": [111, 9], "end": [112, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.eventuallyEq_principal", "start": [1615, 1], "end": [1616, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "full_name": "ContinuousLinearMap.coe_restrictScalars", "start": [1748, 1], "end": [1750, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Fold.lean", "full_name": "Finset.fold_singleton", "start": [59, 1], "end": [60, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Pi/Algebra.lean", "full_name": "Function.extend_div", "start": [393, 1], "end": [397, 42], "traced_tactics": [{"tactic": "classical\nfunext x\nsimp [Function.extend_def, apply_dite\u2082]", "annotated_tactic": ["classical\n  funext x\n  simp [<a>Function.extend_def</a>, <a>apply_dite\u2082</a>]", [{"full_name": "Function.extend_def", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [720, 9], "def_end_pos": [720, 19]}, {"full_name": "apply_dite\u2082", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 20]}]], "state_before": "I : Type u\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf\u271d : I \u2192 Type v\u2081\ng : I \u2192 Type v\u2082\nh : I \u2192 Type v\u2083\nx y : (i : I) \u2192 f\u271d i\ni : I\ninst\u271d : Div \u03b3\nf : \u03b1 \u2192 \u03b2\ng\u2081 g\u2082 : \u03b1 \u2192 \u03b3\ne\u2081 e\u2082 : \u03b2 \u2192 \u03b3\n\u22a2 extend f (g\u2081 / g\u2082) (e\u2081 / e\u2082) = extend f g\u2081 e\u2081 / extend f g\u2082 e\u2082", "state_after": "no goals"}, {"tactic": "funext x", "annotated_tactic": ["funext x", []], "state_before": "I : Type u\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf\u271d : I \u2192 Type v\u2081\ng : I \u2192 Type v\u2082\nh : I \u2192 Type v\u2083\nx y : (i : I) \u2192 f\u271d i\ni : I\ninst\u271d : Div \u03b3\nf : \u03b1 \u2192 \u03b2\ng\u2081 g\u2082 : \u03b1 \u2192 \u03b3\ne\u2081 e\u2082 : \u03b2 \u2192 \u03b3\n\u22a2 extend f (g\u2081 / g\u2082) (e\u2081 / e\u2082) = extend f g\u2081 e\u2081 / extend f g\u2082 e\u2082", "state_after": "case h\nI : Type u\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf\u271d : I \u2192 Type v\u2081\ng : I \u2192 Type v\u2082\nh : I \u2192 Type v\u2083\nx\u271d y : (i : I) \u2192 f\u271d i\ni : I\ninst\u271d : Div \u03b3\nf : \u03b1 \u2192 \u03b2\ng\u2081 g\u2082 : \u03b1 \u2192 \u03b3\ne\u2081 e\u2082 : \u03b2 \u2192 \u03b3\nx : \u03b2\n\u22a2 extend f (g\u2081 / g\u2082) (e\u2081 / e\u2082) x = (extend f g\u2081 e\u2081 / extend f g\u2082 e\u2082) x"}, {"tactic": "simp [Function.extend_def, apply_dite\u2082]", "annotated_tactic": ["simp [<a>Function.extend_def</a>, <a>apply_dite\u2082</a>]", [{"full_name": "Function.extend_def", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [720, 9], "def_end_pos": [720, 19]}, {"full_name": "apply_dite\u2082", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 20]}]], "state_before": "case h\nI : Type u\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf\u271d : I \u2192 Type v\u2081\ng : I \u2192 Type v\u2082\nh : I \u2192 Type v\u2083\nx\u271d y : (i : I) \u2192 f\u271d i\ni : I\ninst\u271d : Div \u03b3\nf : \u03b1 \u2192 \u03b2\ng\u2081 g\u2082 : \u03b1 \u2192 \u03b3\ne\u2081 e\u2082 : \u03b2 \u2192 \u03b3\nx : \u03b2\n\u22a2 extend f (g\u2081 / g\u2082) (e\u2081 / e\u2082) x = (extend f g\u2081 e\u2081 / extend f g\u2082 e\u2082) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Parity.lean", "full_name": "Int.four_dvd_add_or_sub_of_odd", "start": [228, 1], "end": [245, 8], "traced_tactics": [{"tactic": "obtain \u27e8m, rfl\u27e9 := ha", "annotated_tactic": ["obtain \u27e8m, rfl\u27e9 := ha", []], "state_before": "m n a b : \u2124\nha : Odd a\nhb : Odd b\n\u22a2 4 \u2223 a + b \u2228 4 \u2223 a - b", "state_after": "case intro\nm\u271d n b : \u2124\nhb : Odd b\nm : \u2124\n\u22a2 4 \u2223 2 * m + 1 + b \u2228 4 \u2223 2 * m + 1 - b"}, {"tactic": "obtain \u27e8n, rfl\u27e9 := hb", "annotated_tactic": ["obtain \u27e8n, rfl\u27e9 := hb", []], "state_before": "case intro\nm\u271d n b : \u2124\nhb : Odd b\nm : \u2124\n\u22a2 4 \u2223 2 * m + 1 + b \u2228 4 \u2223 2 * m + 1 - b", "state_after": "case intro.intro\nm\u271d n\u271d m n : \u2124\n\u22a2 4 \u2223 2 * m + 1 + (2 * n + 1) \u2228 4 \u2223 2 * m + 1 - (2 * n + 1)"}, {"tactic": "obtain h | h := Int.even_or_odd (m + n)", "annotated_tactic": ["obtain h | h := <a>Int.even_or_odd</a> (m + n)", [{"full_name": "Int.even_or_odd", "def_path": "Mathlib/Data/Int/Parity.lean", "def_pos": [63, 9], "def_end_pos": [63, 20]}]], "state_before": "case intro.intro\nm\u271d n\u271d m n : \u2124\n\u22a2 4 \u2223 2 * m + 1 + (2 * n + 1) \u2228 4 \u2223 2 * m + 1 - (2 * n + 1)", "state_after": "case intro.intro.inl\nm\u271d n\u271d m n : \u2124\nh : Even (m + n)\n\u22a2 4 \u2223 2 * m + 1 + (2 * n + 1) \u2228 4 \u2223 2 * m + 1 - (2 * n + 1)\n\ncase intro.intro.inr\nm\u271d n\u271d m n : \u2124\nh : Odd (m + n)\n\u22a2 4 \u2223 2 * m + 1 + (2 * n + 1) \u2228 4 \u2223 2 * m + 1 - (2 * n + 1)"}, {"tactic": "right", "annotated_tactic": ["right", []], "state_before": "case intro.intro.inl\nm\u271d n\u271d m n : \u2124\nh : Even (m + n)\n\u22a2 4 \u2223 2 * m + 1 + (2 * n + 1) \u2228 4 \u2223 2 * m + 1 - (2 * n + 1)", "state_after": "case intro.intro.inl.h\nm\u271d n\u271d m n : \u2124\nh : Even (m + n)\n\u22a2 4 \u2223 2 * m + 1 - (2 * n + 1)"}, {"tactic": "rw [Int.even_add, \u2190 Int.even_sub] at h", "annotated_tactic": ["rw [<a>Int.even_add</a>, \u2190 <a>Int.even_sub</a>] at h", [{"full_name": "Int.even_add", "def_path": "Mathlib/Data/Int/Parity.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "Int.even_sub", "def_path": "Mathlib/Data/Int/Parity.lean", "def_pos": [122, 9], "def_end_pos": [122, 17]}]], "state_before": "case intro.intro.inl.h\nm\u271d n\u271d m n : \u2124\nh : Even (m + n)\n\u22a2 4 \u2223 2 * m + 1 - (2 * n + 1)", "state_after": "case intro.intro.inl.h\nm\u271d n\u271d m n : \u2124\nh : Even (m - n)\n\u22a2 4 \u2223 2 * m + 1 - (2 * n + 1)"}, {"tactic": "obtain \u27e8k, hk\u27e9 := h", "annotated_tactic": ["obtain \u27e8k, hk\u27e9 := h", []], "state_before": "case intro.intro.inl.h\nm\u271d n\u271d m n : \u2124\nh : Even (m - n)\n\u22a2 4 \u2223 2 * m + 1 - (2 * n + 1)", "state_after": "case intro.intro.inl.h.intro\nm\u271d n\u271d m n k : \u2124\nhk : m - n = k + k\n\u22a2 4 \u2223 2 * m + 1 - (2 * n + 1)"}, {"tactic": "convert dvd_mul_right 4 k using 1", "annotated_tactic": ["convert <a>dvd_mul_right</a> 4 k using 1", [{"full_name": "dvd_mul_right", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "case intro.intro.inl.h.intro\nm\u271d n\u271d m n k : \u2124\nhk : m - n = k + k\n\u22a2 4 \u2223 2 * m + 1 - (2 * n + 1)", "state_after": "case h.e'_4\nm\u271d n\u271d m n k : \u2124\nhk : m - n = k + k\n\u22a2 2 * m + 1 - (2 * n + 1) = 4 * k"}, {"tactic": "rw [eq_add_of_sub_eq hk, mul_add, add_assoc, add_sub_cancel, \u2190 two_mul, \u2190 mul_assoc]", "annotated_tactic": ["rw [<a>eq_add_of_sub_eq</a> hk, <a>mul_add</a>, <a>add_assoc</a>, <a>add_sub_cancel</a>, \u2190 <a>two_mul</a>, \u2190 <a>mul_assoc</a>]", [{"full_name": "eq_add_of_sub_eq", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [766, 3], "def_end_pos": [766, 14]}, {"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "add_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [739, 30], "def_end_pos": [739, 44]}, {"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "case h.e'_4\nm\u271d n\u271d m n k : \u2124\nhk : m - n = k + k\n\u22a2 2 * m + 1 - (2 * n + 1) = 4 * k", "state_after": "case h.e'_4\nm\u271d n\u271d m n k : \u2124\nhk : m - n = k + k\n\u22a2 2 * 2 * k = 4 * k"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.e'_4\nm\u271d n\u271d m n k : \u2124\nhk : m - n = k + k\n\u22a2 2 * 2 * k = 4 * k", "state_after": "no goals"}, {"tactic": "left", "annotated_tactic": ["left", []], "state_before": "case intro.intro.inr\nm\u271d n\u271d m n : \u2124\nh : Odd (m + n)\n\u22a2 4 \u2223 2 * m + 1 + (2 * n + 1) \u2228 4 \u2223 2 * m + 1 - (2 * n + 1)", "state_after": "case intro.intro.inr.h\nm\u271d n\u271d m n : \u2124\nh : Odd (m + n)\n\u22a2 4 \u2223 2 * m + 1 + (2 * n + 1)"}, {"tactic": "obtain \u27e8k, hk\u27e9 := h", "annotated_tactic": ["obtain \u27e8k, hk\u27e9 := h", []], "state_before": "case intro.intro.inr.h\nm\u271d n\u271d m n : \u2124\nh : Odd (m + n)\n\u22a2 4 \u2223 2 * m + 1 + (2 * n + 1)", "state_after": "case intro.intro.inr.h.intro\nm\u271d n\u271d m n k : \u2124\nhk : m + n = 2 * k + 1\n\u22a2 4 \u2223 2 * m + 1 + (2 * n + 1)"}, {"tactic": "convert dvd_mul_right 4 (k + 1) using 1", "annotated_tactic": ["convert <a>dvd_mul_right</a> 4 (k + 1) using 1", [{"full_name": "dvd_mul_right", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [82, 9], "def_end_pos": [82, 22]}]], "state_before": "case intro.intro.inr.h.intro\nm\u271d n\u271d m n k : \u2124\nhk : m + n = 2 * k + 1\n\u22a2 4 \u2223 2 * m + 1 + (2 * n + 1)", "state_after": "case h.e'_4\nm\u271d n\u271d m n k : \u2124\nhk : m + n = 2 * k + 1\n\u22a2 2 * m + 1 + (2 * n + 1) = 4 * (k + 1)"}, {"tactic": "rw [eq_sub_of_add_eq hk, add_right_comm, \u2190 add_sub, mul_add, mul_sub, add_assoc, add_assoc,\n  sub_add, add_assoc, \u2190 sub_sub (2 * n), sub_self, zero_sub, sub_neg_eq_add, \u2190 mul_assoc,\n  mul_add]", "annotated_tactic": ["rw [<a>eq_sub_of_add_eq</a> hk, <a>add_right_comm</a>, \u2190 <a>add_sub</a>, <a>mul_add</a>, <a>mul_sub</a>, <a>add_assoc</a>, <a>add_assoc</a>,\n      <a>sub_add</a>, <a>add_assoc</a>, \u2190 <a>sub_sub</a> (2 * n), <a>sub_self</a>, <a>zero_sub</a>, <a>sub_neg_eq_add</a>, \u2190 <a>mul_assoc</a>,\n      <a>mul_add</a>]", [{"full_name": "eq_sub_of_add_eq", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [756, 15], "def_end_pos": [756, 31]}, {"full_name": "add_right_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [101, 3], "def_end_pos": [101, 14]}, {"full_name": "add_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [323, 3], "def_end_pos": [323, 14]}, {"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}, {"full_name": "mul_sub", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [365, 7], "def_end_pos": [365, 14]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "sub_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [531, 3], "def_end_pos": [531, 14]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "sub_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [526, 3], "def_end_pos": [526, 14]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}, {"full_name": "zero_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [317, 3], "def_end_pos": [317, 14]}, {"full_name": "sub_neg_eq_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [453, 3], "def_end_pos": [453, 14]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_add", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 14]}]], "state_before": "case h.e'_4\nm\u271d n\u271d m n k : \u2124\nhk : m + n = 2 * k + 1\n\u22a2 2 * m + 1 + (2 * n + 1) = 4 * (k + 1)", "state_after": "case h.e'_4\nm\u271d n\u271d m n k : \u2124\nhk : m + n = 2 * k + 1\n\u22a2 2 * 2 * k + (2 * 1 + (1 + 1)) = 4 * k + 4 * 1"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.e'_4\nm\u271d n\u271d m n k : \u2124\nhk : m + n = 2 * k + 1\n\u22a2 2 * 2 * k + (2 * 1 + (1 + 1)) = 4 * k + 4 * 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/FieldTheory/Galois.lean", "full_name": "IsGalois.of_separable_splitting_field_aux", "start": [359, 1], "end": [392, 20], "traced_tactics": [{"tactic": "have h : IsIntegral K x :=\n  isIntegral_of_isScalarTower (isIntegral_of_noetherian (IsNoetherian.iff_fg.2 hFE) x)", "annotated_tactic": ["have h : <a>IsIntegral</a> K x :=\n    <a>isIntegral_of_isScalarTower</a> (<a>isIntegral_of_noetherian</a> (<a>IsNoetherian.iff_fg</a>.2 hFE) x)", [{"full_name": "IsIntegral", "def_path": "Mathlib/RingTheory/IntegralClosure.lean", "def_pos": [61, 5], "def_end_pos": [61, 15]}, {"full_name": "isIntegral_of_isScalarTower", "def_path": "Mathlib/RingTheory/IntegralClosure.lean", "def_pos": [179, 9], "def_end_pos": [179, 36]}, {"full_name": "isIntegral_of_noetherian", "def_path": "Mathlib/RingTheory/IntegralClosure.lean", "def_pos": [82, 9], "def_end_pos": [82, 33]}, {"full_name": "IsNoetherian.iff_fg", "def_path": "Mathlib/FieldTheory/Finiteness.lean", "def_pos": [109, 9], "def_end_pos": [109, 15]}]], "state_before": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\n\u22a2 Fintype.card ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) =\n    Fintype.card (K \u2192\u2090[F] E) * finrank K { x_1 // x_1 \u2208 K\u27eex\u27ef }", "state_after": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\n\u22a2 Fintype.card ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) =\n    Fintype.card (K \u2192\u2090[F] E) * finrank K { x_1 // x_1 \u2208 K\u27eex\u27ef }"}, {"tactic": "have h1 : p \u2260 0 := fun hp => by\n  rw [hp, Polynomial.aroots_zero] at hx\n  exact Multiset.not_mem_zero x hx", "annotated_tactic": ["have h1 : p \u2260 0 := fun hp => by\n    rw [hp, <a>Polynomial.aroots_zero</a>] at hx\n    exact <a>Multiset.not_mem_zero</a> x hx", [{"full_name": "Polynomial.aroots_zero", "def_path": "Mathlib/Data/Polynomial/RingDivision.lean", "def_pos": [956, 9], "def_end_pos": [956, 20]}, {"full_name": "Multiset.not_mem_zero", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [260, 9], "def_end_pos": [260, 21]}]], "state_before": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\n\u22a2 Fintype.card ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) =\n    Fintype.card (K \u2192\u2090[F] E) * finrank K { x_1 // x_1 \u2208 K\u27eex\u27ef }", "state_after": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\n\u22a2 Fintype.card ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) =\n    Fintype.card (K \u2192\u2090[F] E) * finrank K { x_1 // x_1 \u2208 K\u27eex\u27ef }"}, {"tactic": "have h2 : minpoly K x \u2223 p.map (algebraMap F K) := by\n  apply minpoly.dvd\n  rw [Polynomial.aeval_def, Polynomial.eval\u2082_map, \u2190 Polynomial.eval_map, \u2190\n    IsScalarTower.algebraMap_eq]\n  exact (Polynomial.mem_roots (Polynomial.map_ne_zero h1)).mp hx", "annotated_tactic": ["have h2 : <a>minpoly</a> K x \u2223 p.map (<a>algebraMap</a> F K) := by\n    apply <a>minpoly.dvd</a>\n    rw [<a>Polynomial.aeval_def</a>, <a>Polynomial.eval\u2082_map</a>, \u2190 <a>Polynomial.eval_map</a>, \u2190\n      <a>IsScalarTower.algebraMap_eq</a>]\n    exact (<a>Polynomial.mem_roots</a> (<a>Polynomial.map_ne_zero</a> h1)).<a>mp</a> hx", [{"full_name": "minpoly", "def_path": "Mathlib/FieldTheory/Minpoly/Basic.lean", "def_pos": [37, 19], "def_end_pos": [37, 26]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "minpoly.dvd", "def_path": "Mathlib/FieldTheory/Minpoly/Field.lean", "def_pos": [69, 9], "def_end_pos": [69, 12]}, {"full_name": "Polynomial.aeval_def", "def_path": "Mathlib/Data/Polynomial/AlgebraMap.lean", "def_pos": [187, 9], "def_end_pos": [187, 18]}, {"full_name": "Polynomial.eval\u2082_map", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [956, 9], "def_end_pos": [956, 18]}, {"full_name": "Polynomial.eval_map", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [960, 9], "def_end_pos": [960, 17]}, {"full_name": "IsScalarTower.algebraMap_eq", "def_path": "Mathlib/Algebra/Algebra/Tower.lean", "def_pos": [122, 9], "def_end_pos": [122, 22]}, {"full_name": "Polynomial.mem_roots", "def_path": "Mathlib/Data/Polynomial/RingDivision.lean", "def_pos": [599, 9], "def_end_pos": [599, 18]}, {"full_name": "Polynomial.map_ne_zero", "def_path": "Mathlib/Data/Polynomial/FieldDivision.lean", "def_pos": [142, 9], "def_end_pos": [142, 20]}, {"full_name": "Iff.mp", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [90, 3], "def_end_pos": [90, 5]}]], "state_before": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\n\u22a2 Fintype.card ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) =\n    Fintype.card (K \u2192\u2090[F] E) * finrank K { x_1 // x_1 \u2208 K\u27eex\u27ef }", "state_after": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\n\u22a2 Fintype.card ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) =\n    Fintype.card (K \u2192\u2090[F] E) * finrank K { x_1 // x_1 \u2208 K\u27eex\u27ef }"}, {"tactic": "let key_equiv : (K\u27eex\u27ef.restrictScalars F \u2192\u2090[F] E) \u2243\n    \u03a3 f : K \u2192\u2090[F] E, @AlgHom K K\u27eex\u27ef E _ _ _ _ (RingHom.toAlgebra f) := by\n  change (K\u27eex\u27ef \u2192\u2090[F] E) \u2243 \u03a3 f : K \u2192\u2090[F] E, _\n  exact algHomEquivSigma", "annotated_tactic": ["let key_equiv : (K\u27eex\u27ef.<a>restrictScalars</a> F \u2192\u2090[F] E) \u2243\n      \u03a3 f : K \u2192\u2090[F] E, @<a>AlgHom</a> K K\u27eex\u27ef E _ _ _ _ (<a>RingHom.toAlgebra</a> f) := by\n    change (K\u27eex\u27ef \u2192\u2090[F] E) \u2243 \u03a3 f : K \u2192\u2090[F] E, _\n    exact <a>algHomEquivSigma</a>", [{"full_name": "IntermediateField.restrictScalars", "def_path": "Mathlib/FieldTheory/IntermediateField.lean", "def_pos": [620, 5], "def_end_pos": [620, 20]}, {"full_name": "AlgHom", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [34, 11], "def_end_pos": [34, 17]}, {"full_name": "RingHom.toAlgebra", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [267, 5], "def_end_pos": [267, 22]}, {"full_name": "algHomEquivSigma", "def_path": "Mathlib/RingTheory/AlgebraTower.lean", "def_pos": [218, 5], "def_end_pos": [218, 21]}]], "state_before": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\n\u22a2 Fintype.card ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) =\n    Fintype.card (K \u2192\u2090[F] E) * finrank K { x_1 // x_1 \u2208 K\u27eex\u27ef }", "state_after": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\n\u22a2 Fintype.card ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) =\n    Fintype.card (K \u2192\u2090[F] E) * finrank K { x_1 // x_1 \u2208 K\u27eex\u27ef }"}, {"tactic": "haveI : \u2200 f : K \u2192\u2090[F] E, Fintype (@AlgHom K K\u27eex\u27ef E _ _ _ _ (RingHom.toAlgebra f)) := fun f => by\n  have := Fintype.ofEquiv _ key_equiv\n  apply Fintype.ofInjective (Sigma.mk f) fun _ _ H => eq_of_heq (Sigma.ext_iff.mp H).2", "annotated_tactic": ["haveI : \u2200 f : K \u2192\u2090[F] E, <a>Fintype</a> (@<a>AlgHom</a> K K\u27eex\u27ef E _ _ _ _ (<a>RingHom.toAlgebra</a> f)) := fun f => by\n    have := <a>Fintype.ofEquiv</a> _ key_equiv\n    apply <a>Fintype.ofInjective</a> (<a>Sigma.mk</a> f) fun _ _ H => <a>eq_of_heq</a> (Sigma.ext_iff.mp H).2", [{"full_name": "Fintype", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [54, 7], "def_end_pos": [54, 14]}, {"full_name": "AlgHom", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [34, 11], "def_end_pos": [34, 17]}, {"full_name": "RingHom.toAlgebra", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [267, 5], "def_end_pos": [267, 22]}, {"full_name": "Fintype.ofEquiv", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [585, 5], "def_end_pos": [585, 12]}, {"full_name": "Fintype.ofInjective", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [576, 19], "def_end_pos": [576, 30]}, {"full_name": "Sigma.mk", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [140, 3], "def_end_pos": [140, 5]}, {"full_name": "eq_of_heq", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [451, 9], "def_end_pos": [451, 18]}]], "state_before": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\n\u22a2 Fintype.card ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) =\n    Fintype.card (K \u2192\u2090[F] E) * finrank K { x_1 // x_1 \u2208 K\u27eex\u27ef }", "state_after": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\nthis : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)\n\u22a2 Fintype.card ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) =\n    Fintype.card (K \u2192\u2090[F] E) * finrank K { x_1 // x_1 \u2208 K\u27eex\u27ef }"}, {"tactic": "rw [Fintype.card_congr key_equiv, Fintype.card_sigma, IntermediateField.adjoin.finrank h]", "annotated_tactic": ["rw [<a>Fintype.card_congr</a> key_equiv, <a>Fintype.card_sigma</a>, <a>IntermediateField.adjoin.finrank</a> h]", [{"full_name": "Fintype.card_congr", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [149, 9], "def_end_pos": [149, 19]}, {"full_name": "Fintype.card_sigma", "def_path": "Mathlib/Data/Fintype/BigOperators.lean", "def_pos": [123, 16], "def_end_pos": [123, 34]}, {"full_name": "IntermediateField.adjoin.finrank", "def_path": "Mathlib/FieldTheory/Adjoin.lean", "def_pos": [874, 9], "def_end_pos": [874, 23]}]], "state_before": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\nthis : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)\n\u22a2 Fintype.card ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) =\n    Fintype.card (K \u2192\u2090[F] E) * finrank K { x_1 // x_1 \u2208 K\u27eex\u27ef }", "state_after": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\nthis : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)\n\u22a2 (Finset.sum Finset.univ fun a => Fintype.card ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)) =\n    Fintype.card (K \u2192\u2090[F] E) * Polynomial.natDegree (minpoly K x)"}, {"tactic": "apply Finset.sum_const_nat", "annotated_tactic": ["apply <a>Finset.sum_const_nat</a>", [{"full_name": "Finset.sum_const_nat", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1766, 9], "def_end_pos": [1766, 22]}]], "state_before": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\nthis : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)\n\u22a2 (Finset.sum Finset.univ fun a => Fintype.card ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)) =\n    Fintype.card (K \u2192\u2090[F] E) * Polynomial.natDegree (minpoly K x)", "state_after": "case h\u2081\nF : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\nthis : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)\n\u22a2 \u2200 (x_1 : K \u2192\u2090[F] E),\n    x_1 \u2208 Finset.univ \u2192 Fintype.card ({ x_2 // x_2 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) = Polynomial.natDegree (minpoly K x)"}, {"tactic": "intro f _", "annotated_tactic": ["intro f _", []], "state_before": "case h\u2081\nF : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\nthis : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)\n\u22a2 \u2200 (x_1 : K \u2192\u2090[F] E),\n    x_1 \u2208 Finset.univ \u2192 Fintype.card ({ x_2 // x_2 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) = Polynomial.natDegree (minpoly K x)", "state_after": "case h\u2081\nF : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\nthis : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)\nf : K \u2192\u2090[F] E\na\u271d : f \u2208 Finset.univ\n\u22a2 Fintype.card ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) = Polynomial.natDegree (minpoly K x)"}, {"tactic": "rw [\u2190 @IntermediateField.card_algHom_adjoin_integral K _ E _ _ x E _ (RingHom.toAlgebra f) h]", "annotated_tactic": ["rw [\u2190 @<a>IntermediateField.card_algHom_adjoin_integral</a> K _ E _ _ x E _ (<a>RingHom.toAlgebra</a> f) h]", [{"full_name": "IntermediateField.card_algHom_adjoin_integral", "def_path": "Mathlib/FieldTheory/Adjoin.lean", "def_pos": [907, 9], "def_end_pos": [907, 36]}, {"full_name": "RingHom.toAlgebra", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [267, 5], "def_end_pos": [267, 22]}]], "state_before": "case h\u2081\nF : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\nthis : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)\nf : K \u2192\u2090[F] E\na\u271d : f \u2208 Finset.univ\n\u22a2 Fintype.card ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) = Polynomial.natDegree (minpoly K x)", "state_after": "case h\u2081\nF : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\nthis : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)\nf : K \u2192\u2090[F] E\na\u271d : f \u2208 Finset.univ\n\u22a2 Fintype.card ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) = Fintype.card ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)\n\ncase h\u2081.h_sep\nF : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\nthis : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)\nf : K \u2192\u2090[F] E\na\u271d : f \u2208 Finset.univ\n\u22a2 Polynomial.Separable (minpoly K x)\n\ncase h\u2081.h_splits\nF : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\nthis : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)\nf : K \u2192\u2090[F] E\na\u271d : f \u2208 Finset.univ\n\u22a2 Polynomial.Splits (algebraMap K E) (minpoly K x)"}, {"tactic": "rw [hp, Polynomial.aroots_zero] at hx", "annotated_tactic": ["rw [hp, <a>Polynomial.aroots_zero</a>] at hx", [{"full_name": "Polynomial.aroots_zero", "def_path": "Mathlib/Data/Polynomial/RingDivision.lean", "def_pos": [956, 9], "def_end_pos": [956, 20]}]], "state_before": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp\u271d : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nhp : p = 0\n\u22a2 False", "state_after": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp\u271d : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 0\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nhp : p = 0\n\u22a2 False"}, {"tactic": "exact Multiset.not_mem_zero x hx", "annotated_tactic": ["exact <a>Multiset.not_mem_zero</a> x hx", [{"full_name": "Multiset.not_mem_zero", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [260, 9], "def_end_pos": [260, 21]}]], "state_before": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp\u271d : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 0\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nhp : p = 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "apply minpoly.dvd", "annotated_tactic": ["apply <a>minpoly.dvd</a>", [{"full_name": "minpoly.dvd", "def_path": "Mathlib/FieldTheory/Minpoly/Field.lean", "def_pos": [69, 9], "def_end_pos": [69, 12]}]], "state_before": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\n\u22a2 minpoly K x \u2223 Polynomial.map (algebraMap F K) p", "state_after": "case hp\nF : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\n\u22a2 \u2191(Polynomial.aeval x) (Polynomial.map (algebraMap F K) p) = 0"}, {"tactic": "rw [Polynomial.aeval_def, Polynomial.eval\u2082_map, \u2190 Polynomial.eval_map, \u2190\n  IsScalarTower.algebraMap_eq]", "annotated_tactic": ["rw [<a>Polynomial.aeval_def</a>, <a>Polynomial.eval\u2082_map</a>, \u2190 <a>Polynomial.eval_map</a>, \u2190\n      <a>IsScalarTower.algebraMap_eq</a>]", [{"full_name": "Polynomial.aeval_def", "def_path": "Mathlib/Data/Polynomial/AlgebraMap.lean", "def_pos": [187, 9], "def_end_pos": [187, 18]}, {"full_name": "Polynomial.eval\u2082_map", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [956, 9], "def_end_pos": [956, 18]}, {"full_name": "Polynomial.eval_map", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [960, 9], "def_end_pos": [960, 17]}, {"full_name": "IsScalarTower.algebraMap_eq", "def_path": "Mathlib/Algebra/Algebra/Tower.lean", "def_pos": [122, 9], "def_end_pos": [122, 22]}]], "state_before": "case hp\nF : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\n\u22a2 \u2191(Polynomial.aeval x) (Polynomial.map (algebraMap F K) p) = 0", "state_after": "case hp\nF : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\n\u22a2 Polynomial.eval x (Polynomial.map (algebraMap F E) p) = 0"}, {"tactic": "exact (Polynomial.mem_roots (Polynomial.map_ne_zero h1)).mp hx", "annotated_tactic": ["exact (<a>Polynomial.mem_roots</a> (<a>Polynomial.map_ne_zero</a> h1)).<a>mp</a> hx", [{"full_name": "Polynomial.mem_roots", "def_path": "Mathlib/Data/Polynomial/RingDivision.lean", "def_pos": [599, 9], "def_end_pos": [599, 18]}, {"full_name": "Polynomial.map_ne_zero", "def_path": "Mathlib/Data/Polynomial/FieldDivision.lean", "def_pos": [142, 9], "def_end_pos": [142, 20]}, {"full_name": "Iff.mp", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [90, 3], "def_end_pos": [90, 5]}]], "state_before": "case hp\nF : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\n\u22a2 Polynomial.eval x (Polynomial.map (algebraMap F E) p) = 0", "state_after": "no goals"}, {"tactic": "change (K\u27eex\u27ef \u2192\u2090[F] E) \u2243 \u03a3 f : K \u2192\u2090[F] E, _", "annotated_tactic": ["change (K\u27eex\u27ef \u2192\u2090[F] E) \u2243 \u03a3 f : K \u2192\u2090[F] E, _", []], "state_before": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\n\u22a2 ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n    (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)", "state_after": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\n\u22a2 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[F] E) \u2243 (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)"}, {"tactic": "exact algHomEquivSigma", "annotated_tactic": ["exact <a>algHomEquivSigma</a>", [{"full_name": "algHomEquivSigma", "def_path": "Mathlib/RingTheory/AlgebraTower.lean", "def_pos": [218, 5], "def_end_pos": [218, 21]}]], "state_before": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\n\u22a2 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[F] E) \u2243 (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)", "state_after": "no goals"}, {"tactic": "have := Fintype.ofEquiv _ key_equiv", "annotated_tactic": ["have := <a>Fintype.ofEquiv</a> _ key_equiv", [{"full_name": "Fintype.ofEquiv", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [585, 5], "def_end_pos": [585, 12]}]], "state_before": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\nf : K \u2192\u2090[F] E\n\u22a2 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)", "state_after": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\nf : K \u2192\u2090[F] E\nthis : Fintype ((f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E))\n\u22a2 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)"}, {"tactic": "apply Fintype.ofInjective (Sigma.mk f) fun _ _ H => eq_of_heq (Sigma.ext_iff.mp H).2", "annotated_tactic": ["apply <a>Fintype.ofInjective</a> (<a>Sigma.mk</a> f) fun _ _ H => <a>eq_of_heq</a> (Sigma.ext_iff.mp H).2", [{"full_name": "Fintype.ofInjective", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [576, 19], "def_end_pos": [576, 30]}, {"full_name": "Sigma.mk", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [140, 3], "def_end_pos": [140, 5]}, {"full_name": "eq_of_heq", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [451, 9], "def_end_pos": [451, 18]}]], "state_before": "F : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\nf : K \u2192\u2090[F] E\nthis : Fintype ((f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E))\n\u22a2 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)", "state_after": "no goals"}, {"tactic": "congr!", "annotated_tactic": ["congr!", []], "state_before": "case h\u2081\nF : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\nthis : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)\nf : K \u2192\u2090[F] E\na\u271d : f \u2208 Finset.univ\n\u22a2 Fintype.card ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) = Fintype.card ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)", "state_after": "no goals"}, {"tactic": "exact Polynomial.Separable.of_dvd ((Polynomial.separable_map (algebraMap F K)).mpr hp) h2", "annotated_tactic": ["exact <a>Polynomial.Separable.of_dvd</a> ((<a>Polynomial.separable_map</a> (<a>algebraMap</a> F K)).<a>mpr</a> hp) h2", [{"full_name": "Polynomial.Separable.of_dvd", "def_path": "Mathlib/FieldTheory/Separable.lean", "def_pos": [92, 9], "def_end_pos": [92, 25]}, {"full_name": "Polynomial.separable_map", "def_path": "Mathlib/FieldTheory/Separable.lean", "def_pos": [286, 9], "def_end_pos": [286, 22]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "case h\u2081.h_sep\nF : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\nthis : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)\nf : K \u2192\u2090[F] E\na\u271d : f \u2208 Finset.univ\n\u22a2 Polynomial.Separable (minpoly K x)", "state_after": "no goals"}, {"tactic": "refine' Polynomial.splits_of_splits_of_dvd _ (Polynomial.map_ne_zero h1) _ h2", "annotated_tactic": ["refine' <a>Polynomial.splits_of_splits_of_dvd</a> _ (<a>Polynomial.map_ne_zero</a> h1) _ h2", [{"full_name": "Polynomial.splits_of_splits_of_dvd", "def_path": "Mathlib/Data/Polynomial/Splits.lean", "def_pos": [254, 9], "def_end_pos": [254, 32]}, {"full_name": "Polynomial.map_ne_zero", "def_path": "Mathlib/Data/Polynomial/FieldDivision.lean", "def_pos": [142, 9], "def_end_pos": [142, 20]}]], "state_before": "case h\u2081.h_splits\nF : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\nthis : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)\nf : K \u2192\u2090[F] E\na\u271d : f \u2208 Finset.univ\n\u22a2 Polynomial.Splits (algebraMap K E) (minpoly K x)", "state_after": "case h\u2081.h_splits\nF : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\nthis : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)\nf : K \u2192\u2090[F] E\na\u271d : f \u2208 Finset.univ\n\u22a2 Polynomial.Splits (algebraMap K E) (Polynomial.map (algebraMap F K) p)"}, {"tactic": "rw [Polynomial.splits_map_iff, \u2190 @IsScalarTower.algebraMap_eq _ _ _ _ _ _ _ (_) _ _]", "annotated_tactic": ["rw [<a>Polynomial.splits_map_iff</a>, \u2190 @<a>IsScalarTower.algebraMap_eq</a> _ _ _ _ _ _ _ (_) _ _]", [{"full_name": "Polynomial.splits_map_iff", "def_path": "Mathlib/Data/Polynomial/Splits.lean", "def_pos": [131, 9], "def_end_pos": [131, 23]}, {"full_name": "IsScalarTower.algebraMap_eq", "def_path": "Mathlib/Algebra/Algebra/Tower.lean", "def_pos": [122, 9], "def_end_pos": [122, 22]}]], "state_before": "case h\u2081.h_splits\nF : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\nthis : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)\nf : K \u2192\u2090[F] E\na\u271d : f \u2208 Finset.univ\n\u22a2 Polynomial.Splits (algebraMap K E) (Polynomial.map (algebraMap F K) p)", "state_after": "case h\u2081.h_splits\nF : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\nthis : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)\nf : K \u2192\u2090[F] E\na\u271d : f \u2208 Finset.univ\n\u22a2 Polynomial.Splits (algebraMap F E) p"}, {"tactic": "exact sp.splits", "annotated_tactic": ["exact sp.splits", []], "state_before": "case h\u2081.h_splits\nF : Type u_1\ninst\u271d\u2078 : Field F\nE : Type u_2\ninst\u271d\u2077 : Field E\ninst\u271d\u2076 : Algebra F E\np : F[X]\nhFE : FiniteDimensional F E\nsp : Polynomial.IsSplittingField F E p\nhp : Polynomial.Separable p\nK : Type u_3\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsScalarTower F K E\nx : E\nhx : x \u2208 Polynomial.aroots p E\ninst\u271d\u00b9 : Fintype (K \u2192\u2090[F] E)\ninst\u271d : Fintype ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E)\nh : IsIntegral K x\nh1 : p \u2260 0\nh2 : minpoly K x \u2223 Polynomial.map (algebraMap F K) p\nkey_equiv : ({ x_1 // x_1 \u2208 IntermediateField.restrictScalars F K\u27eex\u27ef } \u2192\u2090[F] E) \u2243\n  (f : K \u2192\u2090[F] E) \u00d7 ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E) :=\n  let_fun this := algHomEquivSigma;\n  this\nthis : (f : K \u2192\u2090[F] E) \u2192 Fintype ({ x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E)\nf : K \u2192\u2090[F] E\na\u271d : f \u2208 Finset.univ\n\u22a2 Polynomial.Splits (algebraMap F E) p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Atoms.lean", "full_name": "IsCoatom.lt_iff", "start": [177, 1], "end": [178, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Order/Hom/Basic.lean", "full_name": "ContinuousOrderHom.comp_id", "start": [180, 1], "end": [181, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Manifold/VectorBundle/Basic.lean", "full_name": "Bundle.contMDiffAt_section", "start": [210, 1], "end": [213, 87], "traced_tactics": [{"tactic": "simp_rw [contMDiffAt_totalSpace, and_iff_right_iff_imp]", "annotated_tactic": ["simp_rw [<a>contMDiffAt_totalSpace</a>, <a>and_iff_right_iff_imp</a>]", [{"full_name": "Bundle.contMDiffAt_totalSpace", "def_path": "Mathlib/Geometry/Manifold/VectorBundle/Basic.lean", "def_pos": [202, 9], "def_end_pos": [202, 31]}, {"full_name": "and_iff_right_iff_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [211, 17], "def_end_pos": [211, 38]}]], "state_before": "\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\ns : (x : B) \u2192 E x\nx\u2080 : B\n\u22a2 ContMDiffAt IB (ModelWithCorners.prod IB \ud835\udcd8(\ud835\udd5c, F)) n (fun x => TotalSpace.mk' F x (s x)) x\u2080 \u2194\n    ContMDiffAt IB \ud835\udcd8(\ud835\udd5c, F) n (fun x => (\u2191(trivializationAt F E x\u2080) { proj := x, snd := s x }).2) x\u2080", "state_after": "\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\ns : (x : B) \u2192 E x\nx\u2080 : B\n\u22a2 ContMDiffAt IB \ud835\udcd8(\ud835\udd5c, F) n (fun x => (\u2191(trivializationAt F E x\u2080) (TotalSpace.mk' F x (s x))).2) x\u2080 \u2192\n    ContMDiffAt IB IB n (fun x => x) x\u2080"}, {"tactic": "intro", "annotated_tactic": ["intro", []], "state_before": "\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\ns : (x : B) \u2192 E x\nx\u2080 : B\n\u22a2 ContMDiffAt IB \ud835\udcd8(\ud835\udd5c, F) n (fun x => (\u2191(trivializationAt F E x\u2080) (TotalSpace.mk' F x (s x))).2) x\u2080 \u2192\n    ContMDiffAt IB IB n (fun x => x) x\u2080", "state_after": "\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\ns : (x : B) \u2192 E x\nx\u2080 : B\na\u271d : ContMDiffAt IB \ud835\udcd8(\ud835\udd5c, F) n (fun x => (\u2191(trivializationAt F E x\u2080) (TotalSpace.mk' F x (s x))).2) x\u2080\n\u22a2 ContMDiffAt IB IB n (fun x => x) x\u2080"}, {"tactic": "exact contMDiffAt_id", "annotated_tactic": ["exact <a>contMDiffAt_id</a>", [{"full_name": "contMDiffAt_id", "def_path": "Mathlib/Geometry/Manifold/ContMDiff.lean", "def_pos": [1189, 9], "def_end_pos": [1189, 23]}]], "state_before": "\ud835\udd5c : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B \u2192 Type u_6\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\nEB : Type u_7\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup EB\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c EB\nHB : Type u_8\ninst\u271d\u2079 : TopologicalSpace HB\nIB : ModelWithCorners \ud835\udd5c EB HB\nE' : B \u2192 Type u_9\ninst\u271d\u2078 : (x : B) \u2192 Zero (E' x)\nEM : Type u_10\ninst\u271d\u2077 : NormedAddCommGroup EM\ninst\u271d\u2076 : NormedSpace \ud835\udd5c EM\nHM : Type u_11\ninst\u271d\u2075 : TopologicalSpace HM\nIM : ModelWithCorners \ud835\udd5c EM HM\ninst\u271d\u2074 : TopologicalSpace M\ninst\u271d\u00b3 : ChartedSpace HM M\nIs : SmoothManifoldWithCorners IM M\nn : \u2115\u221e\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : ChartedSpace HB B\ninst\u271d : FiberBundle F E\ns : (x : B) \u2192 E x\nx\u2080 : B\na\u271d : ContMDiffAt IB \ud835\udcd8(\ud835\udd5c, F) n (fun x => (\u2191(trivializationAt F E x\u2080) (TotalSpace.mk' F x (s x))).2) x\u2080\n\u22a2 ContMDiffAt IB IB n (fun x => x) x\u2080", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Init/Order/LinearOrder.lean", "full_name": "le_min", "start": [47, 1], "end": [51, 42], "traced_tactics": [{"tactic": "if h : a \u2264 b\nthen simp [min_def, if_pos h]; exact h\u2081\nelse simp [min_def, if_neg h]; exact h\u2082", "annotated_tactic": ["if h : a \u2264 b\n  then simp [<a>min_def</a>, <a>if_pos</a> h]; exact h\u2081\n  else simp [<a>min_def</a>, <a>if_neg</a> h]; exact h\u2082", [{"full_name": "min_def", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [25, 9], "def_end_pos": [25, 16]}, {"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}, {"full_name": "min_def", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [25, 9], "def_end_pos": [25, 16]}, {"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}]], "state_before": "\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\nh\u2081 : c \u2264 a\nh\u2082 : c \u2264 b\n\u22a2 c \u2264 min a b", "state_after": "no goals"}, {"tactic": "simp [min_def, if_pos h]", "annotated_tactic": ["simp [<a>min_def</a>, <a>if_pos</a> h]", [{"full_name": "min_def", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [25, 9], "def_end_pos": [25, 16]}, {"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}]], "state_before": "\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\nh\u2081 : c \u2264 a\nh\u2082 : c \u2264 b\nh : a \u2264 b\n\u22a2 c \u2264 min a b", "state_after": "\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\nh\u2081 : c \u2264 a\nh\u2082 : c \u2264 b\nh : a \u2264 b\n\u22a2 c \u2264 a"}, {"tactic": "exact h\u2081", "annotated_tactic": ["exact h\u2081", []], "state_before": "\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\nh\u2081 : c \u2264 a\nh\u2082 : c \u2264 b\nh : a \u2264 b\n\u22a2 c \u2264 a", "state_after": "no goals"}, {"tactic": "simp [min_def, if_neg h]", "annotated_tactic": ["simp [<a>min_def</a>, <a>if_neg</a> h]", [{"full_name": "min_def", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [25, 9], "def_end_pos": [25, 16]}, {"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}]], "state_before": "\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\nh\u2081 : c \u2264 a\nh\u2082 : c \u2264 b\nh : \u00aca \u2264 b\n\u22a2 c \u2264 min a b", "state_after": "\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\nh\u2081 : c \u2264 a\nh\u2082 : c \u2264 b\nh : \u00aca \u2264 b\n\u22a2 c \u2264 b"}, {"tactic": "exact h\u2082", "annotated_tactic": ["exact h\u2082", []], "state_before": "\u03b1 : Type u\ninst\u271d : LinearOrder \u03b1\na b c : \u03b1\nh\u2081 : c \u2264 a\nh\u2082 : c \u2264 b\nh : \u00aca \u2264 b\n\u22a2 c \u2264 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Matrix/Block.lean", "full_name": "Matrix.fromBlocks_apply\u2081\u2081", "start": [51, 1], "end": [53, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Clopen.lean", "full_name": "IsClopen.isClosed", "start": [34, 11], "end": [34, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Lie/Basic.lean", "full_name": "LieModuleHom.add_apply", "start": [900, 1], "end": [901, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/LocallyFinite.lean", "full_name": "Finset.Icc_subset_Icc_left", "start": [187, 1], "end": [188, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Complex/Basic.lean", "full_name": "Complex.I_re", "start": [299, 1], "end": [300, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/CharP/Basic.lean", "full_name": "frobenius_sub", "start": [434, 1], "end": [435, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/SesquilinearForm.lean", "full_name": "LinearMap.IsAlt.self_eq_zero", "start": [271, 1], "end": [272, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean", "full_name": "CliffordAlgebra.EquivEven.reverse_e0", "start": [112, 1], "end": [113, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/FieldDivision.lean", "full_name": "Polynomial.rootSet_X_pow", "start": [386, 1], "end": [389, 20], "traced_tactics": [{"tactic": "rw [\u2190 one_mul (X ^ n : R[X]), \u2190 C_1, rootSet_C_mul_X_pow hn]", "annotated_tactic": ["rw [\u2190 <a>one_mul</a> (<a>X</a> ^ n : R[X]), \u2190 <a>C_1</a>, <a>rootSet_C_mul_X_pow</a> hn]", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "Polynomial.X", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [566, 5], "def_end_pos": [566, 6]}, {"full_name": "Polynomial.C_1", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [518, 9], "def_end_pos": [518, 12]}, {"full_name": "Polynomial.rootSet_C_mul_X_pow", "def_path": "Mathlib/Data/Polynomial/FieldDivision.lean", "def_pos": [380, 9], "def_end_pos": [380, 28]}]], "state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b3 : Field R\np q : R[X]\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nn : \u2115\nhn : n \u2260 0\n\u22a2 rootSet (X ^ n) S = {0}", "state_after": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b3 : Field R\np q : R[X]\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nn : \u2115\nhn : n \u2260 0\n\u22a2 1 \u2260 0"}, {"tactic": "exact one_ne_zero", "annotated_tactic": ["exact <a>one_ne_zero</a>", [{"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}]], "state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b3 : Field R\np q : R[X]\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nn : \u2115\nhn : n \u2260 0\n\u22a2 1 \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Sites/Plus.lean", "full_name": "CategoryTheory.GrothendieckTopology.plusMap_comp", "start": [185, 1], "end": [189, 41], "traced_tactics": [{"tactic": "ext : 2", "annotated_tactic": ["ext : 2", []], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP Q R : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\n\u03b3 : Q \u27f6 R\n\u22a2 plusMap J (\u03b7 \u226b \u03b3) = plusMap J \u03b7 \u226b plusMap J \u03b3", "state_after": "case w.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP Q R : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\n\u03b3 : Q \u27f6 R\nx\u271d : C\u1d52\u1d56\n\u22a2 (plusMap J (\u03b7 \u226b \u03b3)).app x\u271d = (plusMap J \u03b7 \u226b plusMap J \u03b3).app x\u271d"}, {"tactic": "refine' colimit.hom_ext (fun S => _)", "annotated_tactic": ["refine' <a>colimit.hom_ext</a> (fun S => _)", [{"full_name": "CategoryTheory.Limits.colimit.hom_ext", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [843, 9], "def_end_pos": [843, 24]}]], "state_before": "case w.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP Q R : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\n\u03b3 : Q \u27f6 R\nx\u271d : C\u1d52\u1d56\n\u22a2 (plusMap J (\u03b7 \u226b \u03b3)).app x\u271d = (plusMap J \u03b7 \u226b plusMap J \u03b3).app x\u271d", "state_after": "case w.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP Q R : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\n\u03b3 : Q \u27f6 R\nx\u271d : C\u1d52\u1d56\nS : (Cover J x\u271d.unop)\u1d52\u1d56\n\u22a2 colimit.\u03b9 (diagram J P x\u271d.unop) S \u226b (plusMap J (\u03b7 \u226b \u03b3)).app x\u271d =\n    colimit.\u03b9 (diagram J P x\u271d.unop) S \u226b (plusMap J \u03b7 \u226b plusMap J \u03b3).app x\u271d"}, {"tactic": "simp [plusMap, J.diagramNatTrans_comp]", "annotated_tactic": ["simp [<a>plusMap</a>, J.diagramNatTrans_comp]", [{"full_name": "CategoryTheory.GrothendieckTopology.plusMap", "def_path": "Mathlib/CategoryTheory/Sites/Plus.lean", "def_pos": [154, 5], "def_end_pos": [154, 12]}]], "state_before": "case w.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\ninst\u271d\u00b9 : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\nP\u271d : C\u1d52\u1d56 \u2964 D\ninst\u271d : \u2200 (X : C), HasColimitsOfShape (Cover J X)\u1d52\u1d56 D\nP Q R : C\u1d52\u1d56 \u2964 D\n\u03b7 : P \u27f6 Q\n\u03b3 : Q \u27f6 R\nx\u271d : C\u1d52\u1d56\nS : (Cover J x\u271d.unop)\u1d52\u1d56\n\u22a2 colimit.\u03b9 (diagram J P x\u271d.unop) S \u226b (plusMap J (\u03b7 \u226b \u03b3)).app x\u271d =\n    colimit.\u03b9 (diagram J P x\u271d.unop) S \u226b (plusMap J \u03b7 \u226b plusMap J \u03b3).app x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean", "full_name": "NonUnitalAlgHom.subtype_comp_codRestrict", "start": [442, 1], "end": [444, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Star/Basic.lean", "full_name": "star_intCast", "start": [343, 1], "end": [344, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/CountableSeparatingOn.lean", "full_name": "Filter.exists_eventuallyEq_const_of_forall_separating", "start": [208, 1], "end": [212, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean", "full_name": "HasStrictDerivAt.cpow", "start": [151, 8], "end": [154, 47], "traced_tactics": [{"tactic": "simpa using (hf.cpow hg h0).hasStrictDerivAt", "annotated_tactic": ["simpa using (hf.cpow hg h0).<a>hasStrictDerivAt</a>", [{"full_name": "HasStrictFDerivAt.hasStrictDerivAt", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [208, 19], "def_end_pos": [208, 53]}]], "state_before": "f g : \u2102 \u2192 \u2102\ns : Set \u2102\nf' g' x c : \u2102\nhf : HasStrictDerivAt f f' x\nhg : HasStrictDerivAt g g' x\nh0 : 0 < (f x).re \u2228 (f x).im \u2260 0\n\u22a2 HasStrictDerivAt (fun x => f x ^ g x) (g x * f x ^ (g x - 1) * f' + f x ^ g x * Complex.log (f x) * g') x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_iSup", "start": [345, 1], "end": [411, 34], "traced_tactics": [{"tactic": "set c : \u211d\u22650 \u2192 \u211d\u22650\u221e := (\u2191)", "annotated_tactic": ["set c : \u211d\u22650 \u2192 \u211d\u22650\u221e := (\u2191)", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 n, f n a \u2202\u03bc = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 n, f n a \u2202\u03bc = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "set F := fun a : \u03b1 => \u2a06 n, f n a", "annotated_tactic": ["set F := fun a : \u03b1 => \u2a06 n, f n a", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\n\u22a2 \u222b\u207b (a : \u03b1), \u2a06 n, f n a \u2202\u03bc = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\n\u22a2 lintegral \u03bc F = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "have _ : Measurable F := measurable_iSup hf", "annotated_tactic": ["have _ : <a>Measurable</a> F := <a>measurable_iSup</a> hf", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "measurable_iSup", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [1360, 9], "def_end_pos": [1360, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\n\u22a2 lintegral \u03bc F = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\n\u22a2 lintegral \u03bc F = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "refine' le_antisymm _ (iSup_lintegral_le _)", "annotated_tactic": ["refine' <a>le_antisymm</a> _ (<a>iSup_lintegral_le</a> _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "MeasureTheory.iSup_lintegral_le", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [241, 9], "def_end_pos": [241, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\n\u22a2 lintegral \u03bc F = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\n\u22a2 lintegral \u03bc F \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "rw [lintegral_eq_nnreal]", "annotated_tactic": ["rw [<a>lintegral_eq_nnreal</a>]", [{"full_name": "MeasureTheory.lintegral_eq_nnreal", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [197, 9], "def_end_pos": [197, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\n\u22a2 lintegral \u03bc F \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\n\u22a2 \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 \u2a06 n, f n x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2264\n    \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "refine' iSup_le fun s => iSup_le fun hsf => _", "annotated_tactic": ["refine' <a>iSup_le</a> fun s => <a>iSup_le</a> fun hsf => _", [{"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}, {"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\n\u22a2 \u2a06 \u03c6, \u2a06 (_ : \u2200 (x : \u03b1), \u2191(\u2191\u03c6 x) \u2264 \u2a06 n, f n x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some \u03c6) \u03bc \u2264\n    \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "refine' ENNReal.le_of_forall_lt_one_mul_le fun a ha => _", "annotated_tactic": ["refine' <a>ENNReal.le_of_forall_lt_one_mul_le</a> fun a ha => _", [{"full_name": "ENNReal.le_of_forall_lt_one_mul_le", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [476, 9], "def_end_pos": [476, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\n\u22a2 SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\na : \u211d\u22650\u221e\nha : a < 1\n\u22a2 a * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "rcases ENNReal.lt_iff_exists_coe.1 ha with \u27e8r, rfl, _\u27e9", "annotated_tactic": ["rcases <a>ENNReal.lt_iff_exists_coe</a>.1 ha with \u27e8r, rfl, _\u27e9", [{"full_name": "ENNReal.lt_iff_exists_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [746, 9], "def_end_pos": [746, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\na : \u211d\u22650\u221e\nha : a < 1\n\u22a2 a * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha : \u2191r < 1\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "have ha : r < 1 := ENNReal.coe_lt_coe.1 ha", "annotated_tactic": ["have ha : r < 1 := <a>ENNReal.coe_lt_coe</a>.1 ha", [{"full_name": "ENNReal.coe_lt_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [352, 28], "def_end_pos": [352, 38]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha : \u2191r < 1\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "let rs := s.map fun a => r * a", "annotated_tactic": ["let rs := s.map fun a => r * a", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "have eq_rs : (const \u03b1 r : \u03b1 \u2192\u209b \u211d\u22650\u221e) * map c s = rs.map c := by\n  ext1 a\n  exact ENNReal.coe_mul.symm", "annotated_tactic": ["have eq_rs : (<a>const</a> \u03b1 r : \u03b1 \u2192\u209b \u211d\u22650\u221e) * <a>map</a> c s = rs.map c := by\n    ext1 a\n    exact ENNReal.coe_mul.symm", [{"full_name": "MeasureTheory.SimpleFunc.const", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [145, 5], "def_end_pos": [145, 10]}, {"full_name": "MeasureTheory.SimpleFunc.map", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [290, 5], "def_end_pos": [290, 8]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "have mono : \u2200 r : \u211d\u22650\u221e, Monotone fun n => rs.map c \u207b\u00b9' {r} \u2229 { a | r \u2264 f n a } := by\n  intro r i j h\n  refine' inter_subset_inter (Subset.refl _) _\n  intro x (hx : r \u2264 f i x)\n  exact le_trans hx (h_mono h x)", "annotated_tactic": ["have mono : \u2200 r : \u211d\u22650\u221e, <a>Monotone</a> fun n => rs.map c \u207b\u00b9' {r} \u2229 { a | r \u2264 f n a } := by\n    intro r i j h\n    refine' <a>inter_subset_inter</a> (<a>Subset.refl</a> _) _\n    intro x (hx : r \u2264 f i x)\n    exact <a>le_trans</a> hx (h_mono h x)", [{"full_name": "Monotone", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [77, 5], "def_end_pos": [77, 13]}, {"full_name": "Set.inter_subset_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1022, 9], "def_end_pos": [1022, 27]}, {"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [354, 9], "def_end_pos": [354, 20]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "have h_meas : \u2200 n, MeasurableSet { a : \u03b1 | (\u21d1(map c rs)) a \u2264 f n a } := fun n =>\n  measurableSet_le (SimpleFunc.measurable _) (hf n)", "annotated_tactic": ["have h_meas : \u2200 n, <a>MeasurableSet</a> { a : \u03b1 | (\u21d1(<a>map</a> c rs)) a \u2264 f n a } := fun n =>\n    <a>measurableSet_le</a> (<a>SimpleFunc.measurable</a> _) (hf n)", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "MeasureTheory.SimpleFunc.map", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [290, 5], "def_end_pos": [290, 8]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "MeasureTheory.SimpleFunc.measurable", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [201, 19], "def_end_pos": [201, 29]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map ENNReal.some s) \u03bc \u2264 \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "ext1 a", "annotated_tactic": ["ext1 a", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\n\u22a2 const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs", "state_after": "case H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\na : \u03b1\n\u22a2 \u2191(const \u03b1 \u2191r * SimpleFunc.map c s) a = \u2191(SimpleFunc.map c rs) a"}, {"tactic": "exact ENNReal.coe_mul.symm", "annotated_tactic": ["exact ENNReal.coe_mul.symm", []], "state_before": "case H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\na : \u03b1\n\u22a2 \u2191(const \u03b1 \u2191r * SimpleFunc.map c s) a = \u2191(SimpleFunc.map c rs) a", "state_after": "no goals"}, {"tactic": "intro p", "annotated_tactic": ["intro p", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\n\u22a2 \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\n\u22a2 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}"}, {"tactic": "rw [\u2190 inter_iUnion]", "annotated_tactic": ["rw [\u2190 <a>inter_iUnion</a>]", [{"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [635, 9], "def_end_pos": [635, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\n\u22a2 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\n\u22a2 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 \u22c3 i, {a | p \u2264 f i a}"}, {"tactic": "nth_rw 1 [\u2190 inter_univ (map c rs \u207b\u00b9' {p})]", "annotated_tactic": ["nth_rw 1 [\u2190 <a>inter_univ</a> (<a>map</a> c rs \u207b\u00b9' {p})]", [{"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 19]}, {"full_name": "MeasureTheory.SimpleFunc.map", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [290, 5], "def_end_pos": [290, 8]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\n\u22a2 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 \u22c3 i, {a | p \u2264 f i a}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\n\u22a2 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 univ = \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 \u22c3 i, {a | p \u2264 f i a}"}, {"tactic": "refine' Set.ext fun x => and_congr_right fun hx => true_iff_iff.2 _", "annotated_tactic": ["refine' <a>Set.ext</a> fun x => <a>and_congr_right</a> fun hx => <a>true_iff_iff</a>.2 _", [{"full_name": "Set.ext", "def_path": "Mathlib/Init/Set.lean", "def_pos": [54, 9], "def_end_pos": [54, 12]}, {"full_name": "and_congr_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [161, 9], "def_end_pos": [161, 24]}, {"full_name": "true_iff_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [198, 9], "def_end_pos": [198, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\n\u22a2 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 univ = \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 \u22c3 i, {a | p \u2264 f i a}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\nx : \u03b1\nhx : x \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p}\n\u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a}"}, {"tactic": "by_cases p_eq : p = 0", "annotated_tactic": ["by_cases p_eq : p = 0", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\nx : \u03b1\nhx : x \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p}\n\u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a}", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\nx : \u03b1\nhx : x \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p}\np_eq : p = 0\n\u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a}\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\nx : \u03b1\nhx : x \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p}\np_eq : \u00acp = 0\n\u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a}"}, {"tactic": "simp only [coe_map, mem_preimage, Function.comp_apply, mem_singleton_iff] at hx", "annotated_tactic": ["simp only [<a>coe_map</a>, <a>mem_preimage</a>, <a>Function.comp_apply</a>, <a>mem_singleton_iff</a>] at hx", [{"full_name": "MeasureTheory.SimpleFunc.coe_map", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [303, 9], "def_end_pos": [303, 16]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\nx : \u03b1\nhx : x \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p}\np_eq : \u00acp = 0\n\u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a}", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\nx : \u03b1\np_eq : \u00acp = 0\nhx : \u2191(r * \u2191s x) = p\n\u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a}"}, {"tactic": "subst hx", "annotated_tactic": ["subst hx", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\nx : \u03b1\np_eq : \u00acp = 0\nhx : \u2191(r * \u2191s x) = p\n\u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a}", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\n\u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a}"}, {"tactic": "have : r * s x \u2260 0 := by rwa [Ne, \u2190 ENNReal.coe_eq_zero]", "annotated_tactic": ["have : r * s x \u2260 0 := by rwa [<a>Ne</a>, \u2190 <a>ENNReal.coe_eq_zero</a>]", [{"full_name": "Ne", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [560, 18], "def_end_pos": [560, 20]}, {"full_name": "ENNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [368, 28], "def_end_pos": [368, 39]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\n\u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a}", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis : r * \u2191s x \u2260 0\n\u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a}"}, {"tactic": "have : s x \u2260 0 := by\n  refine' mt _ this\n  intro h\n  rw [h, mul_zero]", "annotated_tactic": ["have : s x \u2260 0 := by\n      refine' <a>mt</a> _ this\n      intro h\n      rw [h, <a>mul_zero</a>]", [{"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis : r * \u2191s x \u2260 0\n\u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a}", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d : r * \u2191s x \u2260 0\nthis : \u2191s x \u2260 0\n\u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a}"}, {"tactic": "have : (rs.map c) x < \u2a06 n : \u2115, f n x := by\n  refine' lt_of_lt_of_le (ENNReal.coe_lt_coe.2 _) (hsf x)\n  suffices r * s x < 1 * s x by simpa\n  exact mul_lt_mul_of_pos_right ha (pos_iff_ne_zero.2 this)", "annotated_tactic": ["have : (rs.map c) x < \u2a06 n : \u2115, f n x := by\n      refine' <a>lt_of_lt_of_le</a> (<a>ENNReal.coe_lt_coe</a>.2 _) (hsf x)\n      suffices r * s x < 1 * s x by simpa\n      exact <a>mul_lt_mul_of_pos_right</a> ha (<a>pos_iff_ne_zero</a>.2 this)", [{"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}, {"full_name": "ENNReal.coe_lt_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [352, 28], "def_end_pos": [352, 38]}, {"full_name": "mul_lt_mul_of_pos_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [164, 9], "def_end_pos": [164, 32]}, {"full_name": "pos_iff_ne_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [243, 3], "def_end_pos": [243, 14]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d : r * \u2191s x \u2260 0\nthis : \u2191s x \u2260 0\n\u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a}", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d\u00b9 : r * \u2191s x \u2260 0\nthis\u271d : \u2191s x \u2260 0\nthis : \u2191(SimpleFunc.map c rs) x < \u2a06 n, f n x\n\u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a}"}, {"tactic": "rcases lt_iSup_iff.1 this with \u27e8i, hi\u27e9", "annotated_tactic": ["rcases <a>lt_iSup_iff</a>.1 this with \u27e8i, hi\u27e9", [{"full_name": "lt_iSup_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [668, 9], "def_end_pos": [668, 20]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d\u00b9 : r * \u2191s x \u2260 0\nthis\u271d : \u2191s x \u2260 0\nthis : \u2191(SimpleFunc.map c rs) x < \u2a06 n, f n x\n\u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a}", "state_after": "case neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d\u00b9 : r * \u2191s x \u2260 0\nthis\u271d : \u2191s x \u2260 0\nthis : \u2191(SimpleFunc.map c rs) x < \u2a06 n, f n x\ni : \u2115\nhi : \u2191(SimpleFunc.map c rs) x < f i x\n\u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a}"}, {"tactic": "exact mem_iUnion.2 \u27e8i, le_of_lt hi\u27e9", "annotated_tactic": ["exact <a>mem_iUnion</a>.2 \u27e8i, <a>le_of_lt</a> hi\u27e9", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d\u00b9 : r * \u2191s x \u2260 0\nthis\u271d : \u2191s x \u2260 0\nthis : \u2191(SimpleFunc.map c rs) x < \u2a06 n, f n x\ni : \u2115\nhi : \u2191(SimpleFunc.map c rs) x < f i x\n\u22a2 x \u2208 \u22c3 i, {a | \u2191(r * \u2191s x) \u2264 f i a}", "state_after": "no goals"}, {"tactic": "simp [p_eq]", "annotated_tactic": ["simp [p_eq]", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\np : \u211d\u22650\u221e\nx : \u03b1\nhx : x \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p}\np_eq : p = 0\n\u22a2 x \u2208 \u22c3 i, {a | p \u2264 f i a}", "state_after": "no goals"}, {"tactic": "rwa [Ne, \u2190 ENNReal.coe_eq_zero]", "annotated_tactic": ["rwa [<a>Ne</a>, \u2190 <a>ENNReal.coe_eq_zero</a>]", [{"full_name": "Ne", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [560, 18], "def_end_pos": [560, 20]}, {"full_name": "ENNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [368, 28], "def_end_pos": [368, 39]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\n\u22a2 r * \u2191s x \u2260 0", "state_after": "no goals"}, {"tactic": "refine' mt _ this", "annotated_tactic": ["refine' <a>mt</a> _ this", [{"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis : r * \u2191s x \u2260 0\n\u22a2 \u2191s x \u2260 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis : r * \u2191s x \u2260 0\n\u22a2 \u2191s x = 0 \u2192 r * \u2191s x = 0"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis : r * \u2191s x \u2260 0\n\u22a2 \u2191s x = 0 \u2192 r * \u2191s x = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis : r * \u2191s x \u2260 0\nh : \u2191s x = 0\n\u22a2 r * \u2191s x = 0"}, {"tactic": "rw [h, mul_zero]", "annotated_tactic": ["rw [h, <a>mul_zero</a>]", [{"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis : r * \u2191s x \u2260 0\nh : \u2191s x = 0\n\u22a2 r * \u2191s x = 0", "state_after": "no goals"}, {"tactic": "refine' lt_of_lt_of_le (ENNReal.coe_lt_coe.2 _) (hsf x)", "annotated_tactic": ["refine' <a>lt_of_lt_of_le</a> (<a>ENNReal.coe_lt_coe</a>.2 _) (hsf x)", [{"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}, {"full_name": "ENNReal.coe_lt_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [352, 28], "def_end_pos": [352, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d : r * \u2191s x \u2260 0\nthis : \u2191s x \u2260 0\n\u22a2 \u2191(SimpleFunc.map c rs) x < \u2a06 n, f n x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d : r * \u2191s x \u2260 0\nthis : \u2191s x \u2260 0\n\u22a2 \u2191rs x < \u2191s x"}, {"tactic": "suffices r * s x < 1 * s x by simpa", "annotated_tactic": ["suffices r * s x < 1 * s x by simpa", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d : r * \u2191s x \u2260 0\nthis : \u2191s x \u2260 0\n\u22a2 \u2191rs x < \u2191s x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d : r * \u2191s x \u2260 0\nthis : \u2191s x \u2260 0\n\u22a2 r * \u2191s x < 1 * \u2191s x"}, {"tactic": "exact mul_lt_mul_of_pos_right ha (pos_iff_ne_zero.2 this)", "annotated_tactic": ["exact <a>mul_lt_mul_of_pos_right</a> ha (<a>pos_iff_ne_zero</a>.2 this)", [{"full_name": "mul_lt_mul_of_pos_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [164, 9], "def_end_pos": [164, 32]}, {"full_name": "pos_iff_ne_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [243, 3], "def_end_pos": [243, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d : r * \u2191s x \u2260 0\nthis : \u2191s x \u2260 0\n\u22a2 r * \u2191s x < 1 * \u2191s x", "state_after": "no goals"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\nx : \u03b1\np_eq : \u00ac\u2191(r * \u2191s x) = 0\nthis\u271d\u00b9 : r * \u2191s x \u2260 0\nthis\u271d : \u2191s x \u2260 0\nthis : r * \u2191s x < 1 * \u2191s x\n\u22a2 \u2191rs x < \u2191s x", "state_after": "no goals"}, {"tactic": "intro r i j h", "annotated_tactic": ["intro r i j h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\n\u22a2 \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nr : \u211d\u22650\u221e\ni j : \u2115\nh : i \u2264 j\n\u22a2 (fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) i \u2264\n    (fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) j"}, {"tactic": "refine' inter_subset_inter (Subset.refl _) _", "annotated_tactic": ["refine' <a>inter_subset_inter</a> (<a>Subset.refl</a> _) _", [{"full_name": "Set.inter_subset_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1022, 9], "def_end_pos": [1022, 27]}, {"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [354, 9], "def_end_pos": [354, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nr : \u211d\u22650\u221e\ni j : \u2115\nh : i \u2264 j\n\u22a2 (fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) i \u2264\n    (fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) j", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nr : \u211d\u22650\u221e\ni j : \u2115\nh : i \u2264 j\n\u22a2 {a | r \u2264 f i a} \u2286 {a | r \u2264 f j a}"}, {"tactic": "intro x (hx : r \u2264 f i x)", "annotated_tactic": ["intro x (hx : r \u2264 f i x)", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nr : \u211d\u22650\u221e\ni j : \u2115\nh : i \u2264 j\n\u22a2 {a | r \u2264 f i a} \u2286 {a | r \u2264 f j a}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nr : \u211d\u22650\u221e\ni j : \u2115\nh : i \u2264 j\nx : \u03b1\nhx : r \u2264 f i x\n\u22a2 x \u2208 {a | r \u2264 f j a}"}, {"tactic": "exact le_trans hx (h_mono h x)", "annotated_tactic": ["exact <a>le_trans</a> hx (h_mono h x)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nr : \u211d\u22650\u221e\ni j : \u2115\nh : i \u2264 j\nx : \u03b1\nhx : r \u2264 f i x\n\u22a2 x \u2208 {a | r \u2264 f j a}", "state_after": "no goals"}, {"tactic": "rw [\u2190 const_mul_lintegral, eq_rs, SimpleFunc.lintegral]", "annotated_tactic": ["rw [\u2190 <a>const_mul_lintegral</a>, eq_rs, <a>SimpleFunc.lintegral</a>]", [{"full_name": "MeasureTheory.SimpleFunc.const_mul_lintegral", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1003, 9], "def_end_pos": [1003, 28]}, {"full_name": "MeasureTheory.SimpleFunc.lintegral", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [950, 5], "def_end_pos": [950, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2191r * SimpleFunc.lintegral (SimpleFunc.map c s) \u03bc =\n    \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r})", "state_after": "no goals"}, {"tactic": "simp only [(eq _).symm]", "annotated_tactic": ["simp only [(eq _).<a>symm</a>]", [{"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r}) =\n    \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a})", "state_after": "no goals"}, {"tactic": "rw [measure_iUnion_eq_iSup (directed_of_sup <| mono x), ENNReal.mul_iSup]", "annotated_tactic": ["rw [<a>measure_iUnion_eq_iSup</a> (<a>directed_of_sup</a> <| mono x), <a>ENNReal.mul_iSup</a>]", [{"full_name": "MeasureTheory.measure_iUnion_eq_iSup", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [463, 9], "def_end_pos": [463, 31]}, {"full_name": "directed_of_sup", "def_path": "Mathlib/Order/Directed.lean", "def_pos": [107, 9], "def_end_pos": [107, 24]}, {"full_name": "ENNReal.mul_iSup", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [638, 9], "def_end_pos": [638, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d\u00b9 : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nx : \u211d\u22650\u221e\nx\u271d : x \u2208 SimpleFunc.range (SimpleFunc.map c rs)\n\u22a2 x * \u2191\u2191\u03bc (\u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {x} \u2229 {a | x \u2264 f n a}) =\n    \u2a06 n, x * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {x} \u2229 {a | x \u2264 f n a})", "state_after": "no goals"}, {"tactic": "rw [ENNReal.finset_sum_iSup_nat]", "annotated_tactic": ["rw [<a>ENNReal.finset_sum_iSup_nat</a>]", [{"full_name": "ENNReal.finset_sum_iSup_nat", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [627, 9], "def_end_pos": [627, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), \u2a06 n, r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) =\n    \u2a06 n, \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a})", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2200 (a : \u211d\u22650\u221e), Monotone fun n => a * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {a} \u2229 {a_1 | a \u2264 f n a_1})"}, {"tactic": "intro p i j h", "annotated_tactic": ["intro p i j h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2200 (a : \u211d\u22650\u221e), Monotone fun n => a * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {a} \u2229 {a_1 | a \u2264 f n a_1})", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\np : \u211d\u22650\u221e\ni j : \u2115\nh : i \u2264 j\n\u22a2 (fun n => p * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a})) i \u2264\n    (fun n => p * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a})) j"}, {"tactic": "exact mul_le_mul_left' (measure_mono <| mono p h) _", "annotated_tactic": ["exact <a>mul_le_mul_left'</a> (<a>measure_mono</a> <| mono p h) _", [{"full_name": "mul_le_mul_left'", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [50, 9], "def_end_pos": [50, 25]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [193, 9], "def_end_pos": [193, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\np : \u211d\u22650\u221e\ni j : \u2115\nh : i \u2264 j\n\u22a2 (fun n => p * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a})) i \u2264\n    (fun n => p * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a})) j", "state_after": "no goals"}, {"tactic": "refine' iSup_mono fun n => _", "annotated_tactic": ["refine' <a>iSup_mono</a> fun n => _", [{"full_name": "iSup_mono", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [900, 9], "def_end_pos": [900, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2a06 n, \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) \u2264\n    \u2a06 n, SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\n\u22a2 \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) \u2264\n    SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc"}, {"tactic": "rw [restrict_lintegral _ (h_meas n)]", "annotated_tactic": ["rw [<a>restrict_lintegral</a> _ (h_meas n)]", [{"full_name": "MeasureTheory.SimpleFunc.restrict_lintegral", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1048, 9], "def_end_pos": [1048, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\n\u22a2 \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) \u2264\n    SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\n\u22a2 \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) \u2264\n    \u2211 r in SimpleFunc.range (SimpleFunc.map c rs),\n      r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a})"}, {"tactic": "refine' le_of_eq (Finset.sum_congr rfl fun r _ => _)", "annotated_tactic": ["refine' <a>le_of_eq</a> (<a>Finset.sum_congr</a> <a>rfl</a> fun r _ => _)", [{"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Finset.sum_congr", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [376, 3], "def_end_pos": [376, 14]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\n\u22a2 \u2211 r in SimpleFunc.range (SimpleFunc.map c rs), r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) \u2264\n    \u2211 r in SimpleFunc.range (SimpleFunc.map c rs),\n      r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a})", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d\u00b9 : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\nr : \u211d\u22650\u221e\nx\u271d : r \u2208 SimpleFunc.range (SimpleFunc.map c rs)\n\u22a2 r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) =\n    r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a})"}, {"tactic": "congr 2 with a", "annotated_tactic": ["congr 2 with a", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d\u00b9 : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\nr : \u211d\u22650\u221e\nx\u271d : r \u2208 SimpleFunc.range (SimpleFunc.map c rs)\n\u22a2 r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}) =\n    r * \u2191\u2191\u03bc (\u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a})", "state_after": "case e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d\u00b9 : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\nr : \u211d\u22650\u221e\nx\u271d : r \u2208 SimpleFunc.range (SimpleFunc.map c rs)\na : \u03b1\n\u22a2 a \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} \u2194\n    a \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}"}, {"tactic": "refine' and_congr_right _", "annotated_tactic": ["refine' <a>and_congr_right</a> _", [{"full_name": "and_congr_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [161, 9], "def_end_pos": [161, 24]}]], "state_before": "case e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d\u00b9 : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\nr : \u211d\u22650\u221e\nx\u271d : r \u2208 SimpleFunc.range (SimpleFunc.map c rs)\na : \u03b1\n\u22a2 a \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a} \u2194\n    a \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}", "state_after": "case e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d\u00b9 : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\nr : \u211d\u22650\u221e\nx\u271d : r \u2208 SimpleFunc.range (SimpleFunc.map c rs)\na : \u03b1\n\u22a2 a \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2192 (a \u2208 {a | r \u2264 f n a} \u2194 a \u2208 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a})"}, {"tactic": "simp (config := { contextual := true })", "annotated_tactic": ["simp (config := { contextual := <a>true</a> })", [{"full_name": "Bool.true", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [549, 5], "def_end_pos": [549, 9]}]], "state_before": "case e_a.e_a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d\u00b9 : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr\u271d : \u211d\u22650\nright\u271d ha\u271d : \u2191r\u271d < 1\nha : r\u271d < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r\u271d * a) s\neq_rs : const \u03b1 \u2191r\u271d * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\nr : \u211d\u22650\u221e\nx\u271d : r \u2208 SimpleFunc.range (SimpleFunc.map c rs)\na : \u03b1\n\u22a2 a \u2208 \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2192 (a \u2208 {a | r \u2264 f n a} \u2194 a \u2208 {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a})", "state_after": "no goals"}, {"tactic": "refine' iSup_mono fun n => _", "annotated_tactic": ["refine' <a>iSup_mono</a> fun n => _", [{"full_name": "iSup_mono", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [900, 9], "def_end_pos": [900, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\n\u22a2 \u2a06 n, SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc \u2264\n    \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\n\u22a2 SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc \u2264 \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "rw [\u2190 SimpleFunc.lintegral_eq_lintegral]", "annotated_tactic": ["rw [\u2190 <a>SimpleFunc.lintegral_eq_lintegral</a>]", [{"full_name": "MeasureTheory.SimpleFunc.lintegral_eq_lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [80, 9], "def_end_pos": [80, 42]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\n\u22a2 SimpleFunc.lintegral (restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) \u03bc \u2264 \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f n a \u2202\u03bc"}, {"tactic": "refine' lintegral_mono fun a => _", "annotated_tactic": ["refine' <a>lintegral_mono</a> fun a => _", [{"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\n\u22a2 \u222b\u207b (a : \u03b1), \u2191(restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f n a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\na : \u03b1\n\u22a2 \u2191(restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) a \u2264 f n a"}, {"tactic": "simp only [map_apply] at h_meas", "annotated_tactic": ["simp only [<a>map_apply</a>] at h_meas", [{"full_name": "MeasureTheory.SimpleFunc.map_apply", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [294, 9], "def_end_pos": [294, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}\nn : \u2115\na : \u03b1\n\u22a2 \u2191(restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) a \u2264 f n a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(r * \u2191s a) \u2264 f n a}\nn : \u2115\na : \u03b1\n\u22a2 \u2191(restrict (SimpleFunc.map c rs) {a | \u2191(SimpleFunc.map c rs) a \u2264 f n a}) a \u2264 f n a"}, {"tactic": "exact indicator_apply_le id", "annotated_tactic": ["exact <a>indicator_apply_le</a> <a>id</a>", [{"full_name": "Set.indicator_apply_le", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [906, 3], "def_end_pos": [906, 14]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nhf : \u2200 (n : \u2115), Measurable (f n)\nh_mono : Monotone f\nc : \u211d\u22650 \u2192 \u211d\u22650\u221e := ENNReal.some\nF : \u03b1 \u2192 \u211d\u22650\u221e := fun a => \u2a06 n, f n a\nx\u271d : Measurable F\ns : \u03b1 \u2192\u209b \u211d\u22650\nhsf : \u2200 (x : \u03b1), \u2191(\u2191s x) \u2264 \u2a06 n, f n x\nr : \u211d\u22650\nright\u271d ha\u271d : \u2191r < 1\nha : r < 1\nrs : \u03b1 \u2192\u209b \u211d\u22650 := SimpleFunc.map (fun a => r * a) s\neq_rs : const \u03b1 \u2191r * SimpleFunc.map c s = SimpleFunc.map c rs\neq : \u2200 (p : \u211d\u22650\u221e), \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} = \u22c3 n, \u2191(SimpleFunc.map c rs) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}\nmono : \u2200 (r : \u211d\u22650\u221e), Monotone fun n => \u2191(SimpleFunc.map c rs) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}\nh_meas : \u2200 (n : \u2115), MeasurableSet {a | \u2191(r * \u2191s a) \u2264 f n a}\nn : \u2115\na : \u03b1\n\u22a2 indicator {a | \u2191(r * \u2191s a) \u2264 f n a} (fun x => \u2191(r * \u2191s x)) a \u2264 f n a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.OuterMeasure.toMeasure_top", "start": [1067, 1], "end": [1072, 63], "traced_tactics": [{"tactic": "rw [OuterMeasure.top_caratheodory]", "annotated_tactic": ["rw [<a>OuterMeasure.top_caratheodory</a>]", [{"full_name": "MeasureTheory.OuterMeasure.top_caratheodory", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [1095, 9], "def_end_pos": [1095, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 inst\u271d \u2264 OuterMeasure.caratheodory \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 inst\u271d \u2264 \u22a4"}, {"tactic": "exact le_top", "annotated_tactic": ["exact <a>le_top</a>", [{"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [98, 9], "def_end_pos": [98, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 inst\u271d \u2264 \u22a4", "state_after": "no goals"}, {"tactic": "cases' s.eq_empty_or_nonempty with h h <;>\n  simp [h, toMeasure_apply \u22a4 _ hs, OuterMeasure.top_apply]", "annotated_tactic": ["cases' s.eq_empty_or_nonempty with h h <;>\n      simp [h, <a>toMeasure_apply</a> \u22a4 _ hs, <a>OuterMeasure.top_apply</a>]", [{"full_name": "MeasureTheory.toMeasure_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [674, 9], "def_end_pos": [674, 24]}, {"full_name": "MeasureTheory.OuterMeasure.top_apply", "def_path": "Mathlib/MeasureTheory/Measure/OuterMeasure.lean", "def_pos": [619, 9], "def_end_pos": [619, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191\u22a4 s \u2264 \u2191\u2191(OuterMeasure.toMeasure \u22a4 (_ : inst\u271d \u2264 OuterMeasure.caratheodory \u22a4)) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Int/Basic.lean", "full_name": "Int.eq_of_associated_of_nonneg", "start": [128, 1], "end": [130, 98], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.mem_union_left", "start": [1382, 1], "end": [1383, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "full_name": "Fin.val_two", "start": [195, 9], "end": [195, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/Liouville/LiouvilleWith.lean", "full_name": "Liouville.liouvilleWith", "start": [372, 11], "end": [377, 22], "traced_tactics": [{"tactic": "suffices : LiouvilleWith \u2308p\u2309\u208a x", "annotated_tactic": ["suffices : <a>LiouvilleWith</a> \u2308p\u2309\u208a x", [{"full_name": "LiouvilleWith", "def_path": "Mathlib/NumberTheory/Liouville/LiouvilleWith.lean", "def_pos": [49, 5], "def_end_pos": [49, 18]}]], "state_before": "x : \u211d\nhx : Liouville x\np : \u211d\n\u22a2 LiouvilleWith p x", "state_after": "x : \u211d\nhx : Liouville x\np : \u211d\nthis : LiouvilleWith (\u2191\u2308p\u2309\u208a) x\n\u22a2 LiouvilleWith p x\n\ncase this\nx : \u211d\nhx : Liouville x\np : \u211d\n\u22a2 LiouvilleWith (\u2191\u2308p\u2309\u208a) x"}, {"tactic": "exact this.mono (Nat.le_ceil p)", "annotated_tactic": ["exact this.mono (<a>Nat.le_ceil</a> p)", [{"full_name": "Nat.le_ceil", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [299, 9], "def_end_pos": [299, 16]}]], "state_before": "x : \u211d\nhx : Liouville x\np : \u211d\nthis : LiouvilleWith (\u2191\u2308p\u2309\u208a) x\n\u22a2 LiouvilleWith p x\n\ncase this\nx : \u211d\nhx : Liouville x\np : \u211d\n\u22a2 LiouvilleWith (\u2191\u2308p\u2309\u208a) x", "state_after": "case this\nx : \u211d\nhx : Liouville x\np : \u211d\n\u22a2 LiouvilleWith (\u2191\u2308p\u2309\u208a) x"}, {"tactic": "refine \u27e81, ((eventually_gt_atTop 1).and_frequently (hx.frequently_exists_num \u2308p\u2309\u208a)).mono ?_\u27e9", "annotated_tactic": ["refine \u27e81, ((<a>eventually_gt_atTop</a> 1).<a>and_frequently</a> (hx.frequently_exists_num \u2308p\u2309\u208a)).<a>mono</a> ?_\u27e9", [{"full_name": "Filter.eventually_gt_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [197, 9], "def_end_pos": [197, 28]}, {"full_name": "Filter.Eventually.and_frequently", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1297, 9], "def_end_pos": [1297, 34]}, {"full_name": "Filter.Frequently.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1286, 9], "def_end_pos": [1286, 24]}]], "state_before": "case this\nx : \u211d\nhx : Liouville x\np : \u211d\n\u22a2 LiouvilleWith (\u2191\u2308p\u2309\u208a) x", "state_after": "case this\nx : \u211d\nhx : Liouville x\np : \u211d\n\u22a2 \u2200 (x_1 : \u2115),\n    (1 < x_1 \u2227 \u2203 a, x \u2260 \u2191a / \u2191x_1 \u2227 |x - \u2191a / \u2191x_1| < 1 / \u2191x_1 ^ \u2308p\u2309\u208a) \u2192\n      \u2203 m, x \u2260 \u2191m / \u2191x_1 \u2227 |x - \u2191m / \u2191x_1| < 1 / \u2191x_1 ^ \u2191\u2308p\u2309\u208a"}, {"tactic": "rintro b \u27e8_hb, a, hne, hlt\u27e9", "annotated_tactic": ["rintro b \u27e8_hb, a, hne, hlt\u27e9", []], "state_before": "case this\nx : \u211d\nhx : Liouville x\np : \u211d\n\u22a2 \u2200 (x_1 : \u2115),\n    (1 < x_1 \u2227 \u2203 a, x \u2260 \u2191a / \u2191x_1 \u2227 |x - \u2191a / \u2191x_1| < 1 / \u2191x_1 ^ \u2308p\u2309\u208a) \u2192\n      \u2203 m, x \u2260 \u2191m / \u2191x_1 \u2227 |x - \u2191m / \u2191x_1| < 1 / \u2191x_1 ^ \u2191\u2308p\u2309\u208a", "state_after": "case this.intro.intro.intro\nx : \u211d\nhx : Liouville x\np : \u211d\nb : \u2115\n_hb : 1 < b\na : \u2124\nhne : x \u2260 \u2191a / \u2191b\nhlt : |x - \u2191a / \u2191b| < 1 / \u2191b ^ \u2308p\u2309\u208a\n\u22a2 \u2203 m, x \u2260 \u2191m / \u2191b \u2227 |x - \u2191m / \u2191b| < 1 / \u2191b ^ \u2191\u2308p\u2309\u208a"}, {"tactic": "refine \u27e8a, hne, ?_\u27e9", "annotated_tactic": ["refine \u27e8a, hne, ?_\u27e9", []], "state_before": "case this.intro.intro.intro\nx : \u211d\nhx : Liouville x\np : \u211d\nb : \u2115\n_hb : 1 < b\na : \u2124\nhne : x \u2260 \u2191a / \u2191b\nhlt : |x - \u2191a / \u2191b| < 1 / \u2191b ^ \u2308p\u2309\u208a\n\u22a2 \u2203 m, x \u2260 \u2191m / \u2191b \u2227 |x - \u2191m / \u2191b| < 1 / \u2191b ^ \u2191\u2308p\u2309\u208a", "state_after": "case this.intro.intro.intro\nx : \u211d\nhx : Liouville x\np : \u211d\nb : \u2115\n_hb : 1 < b\na : \u2124\nhne : x \u2260 \u2191a / \u2191b\nhlt : |x - \u2191a / \u2191b| < 1 / \u2191b ^ \u2308p\u2309\u208a\n\u22a2 |x - \u2191a / \u2191b| < 1 / \u2191b ^ \u2191\u2308p\u2309\u208a"}, {"tactic": "rwa [rpow_nat_cast]", "annotated_tactic": ["rwa [<a>rpow_nat_cast</a>]", [{"full_name": "Real.rpow_nat_cast", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [357, 9], "def_end_pos": [357, 22]}]], "state_before": "case this.intro.intro.intro\nx : \u211d\nhx : Liouville x\np : \u211d\nb : \u2115\n_hb : 1 < b\na : \u2124\nhne : x \u2260 \u2191a / \u2191b\nhlt : |x - \u2191a / \u2191b| < 1 / \u2191b ^ \u2308p\u2309\u208a\n\u22a2 |x - \u2191a / \u2191b| < 1 / \u2191b ^ \u2191\u2308p\u2309\u208a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean", "full_name": "EuclideanGeometry.oangle_midpoint_rev_left", "start": [587, 1], "end": [588, 43], "traced_tactics": [{"tactic": "rw [midpoint_comm, oangle_midpoint_left]", "annotated_tactic": ["rw [<a>midpoint_comm</a>, <a>oangle_midpoint_left</a>]", [{"full_name": "midpoint_comm", "def_path": "Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean", "def_pos": [73, 9], "def_end_pos": [73, 22]}, {"full_name": "EuclideanGeometry.oangle_midpoint_left", "def_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean", "def_pos": [579, 9], "def_end_pos": [579, 29]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\n\u22a2 \u2221 (midpoint \u211d p\u2082 p\u2081) p\u2082 p\u2083 = \u2221 p\u2081 p\u2082 p\u2083", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.le_max_left", "start": [715, 11], "end": [715, 99], "traced_tactics": [{"tactic": "rw [Int.max_def]", "annotated_tactic": ["rw [<a>Int.max_def</a>]", [{"full_name": "Int.max_def", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [688, 19], "def_end_pos": [688, 26]}]], "state_before": "a b : Int\n\u22a2 a \u2264 max a b", "state_after": "a b : Int\n\u22a2 a \u2264 if a \u2264 b then b else a"}, {"tactic": "split <;> simp [*]", "annotated_tactic": ["split <;> simp [*]", []], "state_before": "a b : Int\n\u22a2 a \u2264 if a \u2264 b then b else a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Bounds/Basic.lean", "full_name": "lowerBounds_mono", "start": [211, 1], "end": [213, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Module/Torsion.lean", "full_name": "Submodule.noZeroSMulDivisors_iff_torsion_eq_bot", "start": [703, 1], "end": [720, 66], "traced_tactics": [{"tactic": "constructor <;> intro h", "annotated_tactic": ["constructor <;> intro h", []], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\n\u22a2 NoZeroSMulDivisors R M \u2194 torsion R M = \u22a5", "state_after": "case mp\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : NoZeroSMulDivisors R M\n\u22a2 torsion R M = \u22a5\n\ncase mpr\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : torsion R M = \u22a5\n\u22a2 NoZeroSMulDivisors R M"}, {"tactic": "haveI : NoZeroSMulDivisors R M := h", "annotated_tactic": ["haveI : <a>NoZeroSMulDivisors</a> R M := h", [{"full_name": "NoZeroSMulDivisors", "def_path": "Mathlib/Algebra/Module/Basic.lean", "def_pos": [583, 7], "def_end_pos": [583, 25]}]], "state_before": "case mp\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : NoZeroSMulDivisors R M\n\u22a2 torsion R M = \u22a5", "state_after": "case mp\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh this : NoZeroSMulDivisors R M\n\u22a2 torsion R M = \u22a5"}, {"tactic": "rw [eq_bot_iff]", "annotated_tactic": ["rw [<a>eq_bot_iff</a>]", [{"full_name": "eq_bot_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [363, 9], "def_end_pos": [363, 19]}]], "state_before": "case mp\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh this : NoZeroSMulDivisors R M\n\u22a2 torsion R M = \u22a5", "state_after": "case mp\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh this : NoZeroSMulDivisors R M\n\u22a2 torsion R M \u2264 \u22a5"}, {"tactic": "rintro x \u27e8a, hax\u27e9", "annotated_tactic": ["rintro x \u27e8a, hax\u27e9", []], "state_before": "case mp\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh this : NoZeroSMulDivisors R M\n\u22a2 torsion R M \u2264 \u22a5", "state_after": "case mp.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh this : NoZeroSMulDivisors R M\nx : M\na : { x // x \u2208 R\u2070 }\nhax : a \u2022 x = 0\n\u22a2 x \u2208 \u22a5"}, {"tactic": "change (a : R) \u2022 x = 0 at hax", "annotated_tactic": ["change (a : R) \u2022 x = 0 at hax", []], "state_before": "case mp.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh this : NoZeroSMulDivisors R M\nx : M\na : { x // x \u2208 R\u2070 }\nhax : a \u2022 x = 0\n\u22a2 x \u2208 \u22a5", "state_after": "case mp.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh this : NoZeroSMulDivisors R M\nx : M\na : { x // x \u2208 R\u2070 }\nhax : \u2191a \u2022 x = 0\n\u22a2 x \u2208 \u22a5"}, {"tactic": "cases' eq_zero_or_eq_zero_of_smul_eq_zero hax with h0 h0", "annotated_tactic": ["cases' <a>eq_zero_or_eq_zero_of_smul_eq_zero</a> hax with h0 h0", [{"full_name": "NoZeroSMulDivisors.eq_zero_or_eq_zero_of_smul_eq_zero", "def_path": "Mathlib/Algebra/Module/Basic.lean", "def_pos": [585, 3], "def_end_pos": [585, 37]}]], "state_before": "case mp.intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh this : NoZeroSMulDivisors R M\nx : M\na : { x // x \u2208 R\u2070 }\nhax : \u2191a \u2022 x = 0\n\u22a2 x \u2208 \u22a5", "state_after": "case mp.intro.inl\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh this : NoZeroSMulDivisors R M\nx : M\na : { x // x \u2208 R\u2070 }\nhax : \u2191a \u2022 x = 0\nh0 : \u2191a = 0\n\u22a2 x \u2208 \u22a5\n\ncase mp.intro.inr\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh this : NoZeroSMulDivisors R M\nx : M\na : { x // x \u2208 R\u2070 }\nhax : \u2191a \u2022 x = 0\nh0 : x = 0\n\u22a2 x \u2208 \u22a5"}, {"tactic": "exfalso", "annotated_tactic": ["exfalso", []], "state_before": "case mp.intro.inl\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh this : NoZeroSMulDivisors R M\nx : M\na : { x // x \u2208 R\u2070 }\nhax : \u2191a \u2022 x = 0\nh0 : \u2191a = 0\n\u22a2 x \u2208 \u22a5", "state_after": "case mp.intro.inl.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh this : NoZeroSMulDivisors R M\nx : M\na : { x // x \u2208 R\u2070 }\nhax : \u2191a \u2022 x = 0\nh0 : \u2191a = 0\n\u22a2 False"}, {"tactic": "exact nonZeroDivisors.coe_ne_zero a h0", "annotated_tactic": ["exact <a>nonZeroDivisors.coe_ne_zero</a> a h0", [{"full_name": "nonZeroDivisors.coe_ne_zero", "def_path": "Mathlib/RingTheory/NonZeroDivisors.lean", "def_pos": [87, 9], "def_end_pos": [87, 36]}]], "state_before": "case mp.intro.inl.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh this : NoZeroSMulDivisors R M\nx : M\na : { x // x \u2208 R\u2070 }\nhax : \u2191a \u2022 x = 0\nh0 : \u2191a = 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exact h0", "annotated_tactic": ["exact h0", []], "state_before": "case mp.intro.inr\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh this : NoZeroSMulDivisors R M\nx : M\na : { x // x \u2208 R\u2070 }\nhax : \u2191a \u2022 x = 0\nh0 : x = 0\n\u22a2 x \u2208 \u22a5", "state_after": "no goals"}, {"tactic": "by_cases ha : a = 0", "annotated_tactic": ["by_cases ha : a = 0", []], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : torsion R M = \u22a5\na : R\nx : M\nhax : a \u2022 x = 0\n\u22a2 a = 0 \u2228 x = 0", "state_after": "case pos\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : torsion R M = \u22a5\na : R\nx : M\nhax : a \u2022 x = 0\nha : a = 0\n\u22a2 a = 0 \u2228 x = 0\n\ncase neg\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : torsion R M = \u22a5\na : R\nx : M\nhax : a \u2022 x = 0\nha : \u00aca = 0\n\u22a2 a = 0 \u2228 x = 0"}, {"tactic": "left", "annotated_tactic": ["left", []], "state_before": "case pos\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : torsion R M = \u22a5\na : R\nx : M\nhax : a \u2022 x = 0\nha : a = 0\n\u22a2 a = 0 \u2228 x = 0", "state_after": "case pos.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : torsion R M = \u22a5\na : R\nx : M\nhax : a \u2022 x = 0\nha : a = 0\n\u22a2 a = 0"}, {"tactic": "exact ha", "annotated_tactic": ["exact ha", []], "state_before": "case pos.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : torsion R M = \u22a5\na : R\nx : M\nhax : a \u2022 x = 0\nha : a = 0\n\u22a2 a = 0", "state_after": "no goals"}, {"tactic": "right", "annotated_tactic": ["right", []], "state_before": "case neg\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : torsion R M = \u22a5\na : R\nx : M\nhax : a \u2022 x = 0\nha : \u00aca = 0\n\u22a2 a = 0 \u2228 x = 0", "state_after": "case neg.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : torsion R M = \u22a5\na : R\nx : M\nhax : a \u2022 x = 0\nha : \u00aca = 0\n\u22a2 x = 0"}, {"tactic": "rw [\u2190 mem_bot R, \u2190 h]", "annotated_tactic": ["rw [\u2190 <a>mem_bot</a> R, \u2190 h]", [{"full_name": "Submodule.mem_bot", "def_path": "Mathlib/Algebra/Module/Submodule/Lattice.lean", "def_pos": [76, 9], "def_end_pos": [76, 16]}]], "state_before": "case neg.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : torsion R M = \u22a5\na : R\nx : M\nhax : a \u2022 x = 0\nha : \u00aca = 0\n\u22a2 x = 0", "state_after": "case neg.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : torsion R M = \u22a5\na : R\nx : M\nhax : a \u2022 x = 0\nha : \u00aca = 0\n\u22a2 x \u2208 torsion R M"}, {"tactic": "exact \u27e8\u27e8a, mem_nonZeroDivisors_of_ne_zero ha\u27e9, hax\u27e9", "annotated_tactic": ["exact \u27e8\u27e8a, <a>mem_nonZeroDivisors_of_ne_zero</a> ha\u27e9, hax\u27e9", [{"full_name": "mem_nonZeroDivisors_of_ne_zero", "def_path": "Mathlib/RingTheory/NonZeroDivisors.lean", "def_pos": [119, 9], "def_end_pos": [119, 39]}]], "state_before": "case neg.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : NoZeroDivisors R\ninst\u271d : Nontrivial R\nh : torsion R M = \u22a5\na : R\nx : M\nhax : a \u2022 x = 0\nha : \u00aca = 0\n\u22a2 x \u2208 torsion R M", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/Pairwise.lean", "full_name": "List.Pairwise.forall", "start": [82, 1], "end": [87, 46], "traced_tactics": [{"tactic": "apply Pairwise.forall_of_forall", "annotated_tactic": ["apply <a>Pairwise.forall_of_forall</a>", [{"full_name": "List.Pairwise.forall_of_forall", "def_path": "Mathlib/Data/List/Pairwise.lean", "def_pos": [77, 9], "def_end_pos": [77, 34]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nhR : Symmetric R\nhl : Pairwise R l\n\u22a2 \u2200 \u2983a : \u03b1\u2984, a \u2208 l \u2192 \u2200 \u2983b : \u03b1\u2984, b \u2208 l \u2192 a \u2260 b \u2192 R a b", "state_after": "case H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nhR : Symmetric R\nhl : Pairwise R l\n\u22a2 Symmetric fun x y => x \u2260 y \u2192 R x y\n\ncase H\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nhR : Symmetric R\nhl : Pairwise R l\n\u22a2 \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2260 x \u2192 R x x\n\ncase H\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nhR : Symmetric R\nhl : Pairwise R l\n\u22a2 Pairwise (fun x y => x \u2260 y \u2192 R x y) l"}, {"tactic": "exact fun a b h hne => hR (h hne.symm)", "annotated_tactic": ["exact fun a b h hne => hR (h hne.symm)", []], "state_before": "case H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nhR : Symmetric R\nhl : Pairwise R l\n\u22a2 Symmetric fun x y => x \u2260 y \u2192 R x y", "state_after": "no goals"}, {"tactic": "exact fun _ _ hx => (hx rfl).elim", "annotated_tactic": ["exact fun _ _ hx => (hx <a>rfl</a>).<a>elim</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case H\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nhR : Symmetric R\nhl : Pairwise R l\n\u22a2 \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2260 x \u2192 R x x", "state_after": "no goals"}, {"tactic": "exact hl.imp (@fun a b h _ => by exact h)", "annotated_tactic": ["exact hl.imp (@fun a b h _ => by exact h)", []], "state_before": "case H\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nhR : Symmetric R\nhl : Pairwise R l\n\u22a2 Pairwise (fun x y => x \u2260 y \u2192 R x y) l", "state_after": "no goals"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\nhR : Symmetric R\nhl : Pairwise R l\na b : \u03b1\nh : R a b\nx\u271d : a \u2260 b\n\u22a2 R a b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Holder.lean", "full_name": "HolderOnWith.ediam_image_inter_le_of_le", "start": [174, 1], "end": [177, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/UniformSpace/Completion.lean", "full_name": "CauchyFilter.nonempty_cauchyFilter_iff", "start": [212, 1], "end": [217, 25], "traced_tactics": [{"tactic": "constructor <;> rintro \u27e8c\u27e9", "annotated_tactic": ["constructor <;> rintro \u27e8c\u27e9", []], "state_before": "\u03b1 : Type u\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\n\u22a2 Nonempty (CauchyFilter \u03b1) \u2194 Nonempty \u03b1", "state_after": "case mp.intro\n\u03b1 : Type u\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nc : CauchyFilter \u03b1\n\u22a2 Nonempty \u03b1\n\ncase mpr.intro\n\u03b1 : Type u\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nc : \u03b1\n\u22a2 Nonempty (CauchyFilter \u03b1)"}, {"tactic": "have := eq_univ_iff_forall.1 denseEmbedding_pureCauchy.toDenseInducing.closure_range c", "annotated_tactic": ["have := <a>eq_univ_iff_forall</a>.1 denseEmbedding_pureCauchy.toDenseInducing.closure_range c", [{"full_name": "Set.eq_univ_iff_forall", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [701, 9], "def_end_pos": [701, 27]}]], "state_before": "case mp.intro\n\u03b1 : Type u\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nc : CauchyFilter \u03b1\n\u22a2 Nonempty \u03b1", "state_after": "case mp.intro\n\u03b1 : Type u\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nc : CauchyFilter \u03b1\nthis : c \u2208 closure (range pureCauchy)\n\u22a2 Nonempty \u03b1"}, {"tactic": "obtain \u27e8_, \u27e8_, a, _\u27e9\u27e9 := mem_closure_iff.1 this _ isOpen_univ trivial", "annotated_tactic": ["obtain \u27e8_, \u27e8_, a, _\u27e9\u27e9 := <a>mem_closure_iff</a>.1 this _ <a>isOpen_univ</a> <a>trivial</a>", [{"full_name": "mem_closure_iff", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [576, 9], "def_end_pos": [576, 24]}, {"full_name": "isOpen_univ", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [123, 17], "def_end_pos": [123, 28]}, {"full_name": "trivial", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [514, 31], "def_end_pos": [514, 38]}]], "state_before": "case mp.intro\n\u03b1 : Type u\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nc : CauchyFilter \u03b1\nthis : c \u2208 closure (range pureCauchy)\n\u22a2 Nonempty \u03b1", "state_after": "case mp.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nc : CauchyFilter \u03b1\nthis : c \u2208 closure (range pureCauchy)\nw\u271d : CauchyFilter \u03b1\nleft\u271d : w\u271d \u2208 univ\na : \u03b1\nh\u271d : pureCauchy a = w\u271d\n\u22a2 Nonempty \u03b1"}, {"tactic": "exact \u27e8a\u27e9", "annotated_tactic": ["exact \u27e8a\u27e9", []], "state_before": "case mp.intro.intro.intro.intro\n\u03b1 : Type u\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nc : CauchyFilter \u03b1\nthis : c \u2208 closure (range pureCauchy)\nw\u271d : CauchyFilter \u03b1\nleft\u271d : w\u271d \u2208 univ\na : \u03b1\nh\u271d : pureCauchy a = w\u271d\n\u22a2 Nonempty \u03b1", "state_after": "no goals"}, {"tactic": "exact \u27e8pureCauchy c\u27e9", "annotated_tactic": ["exact \u27e8<a>pureCauchy</a> c\u27e9", [{"full_name": "CauchyFilter.pureCauchy", "def_path": "Mathlib/Topology/UniformSpace/Completion.lean", "def_pos": [152, 5], "def_end_pos": [152, 15]}]], "state_before": "case mpr.intro\n\u03b1 : Type u\ninst\u271d\u00b2 : UniformSpace \u03b1\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nc : \u03b1\n\u22a2 Nonempty (CauchyFilter \u03b1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Basic.lean", "full_name": "edist_dist", "start": [192, 1], "end": [193, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Convex/Function.lean", "full_name": "StrictConcaveOn.inf", "start": [625, 1], "end": [627, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean", "full_name": "CategoryTheory.Limits.BinaryBicone.fstKernelFork_\u03b9", "start": [1779, 1], "end": [1779, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/TensorProduct.lean", "full_name": "TensorProduct.lid_tmul", "start": [667, 1], "end": [668, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Lift.lean", "full_name": "Filter.tendsto_lift", "start": [104, 1], "end": [106, 40], "traced_tactics": [{"tactic": "simp only [Filter.lift, tendsto_iInf]", "annotated_tactic": ["simp only [<a>Filter.lift</a>, <a>tendsto_iInf</a>]", [{"full_name": "Filter.lift", "def_path": "Mathlib/Order/Filter/Lift.lean", "def_pos": [25, 15], "def_end_pos": [25, 19]}, {"full_name": "Filter.tendsto_iInf", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3121, 9], "def_end_pos": [3121, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Set \u03b1 \u2192 Filter \u03b2\nm : \u03b3 \u2192 \u03b2\nl : Filter \u03b3\n\u22a2 Tendsto m l (Filter.lift f g) \u2194 \u2200 (s : Set \u03b1), s \u2208 f \u2192 Tendsto m l (g s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "full_name": "MeasurableSpace.generateFrom_sup_generateFrom", "start": [442, 1], "end": [444, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean", "full_name": "ContinuousAt.rpow_const", "start": [313, 8], "end": [315, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Span.lean", "full_name": "Submodule.comap_map_eq_self", "start": [848, 1], "end": [849, 78], "traced_tactics": [{"tactic": "rw [Submodule.comap_map_eq, sup_of_le_left h]", "annotated_tactic": ["rw [<a>Submodule.comap_map_eq</a>, <a>sup_of_le_left</a> h]", [{"full_name": "Submodule.comap_map_eq", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [842, 9], "def_end_pos": [842, 21]}, {"full_name": "sup_of_le_left", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [197, 11], "def_end_pos": [197, 25]}]], "state_before": "R : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring R\u2082\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nF : Type u_8\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\np : Submodule R M\nh : LinearMap.ker f \u2264 p\n\u22a2 comap f (map f p) = p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/ToIntervalMod.lean", "full_name": "toIocDiv_sub", "start": [346, 1], "end": [347, 59], "traced_tactics": [{"tactic": "simpa only [one_zsmul] using toIocDiv_sub_zsmul hp a b 1", "annotated_tactic": ["simpa only [<a>one_zsmul</a>] using <a>toIocDiv_sub_zsmul</a> hp a b 1", [{"full_name": "one_zsmul", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [284, 30], "def_end_pos": [284, 39]}, {"full_name": "toIocDiv_sub_zsmul", "def_path": "Mathlib/Algebra/Order/ToIntervalMod.lean", "def_pos": [286, 9], "def_end_pos": [286, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nh\u03b1 : Archimedean \u03b1\np : \u03b1\nhp : 0 < p\na\u271d b\u271d c : \u03b1\nn : \u2124\na b : \u03b1\n\u22a2 toIocDiv hp a (b - p) = toIocDiv hp a b - 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/GCD/Basic.lean", "full_name": "Nat.coprime_mul_left_add_left", "start": [219, 1], "end": [220, 47], "traced_tactics": [{"tactic": "rw [Coprime, Coprime, gcd_mul_left_add_left]", "annotated_tactic": ["rw [<a>Coprime</a>, <a>Coprime</a>, <a>gcd_mul_left_add_left</a>]", [{"full_name": "Nat.Coprime", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [33, 18], "def_end_pos": [33, 25]}, {"full_name": "Nat.Coprime", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [33, 18], "def_end_pos": [33, 25]}, {"full_name": "Nat.gcd_mul_left_add_left", "def_path": "Mathlib/Data/Nat/GCD/Basic.lean", "def_pos": [68, 9], "def_end_pos": [68, 30]}]], "state_before": "m n k : \u2115\n\u22a2 Coprime (n * k + m) n \u2194 Coprime m n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/MinMax.lean", "full_name": "List.not_of_mem_foldl_argAux", "start": [69, 1], "end": [86, 13], "traced_tactics": [{"tactic": "induction' l using List.reverseRecOn with tl a ih", "annotated_tactic": ["induction' l using <a>List.reverseRecOn</a> with tl a ih", [{"full_name": "List.reverseRecOn", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [954, 5], "def_end_pos": [954, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no : Option \u03b1\na m : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\n\u22a2 \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 l \u2192 m \u2208 foldl (argAux r) o l \u2192 \u00acr a m", "state_after": "case H0\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no : Option \u03b1\na m : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\n\u22a2 \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 [] \u2192 m \u2208 foldl (argAux r) o [] \u2192 \u00acr a m\n\ncase H1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no : Option \u03b1\na\u271d m : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\n\u22a2 \u2200 {a_1 m : \u03b1} {o : Option \u03b1}, a_1 \u2208 tl ++ [a] \u2192 m \u2208 foldl (argAux r) o (tl ++ [a]) \u2192 \u00acr a_1 m"}, {"tactic": "intro b m o hb ho", "annotated_tactic": ["intro b m o hb ho", []], "state_before": "case H1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no : Option \u03b1\na\u271d m : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\n\u22a2 \u2200 {a_1 m : \u03b1} {o : Option \u03b1}, a_1 \u2208 tl ++ [a] \u2192 m \u2208 foldl (argAux r) o (tl ++ [a]) \u2192 \u00acr a_1 m", "state_after": "case H1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nho : m \u2208 foldl (argAux r) o (tl ++ [a])\n\u22a2 \u00acr b m"}, {"tactic": "rw [foldl_append, foldl_cons, foldl_nil, argAux] at ho", "annotated_tactic": ["rw [<a>foldl_append</a>, <a>foldl_cons</a>, <a>foldl_nil</a>, <a>argAux</a>] at ho", [{"full_name": "List.foldl_append", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [226, 17], "def_end_pos": [226, 29]}, {"full_name": "List.foldl_cons", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [234, 17], "def_end_pos": [234, 27]}, {"full_name": "List.foldl_nil", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [232, 17], "def_end_pos": [232, 26]}, {"full_name": "List.argAux", "def_path": "Mathlib/Data/List/MinMax.lean", "def_pos": [37, 5], "def_end_pos": [37, 11]}]], "state_before": "case H1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nho : m \u2208 foldl (argAux r) o (tl ++ [a])\n\u22a2 \u00acr b m", "state_after": "case H1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nho : m \u2208 Option.casesOn (foldl (argAux r) o tl) (some a) fun c => if r a c then some a else some c\n\u22a2 \u00acr b m"}, {"tactic": "cases' hf : foldl (argAux r) o tl with c", "annotated_tactic": ["cases' hf : <a>foldl</a> (<a>argAux</a> r) o tl with c", [{"full_name": "List.foldl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2212, 5], "def_end_pos": [2212, 15]}, {"full_name": "List.argAux", "def_path": "Mathlib/Data/List/MinMax.lean", "def_pos": [37, 5], "def_end_pos": [37, 11]}]], "state_before": "case H1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nho : m \u2208 Option.casesOn (foldl (argAux r) o tl) (some a) fun c => if r a c then some a else some c\n\u22a2 \u00acr b m", "state_after": "case H1.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nho : m \u2208 Option.casesOn (foldl (argAux r) o tl) (some a) fun c => if r a c then some a else some c\nhf : foldl (argAux r) o tl = none\n\u22a2 \u00acr b m\n\ncase H1.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nho : m \u2208 Option.casesOn (foldl (argAux r) o tl) (some a) fun c => if r a c then some a else some c\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\n\u22a2 \u00acr b m"}, {"tactic": "rw [hf, Option.mem_def] at ho", "annotated_tactic": ["rw [hf, <a>Option.mem_def</a>] at ho", [{"full_name": "Option.mem_def", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [19, 17], "def_end_pos": [19, 24]}]], "state_before": "case H1.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nho : m \u2208 Option.casesOn (foldl (argAux r) o tl) (some a) fun c => if r a c then some a else some c\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\n\u22a2 \u00acr b m", "state_after": "case H1.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nho : (Option.casesOn (some c) (some a) fun c => if r a c then some a else some c) = some m\nhf : foldl (argAux r) o tl = some c\n\u22a2 \u00acr b m"}, {"tactic": "dsimp only at ho", "annotated_tactic": ["dsimp only at ho", []], "state_before": "case H1.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nho : (Option.casesOn (some c) (some a) fun c => if r a c then some a else some c) = some m\nhf : foldl (argAux r) o tl = some c\n\u22a2 \u00acr b m", "state_after": "case H1.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nho : (if r a c then some a else some c) = some m\nhf : foldl (argAux r) o tl = some c\n\u22a2 \u00acr b m"}, {"tactic": "split_ifs at ho with hac <;> cases' mem_append.1 hb with h h <;>\n  injection ho with ho <;> subst ho", "annotated_tactic": ["split_ifs at ho with hac <;> cases' <a>mem_append</a>.1 hb with h h <;>\n    injection ho with ho <;> subst ho", [{"full_name": "List.mem_append", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [137, 17], "def_end_pos": [137, 27]}]], "state_before": "case H1.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nho : (if r a c then some a else some c) = some m\nhf : foldl (argAux r) o tl = some c\n\u22a2 \u00acr b m", "state_after": "case pos.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\nhac : r a c\nh : b \u2208 tl\n\u22a2 \u00acr b a\n\ncase pos.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\nhac : r a c\nh : b \u2208 [a]\n\u22a2 \u00acr b a\n\ncase neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\nhac : \u00acr a c\nh : b \u2208 tl\n\u22a2 \u00acr b c\n\ncase neg.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\nhac : \u00acr a c\nh : b \u2208 [a]\n\u22a2 \u00acr b c"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case H0\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no : Option \u03b1\na m : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\n\u22a2 \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 [] \u2192 m \u2208 foldl (argAux r) o [] \u2192 \u00acr a m", "state_after": "no goals"}, {"tactic": "rw [hf] at ho", "annotated_tactic": ["rw [hf] at ho", []], "state_before": "case H1.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nho : m \u2208 Option.casesOn (foldl (argAux r) o tl) (some a) fun c => if r a c then some a else some c\nhf : foldl (argAux r) o tl = none\n\u22a2 \u00acr b m", "state_after": "case H1.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nho : m \u2208 Option.casesOn none (some a) fun c => if r a c then some a else some c\nhf : foldl (argAux r) o tl = none\n\u22a2 \u00acr b m"}, {"tactic": "rw [foldl_argAux_eq_none] at hf", "annotated_tactic": ["rw [<a>foldl_argAux_eq_none</a>] at hf", [{"full_name": "List.foldl_argAux_eq_none", "def_path": "Mathlib/Data/List/MinMax.lean", "def_pos": [42, 9], "def_end_pos": [42, 29]}]], "state_before": "case H1.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nho : m \u2208 Option.casesOn none (some a) fun c => if r a c then some a else some c\nhf : foldl (argAux r) o tl = none\n\u22a2 \u00acr b m", "state_after": "case H1.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nho : m \u2208 Option.casesOn none (some a) fun c => if r a c then some a else some c\nhf : tl = [] \u2227 o = none\n\u22a2 \u00acr b m"}, {"tactic": "simp_all [hf.1, hf.2, hr\u2080 _]", "annotated_tactic": ["simp_all [hf.1, hf.2, hr\u2080 _]", []], "state_before": "case H1.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m\u271d : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb m : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nho : m \u2208 Option.casesOn none (some a) fun c => if r a c then some a else some c\nhf : tl = [] \u2227 o = none\n\u22a2 \u00acr b m", "state_after": "no goals"}, {"tactic": "exact fun hba => ih h hf (hr\u2081 hba hac)", "annotated_tactic": ["exact fun hba => ih h hf (hr\u2081 hba hac)", []], "state_before": "case pos.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\nhac : r a c\nh : b \u2208 tl\n\u22a2 \u00acr b a", "state_after": "no goals"}, {"tactic": "simp_all [hr\u2080 _]", "annotated_tactic": ["simp_all [hr\u2080 _]", []], "state_before": "case pos.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\nhac : r a c\nh : b \u2208 [a]\n\u22a2 \u00acr b a", "state_after": "no goals"}, {"tactic": "exact ih h hf", "annotated_tactic": ["exact ih h hf", []], "state_before": "case neg.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\nhac : \u00acr a c\nh : b \u2208 tl\n\u22a2 \u00acr b c", "state_after": "no goals"}, {"tactic": "simp_all", "annotated_tactic": ["simp_all", []], "state_before": "case neg.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel r\nl : List \u03b1\no\u271d : Option \u03b1\na\u271d m : \u03b1\nhr\u2080 : Irreflexive r\nhr\u2081 : Transitive r\ntl : List \u03b1\na : \u03b1\nih : \u2200 {a m : \u03b1} {o : Option \u03b1}, a \u2208 tl \u2192 m \u2208 foldl (argAux r) o tl \u2192 \u00acr a m\nb : \u03b1\no : Option \u03b1\nhb : b \u2208 tl ++ [a]\nc : \u03b1\nhf : foldl (argAux r) o tl = some c\nhac : \u00acr a c\nh : b \u2208 [a]\n\u22a2 \u00acr b c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "full_name": "String.find_of_valid", "start": [349, 1], "end": [350, 43], "traced_tactics": [{"tactic": "simpa using findAux_of_valid p [] s.1 []", "annotated_tactic": ["simpa using <a>findAux_of_valid</a> p [] s.1 []", [{"full_name": "String.findAux_of_valid", "def_path": "lake-packages/std/Std/Data/String/Lemmas.lean", "def_pos": [335, 9], "def_end_pos": [335, 25]}]], "state_before": "p : Char \u2192 Bool\ns : String\n\u22a2 find s p = { byteIdx := utf8Len (List.takeWhile (fun x => !p x) s.data) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/EventuallyConst.lean", "full_name": "Filter.HasBasis.eventuallyConst_iff", "start": [33, 1], "end": [35, 86], "traced_tactics": [{"tactic": "simp only [Set.Subsingleton, ball_image_iff]", "annotated_tactic": ["simp only [<a>Set.Subsingleton</a>, <a>ball_image_iff</a>]", [{"full_name": "Set.Subsingleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2356, 15], "def_end_pos": [2356, 27]}, {"full_name": "Set.ball_image_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [249, 9], "def_end_pos": [249, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\n\u03b9 : Sort u_5\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nh : HasBasis l p s\n\u22a2 (\u2203 i, p i \u2227 Set.Subsingleton (f '' s i)) \u2194 \u2203 i, p i \u2227 \u2200 (x : \u03b1), x \u2208 s i \u2192 \u2200 (y : \u03b1), y \u2208 s i \u2192 f x = f y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Tower.lean", "full_name": "Subalgebra.coe_restrictScalars", "start": [97, 1], "end": [98, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/Derivative.lean", "full_name": "Polynomial.iterate_derivative_comp_one_sub_X", "start": [631, 1], "end": [635, 80], "traced_tactics": [{"tactic": "induction' k with k ih generalizing p", "annotated_tactic": ["induction' k with k ih generalizing p", []], "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommRing R\np : R[X]\nk : \u2115\n\u22a2 (\u2191derivative)^[k] (comp p (1 - X)) = (-1) ^ k * comp ((\u2191derivative)^[k] p) (1 - X)", "state_after": "case zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommRing R\np\u271d p : R[X]\n\u22a2 (\u2191derivative)^[Nat.zero] (comp p (1 - X)) = (-1) ^ Nat.zero * comp ((\u2191derivative)^[Nat.zero] p) (1 - X)\n\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommRing R\np\u271d : R[X]\nk : \u2115\nih : \u2200 (p : R[X]), (\u2191derivative)^[k] (comp p (1 - X)) = (-1) ^ k * comp ((\u2191derivative)^[k] p) (1 - X)\np : R[X]\n\u22a2 (\u2191derivative)^[Nat.succ k] (comp p (1 - X)) = (-1) ^ Nat.succ k * comp ((\u2191derivative)^[Nat.succ k] p) (1 - X)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommRing R\np\u271d p : R[X]\n\u22a2 (\u2191derivative)^[Nat.zero] (comp p (1 - X)) = (-1) ^ Nat.zero * comp ((\u2191derivative)^[Nat.zero] p) (1 - X)", "state_after": "no goals"}, {"tactic": "simp [ih (derivative p), iterate_derivative_neg, derivative_comp, pow_succ]", "annotated_tactic": ["simp [ih (<a>derivative</a> p), <a>iterate_derivative_neg</a>, <a>derivative_comp</a>, <a>pow_succ</a>]", [{"full_name": "Polynomial.derivative", "def_path": "Mathlib/Data/Polynomial/Derivative.lean", "def_pos": [39, 5], "def_end_pos": [39, 15]}, {"full_name": "Polynomial.iterate_derivative_neg", "def_path": "Mathlib/Data/Polynomial/Derivative.lean", "def_pos": [581, 9], "def_end_pos": [581, 31]}, {"full_name": "Polynomial.derivative_comp", "def_path": "Mathlib/Data/Polynomial/Derivative.lean", "def_pos": [531, 9], "def_end_pos": [531, 24]}, {"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [645, 9], "def_end_pos": [645, 17]}]], "state_before": "case succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : CommRing R\np\u271d : R[X]\nk : \u2115\nih : \u2200 (p : R[X]), (\u2191derivative)^[k] (comp p (1 - X)) = (-1) ^ k * comp ((\u2191derivative)^[k] p) (1 - X)\np : R[X]\n\u22a2 (\u2191derivative)^[Nat.succ k] (comp p (1 - X)) = (-1) ^ Nat.succ k * comp ((\u2191derivative)^[Nat.succ k] p) (1 - X)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/TrivSqZeroExt.lean", "full_name": "TrivSqZeroExt.inl_add", "start": [314, 1], "end": [316, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Image.lean", "full_name": "Finset.map_inter", "start": [220, 1], "end": [222, 82], "traced_tactics": [{"tactic": "simp only [coe_map, coe_inter, Set.image_inter f.injective]", "annotated_tactic": ["simp only [<a>coe_map</a>, <a>coe_inter</a>, <a>Set.image_inter</a> f.injective]", [{"full_name": "Finset.coe_map", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [107, 9], "def_end_pos": [107, 16]}, {"full_name": "Finset.coe_inter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1628, 9], "def_end_pos": [1628, 18]}, {"full_name": "Set.image_inter", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [354, 9], "def_end_pos": [354, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u21aa \u03b2\ns : Finset \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u21aa \u03b2\ns\u2081 s\u2082 : Finset \u03b1\n\u22a2 \u2191(map f (s\u2081 \u2229 s\u2082)) = \u2191(map f s\u2081 \u2229 map f s\u2082)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Type.lean", "full_name": "Equiv.Perm.VectorsProdEqOne.rotate_zero", "start": [457, 1], "end": [458, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.le_sub_of_le", "start": [572, 1], "end": [573, 63], "traced_tactics": [{"tactic": "rw [\u2190 add_le_add_iff_left b, Ordinal.add_sub_cancel_of_le h]", "annotated_tactic": ["rw [\u2190 <a>add_le_add_iff_left</a> b, <a>Ordinal.add_sub_cancel_of_le</a> h]", [{"full_name": "add_le_add_iff_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "Ordinal.add_sub_cancel_of_le", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [562, 19], "def_end_pos": [562, 39]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\na b c : Ordinal.{u_4}\nh : b \u2264 a\n\u22a2 c \u2264 a - b \u2194 b + c \u2264 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "full_name": "norm_inner_div_norm_mul_norm_eq_one_iff", "start": [1627, 1], "end": [1637, 78], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1 \u2194 x \u2260 0 \u2227 \u2203 r, r \u2260 0 \u2227 y = r \u2022 x", "state_after": "case mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1 \u2192 x \u2260 0 \u2227 \u2203 r, r \u2260 0 \u2227 y = r \u2022 x\n\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 (x \u2260 0 \u2227 \u2203 r, r \u2260 0 \u2227 y = r \u2022 x) \u2192 \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1 \u2192 x \u2260 0 \u2227 \u2203 r, r \u2260 0 \u2227 y = r \u2022 x", "state_after": "case mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1\n\u22a2 x \u2260 0 \u2227 \u2203 r, r \u2260 0 \u2227 y = r \u2022 x"}, {"tactic": "have hx\u2080 : x \u2260 0 := fun h\u2080 => by simp [h\u2080] at h", "annotated_tactic": ["have hx\u2080 : x \u2260 0 := fun h\u2080 => by simp [h\u2080] at h", []], "state_before": "case mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1\n\u22a2 x \u2260 0 \u2227 \u2203 r, r \u2260 0 \u2227 y = r \u2022 x", "state_after": "case mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1\nhx\u2080 : x \u2260 0\n\u22a2 x \u2260 0 \u2227 \u2203 r, r \u2260 0 \u2227 y = r \u2022 x"}, {"tactic": "have hy\u2080 : y \u2260 0 := fun h\u2080 => by simp [h\u2080] at h", "annotated_tactic": ["have hy\u2080 : y \u2260 0 := fun h\u2080 => by simp [h\u2080] at h", []], "state_before": "case mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1\nhx\u2080 : x \u2260 0\n\u22a2 x \u2260 0 \u2227 \u2203 r, r \u2260 0 \u2227 y = r \u2022 x", "state_after": "case mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1\nhx\u2080 : x \u2260 0\nhy\u2080 : y \u2260 0\n\u22a2 x \u2260 0 \u2227 \u2203 r, r \u2260 0 \u2227 y = r \u2022 x"}, {"tactic": "refine' \u27e8hx\u2080, (norm_inner_eq_norm_iff hx\u2080 hy\u2080).1 <| eq_of_div_eq_one _\u27e9", "annotated_tactic": ["refine' \u27e8hx\u2080, (<a>norm_inner_eq_norm_iff</a> hx\u2080 hy\u2080).1 <| <a>eq_of_div_eq_one</a> _\u27e9", [{"full_name": "norm_inner_eq_norm_iff", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [1612, 9], "def_end_pos": [1612, 31]}, {"full_name": "eq_of_div_eq_one", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [377, 9], "def_end_pos": [377, 25]}]], "state_before": "case mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1\nhx\u2080 : x \u2260 0\nhy\u2080 : y \u2260 0\n\u22a2 x \u2260 0 \u2227 \u2203 r, r \u2260 0 \u2227 y = r \u2022 x", "state_after": "case mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1\nhx\u2080 : x \u2260 0\nhy\u2080 : y \u2260 0\n\u22a2 \u2016inner x y\u2016 / (\u2016x\u2016 * \u2016y\u2016) = 1"}, {"tactic": "simpa using h", "annotated_tactic": ["simpa using h", []], "state_before": "case mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1\nhx\u2080 : x \u2260 0\nhy\u2080 : y \u2260 0\n\u22a2 \u2016inner x y\u2016 / (\u2016x\u2016 * \u2016y\u2016) = 1", "state_after": "no goals"}, {"tactic": "simp [h\u2080] at h", "annotated_tactic": ["simp [h\u2080] at h", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1\nh\u2080 : x = 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simp [h\u2080] at h", "annotated_tactic": ["simp [h\u2080] at h", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1\nhx\u2080 : x \u2260 0\nh\u2080 : y = 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rintro \u27e8hx, \u27e8r, \u27e8hr, rfl\u27e9\u27e9\u27e9", "annotated_tactic": ["rintro \u27e8hx, \u27e8r, \u27e8hr, rfl\u27e9\u27e9\u27e9", []], "state_before": "case mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 (x \u2260 0 \u2227 \u2203 r, r \u2260 0 \u2227 y = r \u2022 x) \u2192 \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1", "state_after": "case mpr.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\nhx : x \u2260 0\nr : \ud835\udd5c\nhr : r \u2260 0\n\u22a2 \u2016inner x (r \u2022 x) / (\u2191\u2016x\u2016 * \u2191\u2016r \u2022 x\u2016)\u2016 = 1"}, {"tactic": "simp only [norm_div, norm_mul, norm_ofReal, abs_norm]", "annotated_tactic": ["simp only [<a>norm_div</a>, <a>norm_mul</a>, <a>norm_ofReal</a>, <a>abs_norm</a>]", [{"full_name": "norm_div", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [562, 9], "def_end_pos": [562, 17]}, {"full_name": "norm_mul", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}, {"full_name": "IsROrC.norm_ofReal", "def_path": "Mathlib/Data/IsROrC/Basic.lean", "def_pos": [300, 9], "def_end_pos": [300, 20]}, {"full_name": "abs_norm", "def_path": "Mathlib/Analysis/NormedSpace/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "case mpr.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\nhx : x \u2260 0\nr : \ud835\udd5c\nhr : r \u2260 0\n\u22a2 \u2016inner x (r \u2022 x) / (\u2191\u2016x\u2016 * \u2191\u2016r \u2022 x\u2016)\u2016 = 1", "state_after": "case mpr.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\nhx : x \u2260 0\nr : \ud835\udd5c\nhr : r \u2260 0\n\u22a2 \u2016inner x (r \u2022 x)\u2016 / (\u2016x\u2016 * \u2016r \u2022 x\u2016) = 1"}, {"tactic": "exact norm_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul hx hr", "annotated_tactic": ["exact <a>norm_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul</a> hx hr", [{"full_name": "norm_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [1549, 9], "def_end_pos": [1549, 70]}]], "state_before": "case mpr.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\nhx : x \u2260 0\nr : \ud835\udd5c\nhr : r \u2260 0\n\u22a2 \u2016inner x (r \u2022 x)\u2016 / (\u2016x\u2016 * \u2016r \u2022 x\u2016) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Instances/ENNReal.lean", "full_name": "ENNReal.continuous_nnreal_sub", "start": [451, 1], "end": [452, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Ideal/Norm.lean", "full_name": "Ideal.spanNorm_mul_of_bot_or_top", "start": [590, 1], "end": [601, 15], "traced_tactics": [{"tactic": "refine le_antisymm ?_ (spanNorm_mul_spanNorm_le R _ _)", "annotated_tactic": ["refine <a>le_antisymm</a> ?_ (<a>spanNorm_mul_spanNorm_le</a> R _ _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "Ideal.spanNorm_mul_spanNorm_le", "def_path": "Mathlib/RingTheory/Ideal/Norm.lean", "def_pos": [579, 9], "def_end_pos": [579, 33]}]], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\neq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4\nI J : Ideal S\n\u22a2 spanNorm R (I * J) = spanNorm R I * spanNorm R J", "state_after": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\neq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4\nI J : Ideal S\n\u22a2 spanNorm R (I * J) \u2264 spanNorm R I * spanNorm R J"}, {"tactic": "cases' eq_bot_or_top (spanNorm R I) with hI hI", "annotated_tactic": ["cases' eq_bot_or_top (<a>spanNorm</a> R I) with hI hI", [{"full_name": "Ideal.spanNorm", "def_path": "Mathlib/RingTheory/Ideal/Norm.lean", "def_pos": [493, 5], "def_end_pos": [493, 13]}]], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\neq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4\nI J : Ideal S\n\u22a2 spanNorm R (I * J) \u2264 spanNorm R I * spanNorm R J", "state_after": "case inl\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\neq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4\nI J : Ideal S\nhI : spanNorm R I = \u22a5\n\u22a2 spanNorm R (I * J) \u2264 spanNorm R I * spanNorm R J\n\ncase inr\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\neq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4\nI J : Ideal S\nhI : spanNorm R I = \u22a4\n\u22a2 spanNorm R (I * J) \u2264 spanNorm R I * spanNorm R J"}, {"tactic": "rw [hI, Ideal.top_mul]", "annotated_tactic": ["rw [hI, <a>Ideal.top_mul</a>]", [{"full_name": "Ideal.top_mul", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [763, 9], "def_end_pos": [763, 16]}]], "state_before": "case inr\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\neq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4\nI J : Ideal S\nhI : spanNorm R I = \u22a4\n\u22a2 spanNorm R (I * J) \u2264 spanNorm R I * spanNorm R J", "state_after": "case inr\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\neq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4\nI J : Ideal S\nhI : spanNorm R I = \u22a4\n\u22a2 spanNorm R (I * J) \u2264 spanNorm R J"}, {"tactic": "cases' eq_bot_or_top (spanNorm R J) with hJ hJ", "annotated_tactic": ["cases' eq_bot_or_top (<a>spanNorm</a> R J) with hJ hJ", [{"full_name": "Ideal.spanNorm", "def_path": "Mathlib/RingTheory/Ideal/Norm.lean", "def_pos": [493, 5], "def_end_pos": [493, 13]}]], "state_before": "case inr\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\neq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4\nI J : Ideal S\nhI : spanNorm R I = \u22a4\n\u22a2 spanNorm R (I * J) \u2264 spanNorm R J", "state_after": "case inr.inl\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\neq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4\nI J : Ideal S\nhI : spanNorm R I = \u22a4\nhJ : spanNorm R J = \u22a5\n\u22a2 spanNorm R (I * J) \u2264 spanNorm R J\n\ncase inr.inr\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\neq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4\nI J : Ideal S\nhI : spanNorm R I = \u22a4\nhJ : spanNorm R J = \u22a4\n\u22a2 spanNorm R (I * J) \u2264 spanNorm R J"}, {"tactic": "rw [hJ]", "annotated_tactic": ["rw [hJ]", []], "state_before": "case inr.inr\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\neq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4\nI J : Ideal S\nhI : spanNorm R I = \u22a4\nhJ : spanNorm R J = \u22a4\n\u22a2 spanNorm R (I * J) \u2264 spanNorm R J", "state_after": "case inr.inr\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\neq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4\nI J : Ideal S\nhI : spanNorm R I = \u22a4\nhJ : spanNorm R J = \u22a4\n\u22a2 spanNorm R (I * J) \u2264 \u22a4"}, {"tactic": "exact le_top", "annotated_tactic": ["exact <a>le_top</a>", [{"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [98, 9], "def_end_pos": [98, 15]}]], "state_before": "case inr.inr\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\neq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4\nI J : Ideal S\nhI : spanNorm R I = \u22a4\nhJ : spanNorm R J = \u22a4\n\u22a2 spanNorm R (I * J) \u2264 \u22a4", "state_after": "no goals"}, {"tactic": "rw [hI, spanNorm_eq_bot_iff.mp hI, bot_mul, spanNorm_bot]", "annotated_tactic": ["rw [hI, spanNorm_eq_bot_iff.mp hI, <a>bot_mul</a>, <a>spanNorm_bot</a>]", [{"full_name": "Ideal.bot_mul", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [754, 9], "def_end_pos": [754, 16]}, {"full_name": "Ideal.spanNorm_bot", "def_path": "Mathlib/RingTheory/Ideal/Norm.lean", "def_pos": [498, 9], "def_end_pos": [498, 21]}]], "state_before": "case inl\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\neq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4\nI J : Ideal S\nhI : spanNorm R I = \u22a5\n\u22a2 spanNorm R (I * J) \u2264 spanNorm R I * spanNorm R J", "state_after": "case inl\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\neq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4\nI J : Ideal S\nhI : spanNorm R I = \u22a5\n\u22a2 \u22a5 \u2264 \u22a5 * spanNorm R J"}, {"tactic": "exact bot_le", "annotated_tactic": ["exact <a>bot_le</a>", [{"full_name": "bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [256, 9], "def_end_pos": [256, 15]}]], "state_before": "case inl\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\neq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4\nI J : Ideal S\nhI : spanNorm R I = \u22a5\n\u22a2 \u22a5 \u2264 \u22a5 * spanNorm R J", "state_after": "no goals"}, {"tactic": "rw [hJ, spanNorm_eq_bot_iff.mp hJ, mul_bot, spanNorm_bot]", "annotated_tactic": ["rw [hJ, spanNorm_eq_bot_iff.mp hJ, <a>mul_bot</a>, <a>spanNorm_bot</a>]", [{"full_name": "Ideal.mul_bot", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [750, 9], "def_end_pos": [750, 16]}, {"full_name": "Ideal.spanNorm_bot", "def_path": "Mathlib/RingTheory/Ideal/Norm.lean", "def_pos": [498, 9], "def_end_pos": [498, 21]}]], "state_before": "case inr.inl\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nS : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\neq_bot_or_top : \u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4\nI J : Ideal S\nhI : spanNorm R I = \u22a4\nhJ : spanNorm R J = \u22a5\n\u22a2 spanNorm R (I * J) \u2264 spanNorm R J", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Instances/ENNReal.lean", "full_name": "NNReal.not_summable_iff_tendsto_nat_atTop", "start": [1135, 1], "end": [1142, 81], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u2115 \u2192 \u211d\u22650\n\u22a2 \u00acSummable f \u2194 Tendsto (fun n => \u2211 i in Finset.range n, f i) atTop atTop", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u2115 \u2192 \u211d\u22650\n\u22a2 \u00acSummable f \u2192 Tendsto (fun n => \u2211 i in Finset.range n, f i) atTop atTop\n\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u2115 \u2192 \u211d\u22650\n\u22a2 Tendsto (fun n => \u2211 i in Finset.range n, f i) atTop atTop \u2192 \u00acSummable f"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u2115 \u2192 \u211d\u22650\n\u22a2 \u00acSummable f \u2192 Tendsto (fun n => \u2211 i in Finset.range n, f i) atTop atTop", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u2115 \u2192 \u211d\u22650\nh : \u00acSummable f\n\u22a2 Tendsto (fun n => \u2211 i in Finset.range n, f i) atTop atTop"}, {"tactic": "refine' ((tendsto_of_monotone _).resolve_right h).comp _", "annotated_tactic": ["refine' ((<a>tendsto_of_monotone</a> _).<a>resolve_right</a> h).<a>comp</a> _", [{"full_name": "tendsto_of_monotone", "def_path": "Mathlib/Topology/Algebra/Order/MonotoneConvergence.lean", "def_pos": [223, 9], "def_end_pos": [223, 28]}, {"full_name": "Or.resolve_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [272, 9], "def_end_pos": [272, 25]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3032, 9], "def_end_pos": [3032, 21]}]], "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u2115 \u2192 \u211d\u22650\nh : \u00acSummable f\n\u22a2 Tendsto (fun n => \u2211 i in Finset.range n, f i) atTop atTop", "state_after": "case mp.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u2115 \u2192 \u211d\u22650\nh : \u00acSummable f\n\u22a2 Monotone fun s => \u2211 b in s, f b\n\ncase mp.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u2115 \u2192 \u211d\u22650\nh : \u00acSummable f\n\u22a2 Tendsto (fun n => Finset.range n) atTop atTop"}, {"tactic": "exacts [Finset.sum_mono_set _, tendsto_finset_range]", "annotated_tactic": ["exacts [<a>Finset.sum_mono_set</a> _, <a>tendsto_finset_range</a>]", [{"full_name": "Finset.sum_mono_set", "def_path": "Mathlib/Algebra/BigOperators/Order.lean", "def_pos": [409, 15], "def_end_pos": [409, 27]}, {"full_name": "Filter.tendsto_finset_range", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1416, 9], "def_end_pos": [1416, 29]}]], "state_before": "case mp.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u2115 \u2192 \u211d\u22650\nh : \u00acSummable f\n\u22a2 Monotone fun s => \u2211 b in s, f b\n\ncase mp.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u2115 \u2192 \u211d\u22650\nh : \u00acSummable f\n\u22a2 Tendsto (fun n => Finset.range n) atTop atTop", "state_after": "no goals"}, {"tactic": "rintro hnat \u27e8r, hr\u27e9", "annotated_tactic": ["rintro hnat \u27e8r, hr\u27e9", []], "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u2115 \u2192 \u211d\u22650\n\u22a2 Tendsto (fun n => \u2211 i in Finset.range n, f i) atTop atTop \u2192 \u00acSummable f", "state_after": "case mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u2115 \u2192 \u211d\u22650\nhnat : Tendsto (fun n => \u2211 i in Finset.range n, f i) atTop atTop\nr : \u211d\u22650\nhr : HasSum f r\n\u22a2 False"}, {"tactic": "exact not_tendsto_nhds_of_tendsto_atTop hnat _ (hasSum_iff_tendsto_nat.1 hr)", "annotated_tactic": ["exact <a>not_tendsto_nhds_of_tendsto_atTop</a> hnat _ (<a>hasSum_iff_tendsto_nat</a>.1 hr)", [{"full_name": "not_tendsto_nhds_of_tendsto_atTop", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [1626, 9], "def_end_pos": [1626, 42]}, {"full_name": "NNReal.hasSum_iff_tendsto_nat", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1128, 9], "def_end_pos": [1128, 31]}]], "state_before": "case mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u2115 \u2192 \u211d\u22650\nhnat : Tendsto (fun n => \u2211 i in Finset.range n, f i) atTop atTop\nr : \u211d\u22650\nhr : HasSum f r\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean", "full_name": "integral_comp_neg_Iic", "start": [91, 1], "end": [97, 90], "traced_tactics": [{"tactic": "have A : MeasurableEmbedding fun x : \u211d => -x :=\n  (Homeomorph.neg \u211d).closedEmbedding.measurableEmbedding", "annotated_tactic": ["have A : <a>MeasurableEmbedding</a> fun x : \u211d => -x :=\n    (<a>Homeomorph.neg</a> \u211d).closedEmbedding.measurableEmbedding", [{"full_name": "MeasurableEmbedding", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [1178, 11], "def_end_pos": [1178, 30]}, {"full_name": "Homeomorph.neg", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [330, 3], "def_end_pos": [330, 14]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\nA : MeasurableEmbedding fun x => -x\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x"}, {"tactic": "have := MeasurableEmbedding.set_integral_map (\u03bc := volume) A f (Ici (-c))", "annotated_tactic": ["have := <a>MeasurableEmbedding.set_integral_map</a> (\u03bc := <a>volume</a>) A f (<a>Ici</a> (-c))", [{"full_name": "MeasurableEmbedding.set_integral_map", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [519, 9], "def_end_pos": [519, 52]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "Set.Ici", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [74, 5], "def_end_pos": [74, 8]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\nA : MeasurableEmbedding fun x => -x\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\nA : MeasurableEmbedding fun x => -x\nthis : \u222b (y : \u211d) in Ici (-c), f y \u2202Measure.map (fun x => -x) volume = \u222b (x : \u211d) in (fun x => -x) \u207b\u00b9' Ici (-c), f (-x)\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x"}, {"tactic": "rw [Measure.map_neg_eq_self (volume : Measure \u211d)] at this", "annotated_tactic": ["rw [<a>Measure.map_neg_eq_self</a> (<a>volume</a> : <a>Measure</a> \u211d)] at this", [{"full_name": "MeasureTheory.Measure.map_neg_eq_self", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [421, 3], "def_end_pos": [421, 14]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "MeasureTheory.Measure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [74, 11], "def_end_pos": [74, 18]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\nA : MeasurableEmbedding fun x => -x\nthis : \u222b (y : \u211d) in Ici (-c), f y \u2202Measure.map (fun x => -x) volume = \u222b (x : \u211d) in (fun x => -x) \u207b\u00b9' Ici (-c), f (-x)\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x", "state_after": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\nA : MeasurableEmbedding fun x => -x\nthis : \u222b (y : \u211d) in Ici (-c), f y = \u222b (x : \u211d) in (fun x => -x) \u207b\u00b9' Ici (-c), f (-x)\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x"}, {"tactic": "simp_rw [\u2190 integral_Ici_eq_integral_Ioi, this, neg_preimage, preimage_neg_Ici, neg_neg]", "annotated_tactic": ["simp_rw [\u2190 <a>integral_Ici_eq_integral_Ioi</a>, this, <a>neg_preimage</a>, <a>preimage_neg_Ici</a>, <a>neg_neg</a>]", [{"full_name": "MeasureTheory.integral_Ici_eq_integral_Ioi", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [699, 9], "def_end_pos": [699, 37]}, {"full_name": "Set.neg_preimage", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [185, 3], "def_end_pos": [185, 14]}, {"full_name": "Set.preimage_neg_Ici", "def_path": "Mathlib/Data/Set/Pointwise/Interval.lean", "def_pos": [130, 9], "def_end_pos": [130, 25]}, {"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [799, 3], "def_end_pos": [799, 14]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d E\nc : \u211d\nf : \u211d \u2192 E\nA : MeasurableEmbedding fun x => -x\nthis : \u222b (y : \u211d) in Ici (-c), f y = \u222b (x : \u211d) in (fun x => -x) \u207b\u00b9' Ici (-c), f (-x)\n\u22a2 \u222b (x : \u211d) in Iic c, f (-x) = \u222b (x : \u211d) in Ioi (-c), f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.mul_univ_of_one_mem", "start": [932, 1], "end": [933, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/LaurentSeries.lean", "full_name": "PowerSeries.coe_X", "start": [235, 1], "end": [236, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Image.lean", "full_name": "Set.image_subtype_val_Ioo_subset", "start": [261, 1], "end": [263, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Gluing.lean", "full_name": "Metric.isometry_inl", "start": [297, 1], "end": [298, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Ideal/Operations.lean", "full_name": "Ideal.le_map_of_comap_le_of_surjective", "start": [1677, 1], "end": [1678, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Maps.lean", "full_name": "embedding_of_embedding_compose", "start": [219, 1], "end": [222, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Lie/Subalgebra.lean", "full_name": "LieSubalgebra.span_empty", "start": [731, 1], "end": [732, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/UnitaryGroup.lean", "full_name": "Matrix.mem_orthogonalGroup_iff", "start": [215, 1], "end": [218, 40], "traced_tactics": [{"tactic": "refine' \u27e8And.right, fun hA => \u27e8_, hA\u27e9\u27e9", "annotated_tactic": ["refine' \u27e8<a>And.right</a>, fun hA => \u27e8_, hA\u27e9\u27e9", [{"full_name": "And.right", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [507, 3], "def_end_pos": [507, 8]}]], "state_before": "n : Type u\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\n\u03b1 : Type v\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : StarRing \u03b1\nA\u271d : Matrix n n \u03b1\n\u03b2 : Type v\ninst\u271d : CommRing \u03b2\nA : Matrix n n \u03b2\n\u22a2 A \u2208 orthogonalGroup n \u03b2 \u2194 A * star A = 1", "state_after": "n : Type u\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\n\u03b1 : Type v\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : StarRing \u03b1\nA\u271d : Matrix n n \u03b1\n\u03b2 : Type v\ninst\u271d : CommRing \u03b2\nA : Matrix n n \u03b2\nhA : A * star A = 1\n\u22a2 star A * A = 1"}, {"tactic": "simpa only [mul_eq_one_comm] using hA", "annotated_tactic": ["simpa only [<a>mul_eq_one_comm</a>] using hA", [{"full_name": "Matrix.mul_eq_one_comm", "def_path": "Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean", "def_pos": [120, 9], "def_end_pos": [120, 24]}]], "state_before": "n : Type u\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\n\u03b1 : Type v\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : StarRing \u03b1\nA\u271d : Matrix n n \u03b1\n\u03b2 : Type v\ninst\u271d : CommRing \u03b2\nA : Matrix n n \u03b2\nhA : A * star A = 1\n\u22a2 star A * A = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/InformationTheory/Hamming.lean", "full_name": "hammingNorm_lt_one", "start": [204, 1], "end": [205, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/GroupAction/SubMulAction.lean", "full_name": "SubMulAction.mem_carrier", "start": [155, 1], "end": [156, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.mk_eq_two_iff'", "start": [2381, 1], "end": [2388, 84], "traced_tactics": [{"tactic": "rw [mk_eq_two_iff]", "annotated_tactic": ["rw [<a>mk_eq_two_iff</a>]", [{"full_name": "Cardinal.mk_eq_two_iff", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [2372, 9], "def_end_pos": [2372, 22]}]], "state_before": "\u03b1 \u03b2 : Type u\nx : \u03b1\n\u22a2 #\u03b1 = 2 \u2194 \u2203! y, y \u2260 x", "state_after": "\u03b1 \u03b2 : Type u\nx : \u03b1\n\u22a2 (\u2203 x y, x \u2260 y \u2227 {x, y} = univ) \u2194 \u2203! y, y \u2260 x"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1 \u03b2 : Type u\nx : \u03b1\n\u22a2 (\u2203 x y, x \u2260 y \u2227 {x, y} = univ) \u2194 \u2203! y, y \u2260 x", "state_after": "case mp\n\u03b1 \u03b2 : Type u\nx : \u03b1\n\u22a2 (\u2203 x y, x \u2260 y \u2227 {x, y} = univ) \u2192 \u2203! y, y \u2260 x\n\ncase mpr\n\u03b1 \u03b2 : Type u\nx : \u03b1\n\u22a2 (\u2203! y, y \u2260 x) \u2192 \u2203 x y, x \u2260 y \u2227 {x, y} = univ"}, {"tactic": "rintro \u27e8a, b, hne, h\u27e9", "annotated_tactic": ["rintro \u27e8a, b, hne, h\u27e9", []], "state_before": "case mp\n\u03b1 \u03b2 : Type u\nx : \u03b1\n\u22a2 (\u2203 x y, x \u2260 y \u2227 {x, y} = univ) \u2192 \u2203! y, y \u2260 x", "state_after": "case mp.intro.intro.intro\n\u03b1 \u03b2 : Type u\nx a b : \u03b1\nhne : a \u2260 b\nh : {a, b} = univ\n\u22a2 \u2203! y, y \u2260 x"}, {"tactic": "simp only [eq_univ_iff_forall, mem_insert_iff, mem_singleton_iff] at h", "annotated_tactic": ["simp only [<a>eq_univ_iff_forall</a>, <a>mem_insert_iff</a>, <a>mem_singleton_iff</a>] at h", [{"full_name": "Set.eq_univ_iff_forall", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [701, 9], "def_end_pos": [701, 27]}, {"full_name": "Set.mem_insert_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1147, 9], "def_end_pos": [1147, 23]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1273, 9], "def_end_pos": [1273, 26]}]], "state_before": "case mp.intro.intro.intro\n\u03b1 \u03b2 : Type u\nx a b : \u03b1\nhne : a \u2260 b\nh : {a, b} = univ\n\u22a2 \u2203! y, y \u2260 x", "state_after": "case mp.intro.intro.intro\n\u03b1 \u03b2 : Type u\nx a b : \u03b1\nhne : a \u2260 b\nh : \u2200 (x : \u03b1), x = a \u2228 x = b\n\u22a2 \u2203! y, y \u2260 x"}, {"tactic": "rcases h x with (rfl | rfl)", "annotated_tactic": ["rcases h x with (rfl | rfl)", []], "state_before": "case mp.intro.intro.intro\n\u03b1 \u03b2 : Type u\nx a b : \u03b1\nhne : a \u2260 b\nh : \u2200 (x : \u03b1), x = a \u2228 x = b\n\u22a2 \u2203! y, y \u2260 x", "state_after": "case mp.intro.intro.intro.inl\n\u03b1 \u03b2 : Type u\nx b : \u03b1\nhne : x \u2260 b\nh : \u2200 (x_1 : \u03b1), x_1 = x \u2228 x_1 = b\n\u22a2 \u2203! y, y \u2260 x\n\ncase mp.intro.intro.intro.inr\n\u03b1 \u03b2 : Type u\nx a : \u03b1\nhne : a \u2260 x\nh : \u2200 (x_1 : \u03b1), x_1 = a \u2228 x_1 = x\n\u22a2 \u2203! y, y \u2260 x"}, {"tactic": "exacts [\u27e8b, hne.symm, fun z => (h z).resolve_left\u27e9, \u27e8a, hne, fun z => (h z).resolve_right\u27e9]", "annotated_tactic": ["exacts [\u27e8b, hne.symm, fun z => (h z).<a>resolve_left</a>\u27e9, \u27e8a, hne, fun z => (h z).<a>resolve_right</a>\u27e9]", [{"full_name": "Or.resolve_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [268, 9], "def_end_pos": [268, 24]}, {"full_name": "Or.resolve_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [272, 9], "def_end_pos": [272, 25]}]], "state_before": "case mp.intro.intro.intro.inl\n\u03b1 \u03b2 : Type u\nx b : \u03b1\nhne : x \u2260 b\nh : \u2200 (x_1 : \u03b1), x_1 = x \u2228 x_1 = b\n\u22a2 \u2203! y, y \u2260 x\n\ncase mp.intro.intro.intro.inr\n\u03b1 \u03b2 : Type u\nx a : \u03b1\nhne : a \u2260 x\nh : \u2200 (x_1 : \u03b1), x_1 = a \u2228 x_1 = x\n\u22a2 \u2203! y, y \u2260 x", "state_after": "no goals"}, {"tactic": "rintro \u27e8y, hne, hy\u27e9", "annotated_tactic": ["rintro \u27e8y, hne, hy\u27e9", []], "state_before": "case mpr\n\u03b1 \u03b2 : Type u\nx : \u03b1\n\u22a2 (\u2203! y, y \u2260 x) \u2192 \u2203 x y, x \u2260 y \u2227 {x, y} = univ", "state_after": "case mpr.intro.intro\n\u03b1 \u03b2 : Type u\nx y : \u03b1\nhne : y \u2260 x\nhy : \u2200 (y_1 : \u03b1), (fun y => y \u2260 x) y_1 \u2192 y_1 = y\n\u22a2 \u2203 x y, x \u2260 y \u2227 {x, y} = univ"}, {"tactic": "exact \u27e8x, y, hne.symm, eq_univ_of_forall fun z => or_iff_not_imp_left.2 (hy z)\u27e9", "annotated_tactic": ["exact \u27e8x, y, hne.symm, <a>eq_univ_of_forall</a> fun z => <a>or_iff_not_imp_left</a>.2 (hy z)\u27e9", [{"full_name": "Set.eq_univ_of_forall", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [705, 9], "def_end_pos": [705, 26]}, {"full_name": "or_iff_not_imp_left", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [360, 9], "def_end_pos": [360, 28]}]], "state_before": "case mpr.intro.intro\n\u03b1 \u03b2 : Type u\nx y : \u03b1\nhne : y \u2260 x\nhy : \u2200 (y_1 : \u03b1), (fun y => y \u2260 x) y_1 \u2192 y_1 = y\n\u22a2 \u2203 x y, x \u2260 y \u2227 {x, y} = univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Hom/Iterate.lean", "full_name": "Monoid.End.coe_pow", "start": [90, 1], "end": [91, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "tendsto_inv_nhdsWithin_Iic_inv", "start": [599, 1], "end": [600, 69], "traced_tactics": [{"tactic": "simpa only [inv_inv] using @tendsto_inv_nhdsWithin_Iic _ _ _ _ a\u207b\u00b9", "annotated_tactic": ["simpa only [<a>inv_inv</a>] using @<a>tendsto_inv_nhdsWithin_Iic</a> _ _ _ _ a\u207b\u00b9", [{"full_name": "inv_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [800, 9], "def_end_pos": [800, 16]}, {"full_name": "tendsto_inv_nhdsWithin_Iic", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [587, 9], "def_end_pos": [587, 35]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nG : Type w\nH : Type x\ninst\u271d\u2076 : TopologicalSpace G\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalGroup G\ninst\u271d\u00b3 : TopologicalSpace \u03b1\nf : \u03b1 \u2192 G\ns : Set \u03b1\nx : \u03b1\ninst\u271d\u00b2 : TopologicalSpace H\ninst\u271d\u00b9 : OrderedCommGroup H\ninst\u271d : ContinuousInv H\na : H\n\u22a2 Tendsto Inv.inv (\ud835\udcdd[Iic a\u207b\u00b9] a\u207b\u00b9) (\ud835\udcdd[Ici a] a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/IsROrC/Basic.lean", "full_name": "IsROrC.I_mul_I_of_nonzero", "start": [599, 1], "end": [600, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/CondDistrib.lean", "full_name": "MeasureTheory.Integrable.integral_condDistrib", "start": [183, 1], "end": [186, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.StronglyMeasurable.div", "start": [437, 11], "end": [439, 98], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean", "full_name": "CategoryTheory.Limits.hasTerminalChangeDiagram", "start": [259, 1], "end": [261, 67], "traced_tactics": [{"tactic": "aesop_cat", "annotated_tactic": ["aesop_cat", []], "state_before": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty.{w + 1} \u2964 C\nF\u2082 : Discrete PEmpty.{w' + 1} \u2964 C\nh : HasLimit F\u2081\n\u22a2 (X : Discrete PEmpty.{w' + 1}) \u2192 ((Functor.const (Discrete PEmpty.{w' + 1})).obj (limit F\u2081)).obj X \u27f6 F\u2082.obj X", "state_after": "no goals"}, {"tactic": "aesop_cat", "annotated_tactic": ["aesop_cat", []], "state_before": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX : C\nF\u2081 : Discrete PEmpty.{w + 1} \u2964 C\nF\u2082 : Discrete PEmpty.{w' + 1} \u2964 C\nh : HasLimit F\u2081\n\u22a2 \u2200 \u2983X Y : Discrete PEmpty.{w' + 1}\u2984 (f : X \u27f6 Y),\n    ((Functor.const (Discrete PEmpty.{w' + 1})).obj (limit F\u2081)).map f \u226b\n        id (Discrete.casesOn Y fun as => (_ : False).elim) =\n      id (Discrete.casesOn X fun as => (_ : False).elim) \u226b F\u2082.map f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.inter_le_right", "start": [1780, 1], "end": [1782, 94], "traced_tactics": [{"tactic": "simpa [h] using cons_le_cons a (IH (t.erase a))", "annotated_tactic": ["simpa [h] using <a>cons_le_cons</a> a (IH (t.erase a))", [{"full_name": "Multiset.cons_le_cons", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [575, 9], "def_end_pos": [575, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns\u271d\u00b9 t\u271d u : Multiset \u03b1\na\u271d b : \u03b1\ns\u271d : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nIH : \u2200 (t : Multiset \u03b1), s \u2229 t \u2264 t\nt : Multiset \u03b1\nh : a \u2208 t\n\u22a2 (a ::\u2098 s) \u2229 t \u2264 t", "state_after": "no goals"}, {"tactic": "simp [h, IH]", "annotated_tactic": ["simp [h, IH]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns\u271d\u00b9 t\u271d u : Multiset \u03b1\na\u271d b : \u03b1\ns\u271d : Multiset \u03b1\na : \u03b1\ns : Multiset \u03b1\nIH : \u2200 (t : Multiset \u03b1), s \u2229 t \u2264 t\nt : Multiset \u03b1\nh : \u00aca \u2208 t\n\u22a2 (a ::\u2098 s) \u2229 t \u2264 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/GroupPower/Ring.lean", "full_name": "pow_ne_zero", "start": [84, 1], "end": [85, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Infsep.lean", "full_name": "Set.einfsep_ne_top", "start": [77, 1], "end": [79, 48], "traced_tactics": [{"tactic": "simp_rw [\u2190 lt_top_iff_ne_top, einfsep_lt_top]", "annotated_tactic": ["simp_rw [\u2190 <a>lt_top_iff_ne_top</a>, <a>einfsep_lt_top</a>]", [{"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}, {"full_name": "Set.einfsep_lt_top", "def_path": "Mathlib/Topology/MetricSpace/Infsep.lean", "def_pos": [72, 9], "def_end_pos": [72, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : EDist \u03b1\nx y : \u03b1\ns t : Set \u03b1\n\u22a2 einfsep s \u2260 \u22a4 \u2194 \u2203 x x_1 y x_2 _hxy, edist x y \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Compactness/Compact.lean", "full_name": "Filter.coprod\u1d62_cocompact", "start": [1023, 1], "end": [1029, 73], "traced_tactics": [{"tactic": "refine' le_antisymm (iSup_le fun i => Filter.comap_cocompact_le (continuous_apply i)) _", "annotated_tactic": ["refine' <a>le_antisymm</a> (<a>iSup_le</a> fun i => <a>Filter.comap_cocompact_le</a> (<a>continuous_apply</a> i)) _", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}, {"full_name": "Filter.comap_cocompact_le", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [803, 9], "def_end_pos": [803, 34]}, {"full_name": "continuous_apply", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 25]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\n\u03b4 : Type u_3\n\u03ba : \u03b4 \u2192 Type u_4\ninst\u271d : (d : \u03b4) \u2192 TopologicalSpace (\u03ba d)\n\u22a2 (Filter.coprod\u1d62 fun d => cocompact (\u03ba d)) = cocompact ((d : \u03b4) \u2192 \u03ba d)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\n\u03b4 : Type u_3\n\u03ba : \u03b4 \u2192 Type u_4\ninst\u271d : (d : \u03b4) \u2192 TopologicalSpace (\u03ba d)\n\u22a2 cocompact ((d : \u03b4) \u2192 \u03ba d) \u2264 Filter.coprod\u1d62 fun d => cocompact (\u03ba d)"}, {"tactic": "refine' compl_surjective.forall.2 fun s H => _", "annotated_tactic": ["refine' compl_surjective.forall.2 fun s H => _", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns t : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\n\u03b4 : Type u_3\n\u03ba : \u03b4 \u2192 Type u_4\ninst\u271d : (d : \u03b4) \u2192 TopologicalSpace (\u03ba d)\n\u22a2 cocompact ((d : \u03b4) \u2192 \u03ba d) \u2264 Filter.coprod\u1d62 fun d => cocompact (\u03ba d)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\n\u03b4 : Type u_3\n\u03ba : \u03b4 \u2192 Type u_4\ninst\u271d : (d : \u03b4) \u2192 TopologicalSpace (\u03ba d)\ns : Set ((i : \u03b4) \u2192 \u03ba i)\nH : s\u1d9c \u2208 Filter.coprod\u1d62 fun d => cocompact (\u03ba d)\n\u22a2 s\u1d9c \u2208 cocompact ((d : \u03b4) \u2192 \u03ba d)"}, {"tactic": "simp only [compl_mem_coprod\u1d62, Filter.mem_cocompact, compl_subset_compl, image_subset_iff] at H \u22a2", "annotated_tactic": ["simp only [<a>compl_mem_coprod\u1d62</a>, <a>Filter.mem_cocompact</a>, <a>compl_subset_compl</a>, <a>image_subset_iff</a>] at H \u22a2", [{"full_name": "Filter.compl_mem_coprod\u1d62", "def_path": "Mathlib/Order/Filter/Pi.lean", "def_pos": [205, 9], "def_end_pos": [205, 26]}, {"full_name": "Filter.mem_cocompact", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [523, 9], "def_end_pos": [523, 22]}, {"full_name": "Set.compl_subset_compl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1756, 9], "def_end_pos": [1756, 27]}, {"full_name": "Set.image_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [497, 9], "def_end_pos": [497, 25]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\n\u03b4 : Type u_3\n\u03ba : \u03b4 \u2192 Type u_4\ninst\u271d : (d : \u03b4) \u2192 TopologicalSpace (\u03ba d)\ns : Set ((i : \u03b4) \u2192 \u03ba i)\nH : s\u1d9c \u2208 Filter.coprod\u1d62 fun d => cocompact (\u03ba d)\n\u22a2 s\u1d9c \u2208 cocompact ((d : \u03b4) \u2192 \u03ba d)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\n\u03b4 : Type u_3\n\u03ba : \u03b4 \u2192 Type u_4\ninst\u271d : (d : \u03b4) \u2192 TopologicalSpace (\u03ba d)\ns : Set ((i : \u03b4) \u2192 \u03ba i)\nH : \u2200 (i : \u03b4), \u2203 t, IsCompact t \u2227 s \u2286 Function.eval i \u207b\u00b9' t\n\u22a2 \u2203 t, IsCompact t \u2227 s \u2286 t"}, {"tactic": "choose K hKc htK using H", "annotated_tactic": ["choose K hKc htK using H", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\n\u03b4 : Type u_3\n\u03ba : \u03b4 \u2192 Type u_4\ninst\u271d : (d : \u03b4) \u2192 TopologicalSpace (\u03ba d)\ns : Set ((i : \u03b4) \u2192 \u03ba i)\nH : \u2200 (i : \u03b4), \u2203 t, IsCompact t \u2227 s \u2286 Function.eval i \u207b\u00b9' t\n\u22a2 \u2203 t, IsCompact t \u2227 s \u2286 t", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\n\u03b4 : Type u_3\n\u03ba : \u03b4 \u2192 Type u_4\ninst\u271d : (d : \u03b4) \u2192 TopologicalSpace (\u03ba d)\ns : Set ((i : \u03b4) \u2192 \u03ba i)\nK : (i : \u03b4) \u2192 Set (\u03ba i)\nhKc : \u2200 (i : \u03b4), IsCompact (K i)\nhtK : \u2200 (i : \u03b4), s \u2286 Function.eval i \u207b\u00b9' K i\n\u22a2 \u2203 t, IsCompact t \u2227 s \u2286 t"}, {"tactic": "exact \u27e8Set.pi univ K, isCompact_univ_pi hKc, fun f hf i _ => htK i hf\u27e9", "annotated_tactic": ["exact \u27e8<a>Set.pi</a> <a>univ</a> K, <a>isCompact_univ_pi</a> hKc, fun f hf i _ => htK i hf\u27e9", [{"full_name": "Set.pi", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [665, 5], "def_end_pos": [665, 7]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}, {"full_name": "isCompact_univ_pi", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [1007, 9], "def_end_pos": [1007, 26]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\n\u03b4 : Type u_3\n\u03ba : \u03b4 \u2192 Type u_4\ninst\u271d : (d : \u03b4) \u2192 TopologicalSpace (\u03ba d)\ns : Set ((i : \u03b4) \u2192 \u03ba i)\nK : (i : \u03b4) \u2192 Set (\u03ba i)\nhKc : \u2200 (i : \u03b4), IsCompact (K i)\nhtK : \u2200 (i : \u03b4), s \u2286 Function.eval i \u207b\u00b9' K i\n\u22a2 \u2203 t, IsCompact t \u2227 s \u2286 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean", "full_name": "MeasurableEquiv.coe_smul\u2080", "start": [75, 1], "end": [76, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean", "full_name": "MeasurableSpace.le_invariants_iterate", "start": [50, 1], "end": [54, 82], "traced_tactics": [{"tactic": "induction n with\n| zero => simp [invariants_le]\n| succ n ihn => exact le_trans (le_inf ihn le_rfl) (inf_le_invariants_comp _ _)", "annotated_tactic": ["induction n with\n  | <a>zero</a> => simp [<a>invariants_le</a>]\n  | <a>succ</a> n ihn => exact <a>le_trans</a> (<a>le_inf</a> ihn <a>le_rfl</a>) (<a>inf_le_invariants_comp</a> _ _)", [{"full_name": "Nat.zero", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1041, 5], "def_end_pos": [1041, 9]}, {"full_name": "MeasurableSpace.invariants_le", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean", "def_pos": [44, 9], "def_end_pos": [44, 22]}, {"full_name": "Nat.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_inf", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [409, 9], "def_end_pos": [409, 15]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "MeasurableSpace.inf_le_invariants_comp", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean", "def_pos": [46, 9], "def_end_pos": [46, 31]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\nn : \u2115\n\u22a2 invariants f \u2264 invariants f^[n]", "state_after": "no goals"}, {"tactic": "simp [invariants_le]", "annotated_tactic": ["simp [<a>invariants_le</a>]", [{"full_name": "MeasurableSpace.invariants_le", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean", "def_pos": [44, 9], "def_end_pos": [44, 22]}]], "state_before": "case zero\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\n\u22a2 invariants f \u2264 invariants f^[Nat.zero]", "state_after": "no goals"}, {"tactic": "exact le_trans (le_inf ihn le_rfl) (inf_le_invariants_comp _ _)", "annotated_tactic": ["exact <a>le_trans</a> (<a>le_inf</a> ihn <a>le_rfl</a>) (<a>inf_le_invariants_comp</a> _ _)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_inf", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [409, 9], "def_end_pos": [409, 15]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "MeasurableSpace.inf_le_invariants_comp", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean", "def_pos": [46, 9], "def_end_pos": [46, 31]}]], "state_before": "case succ\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\nn : \u2115\nihn : invariants f \u2264 invariants f^[n]\n\u22a2 invariants f \u2264 invariants f^[Nat.succ n]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Convolution.lean", "full_name": "integral_convolution", "start": [1006, 1], "end": [1011, 53], "traced_tactics": [{"tactic": "refine' (integral_integral_swap (by apply hf.rst.imnvolution_integrand L hg)).trans _", "annotated_tactic": ["refine' (<a>integral_integral_swap</a> (by apply hf.rst.imnvolution_integrand L hg)).<a>trans</a> _", [{"full_name": "MeasureTheory.integral_integral_swap", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "def_pos": [499, 9], "def_end_pos": [499, 31]}, {"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}]], "state_before": "\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2079 : NormedAddCommGroup E\ninst\u271d\u00b2\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b2\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u00b9 : NormedSpace \u211d F\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2079 : CompleteSpace F\ninst\u271d\u00b9\u2078 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2077 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2076 : NormedSpace \u211d F'\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u2074 : CompleteSpace F'\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F''\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F''\ninst\u271d\u00b9\u2070 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2079 : AddGroup G\ninst\u271d\u2078 : SigmaFinite \u03bc\ninst\u271d\u2077 : SigmaFinite \u03bd\ninst\u271d\u2076 : IsAddRightInvariant \u03bc\ninst\u271d\u2075 : MeasurableAdd\u2082 G\ninst\u271d\u2074 : MeasurableNeg G\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace E'\nhf : Integrable f\nhg : Integrable g\n\u22a2 \u222b (x : G), f \u22c6[L, x] g \u2202\u03bc = \u2191(\u2191L (\u222b (x : G), f x \u2202\u03bd)) (\u222b (x : G), g x \u2202\u03bc)", "state_after": "\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2079 : NormedAddCommGroup E\ninst\u271d\u00b2\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b2\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u00b9 : NormedSpace \u211d F\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2079 : CompleteSpace F\ninst\u271d\u00b9\u2078 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2077 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2076 : NormedSpace \u211d F'\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u2074 : CompleteSpace F'\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F''\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F''\ninst\u271d\u00b9\u2070 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2079 : AddGroup G\ninst\u271d\u2078 : SigmaFinite \u03bc\ninst\u271d\u2077 : SigmaFinite \u03bd\ninst\u271d\u2076 : IsAddRightInvariant \u03bc\ninst\u271d\u2075 : MeasurableAdd\u2082 G\ninst\u271d\u2074 : MeasurableNeg G\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace E'\nhf : Integrable f\nhg : Integrable g\n\u22a2 \u222b (y : G), \u222b (x : G), \u2191(\u2191L (f y)) (g (x - y)) \u2202\u03bc \u2202\u03bd = \u2191(\u2191L (\u222b (x : G), f x \u2202\u03bd)) (\u222b (x : G), g x \u2202\u03bc)"}, {"tactic": "simp_rw [integral_comp_comm _ (hg.comp_sub_right _), integral_sub_right_eq_self]", "annotated_tactic": ["simp_rw [<a>integral_comp_comm</a> _ (hg.comp_sub_right _), <a>integral_sub_right_eq_self</a>]", [{"full_name": "ContinuousLinearMap.integral_comp_comm", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [1120, 9], "def_end_pos": [1120, 27]}, {"full_name": "MeasureTheory.integral_sub_right_eq_self", "def_path": "Mathlib/MeasureTheory/Group/Integral.lean", "def_pos": [80, 3], "def_end_pos": [80, 14]}]], "state_before": "\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2079 : NormedAddCommGroup E\ninst\u271d\u00b2\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b2\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u00b9 : NormedSpace \u211d F\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2079 : CompleteSpace F\ninst\u271d\u00b9\u2078 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2077 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2076 : NormedSpace \u211d F'\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u2074 : CompleteSpace F'\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F''\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F''\ninst\u271d\u00b9\u2070 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2079 : AddGroup G\ninst\u271d\u2078 : SigmaFinite \u03bc\ninst\u271d\u2077 : SigmaFinite \u03bd\ninst\u271d\u2076 : IsAddRightInvariant \u03bc\ninst\u271d\u2075 : MeasurableAdd\u2082 G\ninst\u271d\u2074 : MeasurableNeg G\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace E'\nhf : Integrable f\nhg : Integrable g\n\u22a2 \u222b (y : G), \u222b (x : G), \u2191(\u2191L (f y)) (g (x - y)) \u2202\u03bc \u2202\u03bd = \u2191(\u2191L (\u222b (x : G), f x \u2202\u03bd)) (\u222b (x : G), g x \u2202\u03bc)", "state_after": "\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2079 : NormedAddCommGroup E\ninst\u271d\u00b2\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b2\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u00b9 : NormedSpace \u211d F\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2079 : CompleteSpace F\ninst\u271d\u00b9\u2078 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2077 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2076 : NormedSpace \u211d F'\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u2074 : CompleteSpace F'\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F''\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F''\ninst\u271d\u00b9\u2070 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2079 : AddGroup G\ninst\u271d\u2078 : SigmaFinite \u03bc\ninst\u271d\u2077 : SigmaFinite \u03bd\ninst\u271d\u2076 : IsAddRightInvariant \u03bc\ninst\u271d\u2075 : MeasurableAdd\u2082 G\ninst\u271d\u2074 : MeasurableNeg G\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace E'\nhf : Integrable f\nhg : Integrable g\n\u22a2 \u222b (y : G), \u2191(\u2191L (f y)) (\u222b (x : G), g x \u2202\u03bc) \u2202\u03bd = \u2191(\u2191L (\u222b (x : G), f x \u2202\u03bd)) (\u222b (x : G), g x \u2202\u03bc)"}, {"tactic": "exact (L.flip (\u222b x, g x \u2202\u03bc)).integral_comp_comm hf", "annotated_tactic": ["exact (L.flip (\u222b x, g x \u2202\u03bc)).<a>integral_comp_comm</a> hf", [{"full_name": "ContinuousLinearMap.integral_comp_comm", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [1120, 9], "def_end_pos": [1120, 27]}]], "state_before": "\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2079 : NormedAddCommGroup E\ninst\u271d\u00b2\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b2\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u00b9 : NormedSpace \u211d F\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2079 : CompleteSpace F\ninst\u271d\u00b9\u2078 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2077 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2076 : NormedSpace \u211d F'\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u2074 : CompleteSpace F'\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F''\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F''\ninst\u271d\u00b9\u2070 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2079 : AddGroup G\ninst\u271d\u2078 : SigmaFinite \u03bc\ninst\u271d\u2077 : SigmaFinite \u03bd\ninst\u271d\u2076 : IsAddRightInvariant \u03bc\ninst\u271d\u2075 : MeasurableAdd\u2082 G\ninst\u271d\u2074 : MeasurableNeg G\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace E'\nhf : Integrable f\nhg : Integrable g\n\u22a2 \u222b (y : G), \u2191(\u2191L (f y)) (\u222b (x : G), g x \u2202\u03bc) \u2202\u03bd = \u2191(\u2191L (\u222b (x : G), f x \u2202\u03bd)) (\u222b (x : G), g x \u2202\u03bc)", "state_after": "no goals"}, {"tactic": "apply hf.rst.imnvolution_integrand L hg", "annotated_tactic": ["apply hf.rst.imnvolution_integrand L hg", []], "state_before": "\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2079 : NormedAddCommGroup E\ninst\u271d\u00b2\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b2\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u00b9 : NormedSpace \u211d F\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2079 : CompleteSpace F\ninst\u271d\u00b9\u2078 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2077 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2076 : NormedSpace \u211d F'\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u2074 : CompleteSpace F'\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F''\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F''\ninst\u271d\u00b9\u2070 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2079 : AddGroup G\ninst\u271d\u2078 : SigmaFinite \u03bc\ninst\u271d\u2077 : SigmaFinite \u03bd\ninst\u271d\u2076 : IsAddRightInvariant \u03bc\ninst\u271d\u2075 : MeasurableAdd\u2082 G\ninst\u271d\u2074 : MeasurableNeg G\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace E'\nhf : Integrable f\nhg : Integrable g\n\u22a2 Integrable (uncurry fun x t => \u2191(\u2191L (f t)) (g (x - t)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/CompactlyGenerated.lean", "full_name": "sSup_compact_eq_top", "start": [348, 1], "end": [350, 35], "traced_tactics": [{"tactic": "refine' Eq.trans (congr rfl (Set.ext fun x => _)) (sSup_compact_le_eq \u22a4)", "annotated_tactic": ["refine' <a>Eq.trans</a> (<a>congr</a> <a>rfl</a> (<a>Set.ext</a> fun x => _)) (<a>sSup_compact_le_eq</a> \u22a4)", [{"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}, {"full_name": "congr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [359, 9], "def_end_pos": [359, 14]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Set.ext", "def_path": "Mathlib/Init/Set.lean", "def_pos": [54, 9], "def_end_pos": [54, 12]}, {"full_name": "sSup_compact_le_eq", "def_path": "Mathlib/Order/CompactlyGenerated.lean", "def_pos": [341, 9], "def_end_pos": [341, 27]}]], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\na b : \u03b1\ns : Set \u03b1\n\u22a2 sSup {a | CompleteLattice.IsCompactElement a} = \u22a4", "state_after": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\na b : \u03b1\ns : Set \u03b1\nx : \u03b1\n\u22a2 x \u2208 {a | CompleteLattice.IsCompactElement a} \u2194 x \u2208 {c | CompleteLattice.IsCompactElement c \u2227 c \u2264 \u22a4}"}, {"tactic": "exact (and_iff_left le_top).symm", "annotated_tactic": ["exact (<a>and_iff_left</a> <a>le_top</a>).<a>symm</a>", [{"full_name": "and_iff_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [204, 9], "def_end_pos": [204, 21]}, {"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [98, 9], "def_end_pos": [98, 15]}, {"full_name": "Iff.symm", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [671, 9], "def_end_pos": [671, 17]}]], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : CompleteLattice \u03b1\nf : \u03b9 \u2192 \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\na b : \u03b1\ns : Set \u03b1\nx : \u03b1\n\u22a2 x \u2208 {a | CompleteLattice.IsCompactElement a} \u2194 x \u2208 {c | CompleteLattice.IsCompactElement c \u2227 c \u2264 \u22a4}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Parity.lean", "full_name": "Even.mul_left", "start": [279, 1], "end": [280, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Rearrangement.lean", "full_name": "MonovaryOn.sum_smul_comp_perm_eq_sum_smul_iff", "start": [114, 1], "end": [136, 52], "traced_tactics": [{"tactic": "refine' \u27e8not_imp_not.1 fun h \u21a6 _, fun h \u21a6 (hfg.sum_smul_comp_perm_le_sum_smul h\u03c3).antisymm _\u27e9", "annotated_tactic": ["refine' \u27e8<a>not_imp_not</a>.1 fun h \u21a6 _, fun h \u21a6 (hfg.sum_smul_comp_perm_le_sum_smul h\u03c3).<a>antisymm</a> _\u27e9", [{"full_name": "not_imp_not", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [383, 9], "def_end_pos": [383, 20]}, {"full_name": "LE.le.antisymm", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [128, 7], "def_end_pos": [128, 21]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\n\u22a2 \u2211 i in s, f i \u2022 g (\u2191\u03c3 i) = \u2211 i in s, f i \u2022 g i \u2194 MonovaryOn f (g \u2218 \u2191\u03c3) \u2191s", "state_after": "case refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nh : \u00acMonovaryOn f (g \u2218 \u2191\u03c3) \u2191s\n\u22a2 \u00ac\u2211 i in s, f i \u2022 g (\u2191\u03c3 i) = \u2211 i in s, f i \u2022 g i\n\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nh : MonovaryOn f (g \u2218 \u2191\u03c3) \u2191s\n\u22a2 \u2211 i in s, f i \u2022 g i \u2264 \u2211 i in s, f i \u2022 g (\u2191\u03c3 i)"}, {"tactic": "rw [MonovaryOn] at h", "annotated_tactic": ["rw [<a>MonovaryOn</a>] at h", [{"full_name": "MonovaryOn", "def_path": "Mathlib/Order/Monotone/Monovary.lean", "def_pos": [49, 5], "def_end_pos": [49, 15]}]], "state_before": "case refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nh : \u00acMonovaryOn f (g \u2218 \u2191\u03c3) \u2191s\n\u22a2 \u00ac\u2211 i in s, f i \u2022 g (\u2191\u03c3 i) = \u2211 i in s, f i \u2022 g i", "state_after": "case refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nh : \u00ac\u2200 \u2983i : \u03b9\u2984, i \u2208 \u2191s \u2192 \u2200 \u2983j : \u03b9\u2984, j \u2208 \u2191s \u2192 (g \u2218 \u2191\u03c3) i < (g \u2218 \u2191\u03c3) j \u2192 f i \u2264 f j\n\u22a2 \u00ac\u2211 i in s, f i \u2022 g (\u2191\u03c3 i) = \u2211 i in s, f i \u2022 g i"}, {"tactic": "push_neg at h", "annotated_tactic": ["push_neg at h", []], "state_before": "case refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nh : \u00ac\u2200 \u2983i : \u03b9\u2984, i \u2208 \u2191s \u2192 \u2200 \u2983j : \u03b9\u2984, j \u2208 \u2191s \u2192 (g \u2218 \u2191\u03c3) i < (g \u2218 \u2191\u03c3) j \u2192 f i \u2264 f j\n\u22a2 \u00ac\u2211 i in s, f i \u2022 g (\u2191\u03c3 i) = \u2211 i in s, f i \u2022 g i", "state_after": "case refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nh : Exists fun \u2983i\u2984 => i \u2208 \u2191s \u2227 Exists fun \u2983j\u2984 => j \u2208 \u2191s \u2227 (g \u2218 \u2191\u03c3) i < (g \u2218 \u2191\u03c3) j \u2227 f j < f i\n\u22a2 \u00ac\u2211 i in s, f i \u2022 g (\u2191\u03c3 i) = \u2211 i in s, f i \u2022 g i"}, {"tactic": "obtain \u27e8x, hx, y, hy, hgxy, hfxy\u27e9 := h", "annotated_tactic": ["obtain \u27e8x, hx, y, hy, hgxy, hfxy\u27e9 := h", []], "state_before": "case refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nh : Exists fun \u2983i\u2984 => i \u2208 \u2191s \u2227 Exists fun \u2983j\u2984 => j \u2208 \u2191s \u2227 (g \u2218 \u2191\u03c3) i < (g \u2218 \u2191\u03c3) j \u2227 f j < f i\n\u22a2 \u00ac\u2211 i in s, f i \u2022 g (\u2191\u03c3 i) = \u2211 i in s, f i \u2022 g i", "state_after": "case refine'_1.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nx : \u03b9\nhx : x \u2208 \u2191s\ny : \u03b9\nhy : y \u2208 \u2191s\nhgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y\nhfxy : f y < f x\n\u22a2 \u00ac\u2211 i in s, f i \u2022 g (\u2191\u03c3 i) = \u2211 i in s, f i \u2022 g i"}, {"tactic": "set \u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3", "annotated_tactic": ["set \u03c4 : <a>Perm</a> \u03b9 := (<a>Equiv.swap</a> x y).<a>trans</a> \u03c3", [{"full_name": "Equiv.Perm", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [90, 5], "def_end_pos": [90, 15]}, {"full_name": "Equiv.swap", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1622, 5], "def_end_pos": [1622, 9]}, {"full_name": "Equiv.trans", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [175, 15], "def_end_pos": [175, 20]}]], "state_before": "case refine'_1.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nx : \u03b9\nhx : x \u2208 \u2191s\ny : \u03b9\nhy : y \u2208 \u2191s\nhgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y\nhfxy : f y < f x\n\u22a2 \u00ac\u2211 i in s, f i \u2022 g (\u2191\u03c3 i) = \u2211 i in s, f i \u2022 g i", "state_after": "case refine'_1.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nx : \u03b9\nhx : x \u2208 \u2191s\ny : \u03b9\nhy : y \u2208 \u2191s\nhgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y\nhfxy : f y < f x\n\u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3\n\u22a2 \u00ac\u2211 i in s, f i \u2022 g (\u2191\u03c3 i) = \u2211 i in s, f i \u2022 g i"}, {"tactic": "have h\u03c4s : { x | \u03c4 x \u2260 x } \u2286 s := by\n  refine' (set_support_mul_subset \u03c3 <| swap x y).trans (Set.union_subset h\u03c3 fun z hz \u21a6 _)\n  obtain \u27e8_, rfl | rfl\u27e9 := swap_apply_ne_self_iff.1 hz <;> assumption", "annotated_tactic": ["have h\u03c4s : { x | \u03c4 x \u2260 x } \u2286 s := by\n        refine' (<a>set_support_mul_subset</a> \u03c3 <| <a>swap</a> x y).<a>trans</a> (<a>Set.union_subset</a> h\u03c3 fun z hz \u21a6 _)\n        obtain \u27e8_, rfl | rfl\u27e9 := <a>swap_apply_ne_self_iff</a>.1 hz <;> assumption", [{"full_name": "Equiv.Perm.set_support_mul_subset", "def_path": "Mathlib/GroupTheory/Perm/Support.lean", "def_pos": [274, 9], "def_end_pos": [274, 31]}, {"full_name": "Equiv.swap", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1622, 5], "def_end_pos": [1622, 9]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}, {"full_name": "Set.union_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [836, 9], "def_end_pos": [836, 21]}, {"full_name": "Equiv.swap_apply_ne_self_iff", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1723, 9], "def_end_pos": [1723, 31]}]], "state_before": "case refine'_1.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nx : \u03b9\nhx : x \u2208 \u2191s\ny : \u03b9\nhy : y \u2208 \u2191s\nhgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y\nhfxy : f y < f x\n\u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3\n\u22a2 \u00ac\u2211 i in s, f i \u2022 g (\u2191\u03c3 i) = \u2211 i in s, f i \u2022 g i", "state_after": "case refine'_1.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nx : \u03b9\nhx : x \u2208 \u2191s\ny : \u03b9\nhy : y \u2208 \u2191s\nhgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y\nhfxy : f y < f x\n\u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3\nh\u03c4s : {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s\n\u22a2 \u00ac\u2211 i in s, f i \u2022 g (\u2191\u03c3 i) = \u2211 i in s, f i \u2022 g i"}, {"tactic": "refine' ((hfg.sum_smul_comp_perm_le_sum_smul h\u03c4s).trans_lt' _).ne", "annotated_tactic": ["refine' ((hfg.sum_smul_comp_perm_le_sum_smul h\u03c4s).<a>trans_lt'</a> _).<a>ne</a>", [{"full_name": "LE.le.trans_lt'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [126, 7], "def_end_pos": [126, 22]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [152, 7], "def_end_pos": [152, 15]}]], "state_before": "case refine'_1.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nx : \u03b9\nhx : x \u2208 \u2191s\ny : \u03b9\nhy : y \u2208 \u2191s\nhgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y\nhfxy : f y < f x\n\u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3\nh\u03c4s : {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s\n\u22a2 \u00ac\u2211 i in s, f i \u2022 g (\u2191\u03c3 i) = \u2211 i in s, f i \u2022 g i", "state_after": "case refine'_1.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nx : \u03b9\nhx : x \u2208 \u2191s\ny : \u03b9\nhy : y \u2208 \u2191s\nhgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y\nhfxy : f y < f x\n\u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3\nh\u03c4s : {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s\n\u22a2 \u2211 i in s, f i \u2022 g (\u2191\u03c3 i) < \u2211 i in s, f i \u2022 g (\u2191\u03c4 i)"}, {"tactic": "obtain rfl | hxy := eq_or_ne x y", "annotated_tactic": ["obtain rfl | hxy := <a>eq_or_ne</a> x y", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "case refine'_1.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nx : \u03b9\nhx : x \u2208 \u2191s\ny : \u03b9\nhy : y \u2208 \u2191s\nhgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y\nhfxy : f y < f x\n\u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3\nh\u03c4s : {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s\n\u22a2 \u2211 i in s, f i \u2022 g (\u2191\u03c3 i) < \u2211 i in s, f i \u2022 g (\u2191\u03c4 i)", "state_after": "case refine'_1.intro.intro.intro.intro.intro.inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nx : \u03b9\nhx hy : x \u2208 \u2191s\nhgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) x\nhfxy : f x < f x\n\u03c4 : Perm \u03b9 := (Equiv.swap x x).trans \u03c3\nh\u03c4s : {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s\n\u22a2 \u2211 i in s, f i \u2022 g (\u2191\u03c3 i) < \u2211 i in s, f i \u2022 g (\u2191\u03c4 i)\n\ncase refine'_1.intro.intro.intro.intro.intro.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nx : \u03b9\nhx : x \u2208 \u2191s\ny : \u03b9\nhy : y \u2208 \u2191s\nhgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y\nhfxy : f y < f x\n\u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3\nh\u03c4s : {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s\nhxy : x \u2260 y\n\u22a2 \u2211 i in s, f i \u2022 g (\u2191\u03c3 i) < \u2211 i in s, f i \u2022 g (\u2191\u03c4 i)"}, {"tactic": "simp only [\u2190 s.sum_erase_add _ hx, \u2190 (s.erase x).sum_erase_add _ (mem_erase.2 \u27e8hxy.symm, hy\u27e9),\n  add_assoc, Equiv.coe_trans, Function.comp_apply, swap_apply_right, swap_apply_left]", "annotated_tactic": ["simp only [\u2190 s.sum_erase_add _ hx, \u2190 (s.erase x).<a>sum_erase_add</a> _ (<a>mem_erase</a>.2 \u27e8hxy.symm, hy\u27e9),\n        <a>add_assoc</a>, <a>Equiv.coe_trans</a>, <a>Function.comp_apply</a>, <a>swap_apply_right</a>, <a>swap_apply_left</a>]", [{"full_name": "Finset.sum_erase_add", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1670, 3], "def_end_pos": [1670, 14]}, {"full_name": "Finset.mem_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1887, 9], "def_end_pos": [1887, 18]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}, {"full_name": "Equiv.coe_trans", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [272, 17], "def_end_pos": [272, 26]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "Equiv.swap_apply_right", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1646, 9], "def_end_pos": [1646, 25]}, {"full_name": "Equiv.swap_apply_left", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1641, 9], "def_end_pos": [1641, 24]}]], "state_before": "case refine'_1.intro.intro.intro.intro.intro.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nx : \u03b9\nhx : x \u2208 \u2191s\ny : \u03b9\nhy : y \u2208 \u2191s\nhgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y\nhfxy : f y < f x\n\u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3\nh\u03c4s : {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s\nhxy : x \u2260 y\n\u22a2 \u2211 i in s, f i \u2022 g (\u2191\u03c3 i) < \u2211 i in s, f i \u2022 g (\u2191\u03c4 i)", "state_after": "case refine'_1.intro.intro.intro.intro.intro.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nx : \u03b9\nhx : x \u2208 \u2191s\ny : \u03b9\nhy : y \u2208 \u2191s\nhgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y\nhfxy : f y < f x\n\u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3\nh\u03c4s : {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s\nhxy : x \u2260 y\n\u22a2 \u2211 x in erase (erase s x) y, f x \u2022 g (\u2191\u03c3 x) + (f y \u2022 g (\u2191\u03c3 y) + f x \u2022 g (\u2191\u03c3 x)) <\n    \u2211 x_1 in erase (erase s x) y, f x_1 \u2022 g (\u2191\u03c3 (\u2191(Equiv.swap x y) x_1)) + (f y \u2022 g (\u2191\u03c3 x) + f x \u2022 g (\u2191\u03c3 y))"}, {"tactic": "refine' add_lt_add_of_le_of_lt (Finset.sum_congr rfl fun z hz \u21a6 _).le\n  (smul_add_smul_lt_smul_add_smul hfxy hgxy)", "annotated_tactic": ["refine' <a>add_lt_add_of_le_of_lt</a> (<a>Finset.sum_congr</a> <a>rfl</a> fun z hz \u21a6 _).<a>le</a>\n        (<a>smul_add_smul_lt_smul_add_smul</a> hfxy hgxy)", [{"full_name": "add_lt_add_of_le_of_lt", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [170, 3], "def_end_pos": [170, 14]}, {"full_name": "Finset.sum_congr", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [376, 3], "def_end_pos": [376, 14]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Eq.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [159, 7], "def_end_pos": [159, 12]}, {"full_name": "smul_add_smul_lt_smul_add_smul", "def_path": "Mathlib/Algebra/Order/Module.lean", "def_pos": [133, 9], "def_end_pos": [133, 39]}]], "state_before": "case refine'_1.intro.intro.intro.intro.intro.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nx : \u03b9\nhx : x \u2208 \u2191s\ny : \u03b9\nhy : y \u2208 \u2191s\nhgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y\nhfxy : f y < f x\n\u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3\nh\u03c4s : {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s\nhxy : x \u2260 y\n\u22a2 \u2211 x in erase (erase s x) y, f x \u2022 g (\u2191\u03c3 x) + (f y \u2022 g (\u2191\u03c3 y) + f x \u2022 g (\u2191\u03c3 x)) <\n    \u2211 x_1 in erase (erase s x) y, f x_1 \u2022 g (\u2191\u03c3 (\u2191(Equiv.swap x y) x_1)) + (f y \u2022 g (\u2191\u03c3 x) + f x \u2022 g (\u2191\u03c3 y))", "state_after": "case refine'_1.intro.intro.intro.intro.intro.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nx : \u03b9\nhx : x \u2208 \u2191s\ny : \u03b9\nhy : y \u2208 \u2191s\nhgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y\nhfxy : f y < f x\n\u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3\nh\u03c4s : {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s\nhxy : x \u2260 y\nz : \u03b9\nhz : z \u2208 erase (erase s x) y\n\u22a2 f z \u2022 g (\u2191\u03c3 z) = f z \u2022 g (\u2191\u03c3 (\u2191(Equiv.swap x y) z))"}, {"tactic": "simp_rw [mem_erase] at hz", "annotated_tactic": ["simp_rw [<a>mem_erase</a>] at hz", [{"full_name": "Finset.mem_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1887, 9], "def_end_pos": [1887, 18]}]], "state_before": "case refine'_1.intro.intro.intro.intro.intro.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nx : \u03b9\nhx : x \u2208 \u2191s\ny : \u03b9\nhy : y \u2208 \u2191s\nhgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y\nhfxy : f y < f x\n\u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3\nh\u03c4s : {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s\nhxy : x \u2260 y\nz : \u03b9\nhz : z \u2208 erase (erase s x) y\n\u22a2 f z \u2022 g (\u2191\u03c3 z) = f z \u2022 g (\u2191\u03c3 (\u2191(Equiv.swap x y) z))", "state_after": "case refine'_1.intro.intro.intro.intro.intro.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nx : \u03b9\nhx : x \u2208 \u2191s\ny : \u03b9\nhy : y \u2208 \u2191s\nhgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y\nhfxy : f y < f x\n\u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3\nh\u03c4s : {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s\nhxy : x \u2260 y\nz : \u03b9\nhz : z \u2260 y \u2227 z \u2260 x \u2227 z \u2208 s\n\u22a2 f z \u2022 g (\u2191\u03c3 z) = f z \u2022 g (\u2191\u03c3 (\u2191(Equiv.swap x y) z))"}, {"tactic": "rw [swap_apply_of_ne_of_ne hz.2.1 hz.1]", "annotated_tactic": ["rw [<a>swap_apply_of_ne_of_ne</a> hz.2.1 hz.1]", [{"full_name": "Equiv.swap_apply_of_ne_of_ne", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1650, 9], "def_end_pos": [1650, 31]}]], "state_before": "case refine'_1.intro.intro.intro.intro.intro.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nx : \u03b9\nhx : x \u2208 \u2191s\ny : \u03b9\nhy : y \u2208 \u2191s\nhgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y\nhfxy : f y < f x\n\u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3\nh\u03c4s : {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s\nhxy : x \u2260 y\nz : \u03b9\nhz : z \u2260 y \u2227 z \u2260 x \u2227 z \u2208 s\n\u22a2 f z \u2022 g (\u2191\u03c3 z) = f z \u2022 g (\u2191\u03c3 (\u2191(Equiv.swap x y) z))", "state_after": "no goals"}, {"tactic": "refine' (set_support_mul_subset \u03c3 <| swap x y).trans (Set.union_subset h\u03c3 fun z hz \u21a6 _)", "annotated_tactic": ["refine' (<a>set_support_mul_subset</a> \u03c3 <| <a>swap</a> x y).<a>trans</a> (<a>Set.union_subset</a> h\u03c3 fun z hz \u21a6 _)", [{"full_name": "Equiv.Perm.set_support_mul_subset", "def_path": "Mathlib/GroupTheory/Perm/Support.lean", "def_pos": [274, 9], "def_end_pos": [274, 31]}, {"full_name": "Equiv.swap", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1622, 5], "def_end_pos": [1622, 9]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [664, 7], "def_end_pos": [664, 29]}, {"full_name": "Set.union_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [836, 9], "def_end_pos": [836, 21]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nx : \u03b9\nhx : x \u2208 \u2191s\ny : \u03b9\nhy : y \u2208 \u2191s\nhgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y\nhfxy : f y < f x\n\u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3\n\u22a2 {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nx : \u03b9\nhx : x \u2208 \u2191s\ny : \u03b9\nhy : y \u2208 \u2191s\nhgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y\nhfxy : f y < f x\n\u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3\nz : \u03b9\nhz : z \u2208 {x_1 | \u2191(Equiv.swap x y) x_1 \u2260 x_1}\n\u22a2 z \u2208 \u2191s"}, {"tactic": "obtain \u27e8_, rfl | rfl\u27e9 := swap_apply_ne_self_iff.1 hz <;> assumption", "annotated_tactic": ["obtain \u27e8_, rfl | rfl\u27e9 := <a>swap_apply_ne_self_iff</a>.1 hz <;> assumption", [{"full_name": "Equiv.swap_apply_ne_self_iff", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1723, 9], "def_end_pos": [1723, 31]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nx : \u03b9\nhx : x \u2208 \u2191s\ny : \u03b9\nhy : y \u2208 \u2191s\nhgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) y\nhfxy : f y < f x\n\u03c4 : Perm \u03b9 := (Equiv.swap x y).trans \u03c3\nz : \u03b9\nhz : z \u2208 {x_1 | \u2191(Equiv.swap x y) x_1 \u2260 x_1}\n\u22a2 z \u2208 \u2191s", "state_after": "no goals"}, {"tactic": "cases lt_irrefl _ hfxy", "annotated_tactic": ["cases <a>lt_irrefl</a> _ hfxy", [{"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}]], "state_before": "case refine'_1.intro.intro.intro.intro.intro.inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nx : \u03b9\nhx hy : x \u2208 \u2191s\nhgxy : (g \u2218 \u2191\u03c3) x < (g \u2218 \u2191\u03c3) x\nhfxy : f x < f x\n\u03c4 : Perm \u03b9 := (Equiv.swap x x).trans \u03c3\nh\u03c4s : {x | \u2191\u03c4 x \u2260 x} \u2286 \u2191s\n\u22a2 \u2211 i in s, f i \u2022 g (\u2191\u03c3 i) < \u2211 i in s, f i \u2022 g (\u2191\u03c4 i)", "state_after": "no goals"}, {"tactic": "convert h.sum_smul_comp_perm_le_sum_smul ((set_support_inv_eq _).subset.trans h\u03c3) using 1", "annotated_tactic": ["convert h.sum_smul_comp_perm_le_sum_smul ((<a>set_support_inv_eq</a> _).subset.trans h\u03c3) using 1", [{"full_name": "Equiv.Perm.set_support_inv_eq", "def_path": "Mathlib/GroupTheory/Perm/Support.lean", "def_pos": [257, 9], "def_end_pos": [257, 27]}]], "state_before": "case refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nh : MonovaryOn f (g \u2218 \u2191\u03c3) \u2191s\n\u22a2 \u2211 i in s, f i \u2022 g i \u2264 \u2211 i in s, f i \u2022 g (\u2191\u03c3 i)", "state_after": "case h.e'_3\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nh : MonovaryOn f (g \u2218 \u2191\u03c3) \u2191s\n\u22a2 \u2211 i in s, f i \u2022 g i = \u2211 i in s, f i \u2022 (g \u2218 \u2191\u03c3) (\u2191\u03c3\u207b\u00b9 i)"}, {"tactic": "simp_rw [Function.comp_apply, apply_inv_self]", "annotated_tactic": ["simp_rw [<a>Function.comp_apply</a>, <a>apply_inv_self</a>]", [{"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "Equiv.Perm.apply_inv_self", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [81, 9], "def_end_pos": [81, 23]}]], "state_before": "case h.e'_3\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\nh\u03c3 : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\nh : MonovaryOn f (g \u2218 \u2191\u03c3) \u2191s\n\u22a2 \u2211 i in s, f i \u2022 g i = \u2211 i in s, f i \u2022 (g \u2218 \u2191\u03c3) (\u2191\u03c3\u207b\u00b9 i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/StarSubalgebra.lean", "full_name": "Subalgebra.topologicalClosure_star_comm", "start": [126, 1], "end": [131, 48], "traced_tactics": [{"tactic": "suffices \u2200 t : Subalgebra R A, (star t).topologicalClosure \u2264 star t.topologicalClosure from\n  le_antisymm (this s) (by simpa only [star_star] using Subalgebra.star_mono (this (star s)))", "annotated_tactic": ["suffices \u2200 t : <a>Subalgebra</a> R A, (<a>star</a> t).<a>topologicalClosure</a> \u2264 <a>star</a> t.topologicalClosure from\n    <a>le_antisymm</a> (this s) (by simpa only [<a>star_star</a>] using <a>Subalgebra.star_mono</a> (this (<a>star</a> s)))", [{"full_name": "Subalgebra", "def_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "def_pos": [27, 11], "def_end_pos": [27, 21]}, {"full_name": "Star.star", "def_path": "Mathlib/Algebra/Star/Basic.lean", "def_pos": [43, 3], "def_end_pos": [43, 7]}, {"full_name": "Subalgebra.topologicalClosure", "def_path": "Mathlib/Topology/Algebra/Algebra.lean", "def_pos": [96, 5], "def_end_pos": [96, 34]}, {"full_name": "Star.star", "def_path": "Mathlib/Algebra/Star/Basic.lean", "def_pos": [43, 3], "def_end_pos": [43, 7]}, {"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "star_star", "def_path": "Mathlib/Algebra/Star/Basic.lean", "def_pos": [86, 9], "def_end_pos": [86, 18]}, {"full_name": "Subalgebra.star_mono", "def_path": "Mathlib/Algebra/Star/Subalgebra.lean", "def_pos": [373, 9], "def_end_pos": [373, 18]}, {"full_name": "Star.star", "def_path": "Mathlib/Algebra/Star/Basic.lean", "def_pos": [43, 3], "def_end_pos": [43, 7]}]], "state_before": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : StarRing R\ninst\u271d\u00b9\u2070 : TopologicalSpace A\ninst\u271d\u2079 : Semiring A\ninst\u271d\u2078 : Algebra R A\ninst\u271d\u2077 : StarRing A\ninst\u271d\u2076 : StarModule R A\ninst\u271d\u2075 : TopologicalSemiring A\ninst\u271d\u2074 : ContinuousStar A\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R B\ninst\u271d : StarRing B\ns : Subalgebra R A\n\u22a2 Subalgebra.topologicalClosure (star s) = star (Subalgebra.topologicalClosure s)", "state_after": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : StarRing R\ninst\u271d\u00b9\u2070 : TopologicalSpace A\ninst\u271d\u2079 : Semiring A\ninst\u271d\u2078 : Algebra R A\ninst\u271d\u2077 : StarRing A\ninst\u271d\u2076 : StarModule R A\ninst\u271d\u2075 : TopologicalSemiring A\ninst\u271d\u2074 : ContinuousStar A\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R B\ninst\u271d : StarRing B\ns : Subalgebra R A\n\u22a2 \u2200 (t : Subalgebra R A), Subalgebra.topologicalClosure (star t) \u2264 star (Subalgebra.topologicalClosure t)"}, {"tactic": "exact fun t => (star t).topologicalClosure_minimal (Subalgebra.star_mono subset_closure)\n  (isClosed_closure.preimage continuous_star)", "annotated_tactic": ["exact fun t => (<a>star</a> t).<a>topologicalClosure_minimal</a> (<a>Subalgebra.star_mono</a> <a>subset_closure</a>)\n    (isClosed_closure.preimage <a>continuous_star</a>)", [{"full_name": "Star.star", "def_path": "Mathlib/Algebra/Star/Basic.lean", "def_pos": [43, 3], "def_end_pos": [43, 7]}, {"full_name": "Subalgebra.topologicalClosure_minimal", "def_path": "Mathlib/Topology/Algebra/Algebra.lean", "def_pos": [121, 9], "def_end_pos": [121, 46]}, {"full_name": "Subalgebra.star_mono", "def_path": "Mathlib/Algebra/Star/Subalgebra.lean", "def_pos": [373, 9], "def_end_pos": [373, 18]}, {"full_name": "subset_closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [435, 9], "def_end_pos": [435, 23]}, {"full_name": "ContinuousStar.continuous_star", "def_path": "Mathlib/Topology/Algebra/Star.lean", "def_pos": [27, 3], "def_end_pos": [27, 18]}]], "state_before": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : StarRing R\ninst\u271d\u00b9\u2070 : TopologicalSpace A\ninst\u271d\u2079 : Semiring A\ninst\u271d\u2078 : Algebra R A\ninst\u271d\u2077 : StarRing A\ninst\u271d\u2076 : StarModule R A\ninst\u271d\u2075 : TopologicalSemiring A\ninst\u271d\u2074 : ContinuousStar A\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R B\ninst\u271d : StarRing B\ns : Subalgebra R A\n\u22a2 \u2200 (t : Subalgebra R A), Subalgebra.topologicalClosure (star t) \u2264 star (Subalgebra.topologicalClosure t)", "state_after": "no goals"}, {"tactic": "simpa only [star_star] using Subalgebra.star_mono (this (star s))", "annotated_tactic": ["simpa only [<a>star_star</a>] using <a>Subalgebra.star_mono</a> (this (<a>star</a> s))", [{"full_name": "star_star", "def_path": "Mathlib/Algebra/Star/Basic.lean", "def_pos": [86, 9], "def_end_pos": [86, 18]}, {"full_name": "Subalgebra.star_mono", "def_path": "Mathlib/Algebra/Star/Subalgebra.lean", "def_pos": [373, 9], "def_end_pos": [373, 18]}, {"full_name": "Star.star", "def_path": "Mathlib/Algebra/Star/Basic.lean", "def_pos": [43, 3], "def_end_pos": [43, 7]}]], "state_before": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : StarRing R\ninst\u271d\u00b9\u2070 : TopologicalSpace A\ninst\u271d\u2079 : Semiring A\ninst\u271d\u2078 : Algebra R A\ninst\u271d\u2077 : StarRing A\ninst\u271d\u2076 : StarModule R A\ninst\u271d\u2075 : TopologicalSemiring A\ninst\u271d\u2074 : ContinuousStar A\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R B\ninst\u271d : StarRing B\ns : Subalgebra R A\nthis : \u2200 (t : Subalgebra R A), Subalgebra.topologicalClosure (star t) \u2264 star (Subalgebra.topologicalClosure t)\n\u22a2 star (Subalgebra.topologicalClosure s) \u2264 Subalgebra.topologicalClosure (star s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/BoxIntegral/Partition/Basic.lean", "full_name": "BoxIntegral.Prepartition.le_biUnionIndex", "start": [375, 1], "end": [376, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Basic.lean", "full_name": "Metric.subsingleton_closedBall", "start": [2974, 1], "end": [2979, 33], "traced_tactics": [{"tactic": "rcases hr.lt_or_eq with (hr | rfl)", "annotated_tactic": ["rcases hr.lt_or_eq with (hr | rfl)", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx\u271d : \u03b3\ns : Set \u03b3\nx : \u03b3\nr : \u211d\nhr : r \u2264 0\n\u22a2 Set.Subsingleton (closedBall x r)", "state_after": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx\u271d : \u03b3\ns : Set \u03b3\nx : \u03b3\nr : \u211d\nhr\u271d : r \u2264 0\nhr : r < 0\n\u22a2 Set.Subsingleton (closedBall x r)\n\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx\u271d : \u03b3\ns : Set \u03b3\nx : \u03b3\nhr : 0 \u2264 0\n\u22a2 Set.Subsingleton (closedBall x 0)"}, {"tactic": "rw [closedBall_eq_empty.2 hr]", "annotated_tactic": ["rw [<a>closedBall_eq_empty</a>.2 hr]", [{"full_name": "Metric.closedBall_eq_empty", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [525, 9], "def_end_pos": [525, 28]}]], "state_before": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx\u271d : \u03b3\ns : Set \u03b3\nx : \u03b3\nr : \u211d\nhr\u271d : r \u2264 0\nhr : r < 0\n\u22a2 Set.Subsingleton (closedBall x r)", "state_after": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx\u271d : \u03b3\ns : Set \u03b3\nx : \u03b3\nr : \u211d\nhr\u271d : r \u2264 0\nhr : r < 0\n\u22a2 Set.Subsingleton \u2205"}, {"tactic": "exact subsingleton_empty", "annotated_tactic": ["exact <a>subsingleton_empty</a>", [{"full_name": "Set.subsingleton_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2369, 9], "def_end_pos": [2369, 27]}]], "state_before": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx\u271d : \u03b3\ns : Set \u03b3\nx : \u03b3\nr : \u211d\nhr\u271d : r \u2264 0\nhr : r < 0\n\u22a2 Set.Subsingleton \u2205", "state_after": "no goals"}, {"tactic": "rw [closedBall_zero]", "annotated_tactic": ["rw [<a>closedBall_zero</a>]", [{"full_name": "Metric.closedBall_zero", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2968, 17], "def_end_pos": [2968, 32]}]], "state_before": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx\u271d : \u03b3\ns : Set \u03b3\nx : \u03b3\nhr : 0 \u2264 0\n\u22a2 Set.Subsingleton (closedBall x 0)", "state_after": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx\u271d : \u03b3\ns : Set \u03b3\nx : \u03b3\nhr : 0 \u2264 0\n\u22a2 Set.Subsingleton {x}"}, {"tactic": "exact subsingleton_singleton", "annotated_tactic": ["exact <a>subsingleton_singleton</a>", [{"full_name": "Set.subsingleton_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2373, 9], "def_end_pos": [2373, 31]}]], "state_before": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b3 : Type w\ninst\u271d : MetricSpace \u03b3\nx\u271d : \u03b3\ns : Set \u03b3\nx : \u03b3\nhr : 0 \u2264 0\n\u22a2 Set.Subsingleton {x}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Martingale/Basic.lean", "full_name": "MeasureTheory.Submartingale.sum_mul_sub'", "start": [545, 1], "end": [549, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/FLT/Four.lean", "full_name": "Int.coprime_of_sq_sum", "start": [154, 1], "end": [156, 55], "traced_tactics": [{"tactic": "rw [sq, sq]", "annotated_tactic": ["rw [<a>sq</a>, <a>sq</a>]", [{"full_name": "sq", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [106, 7], "def_end_pos": [106, 9]}, {"full_name": "sq", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [106, 7], "def_end_pos": [106, 9]}]], "state_before": "r s : \u2124\nh2 : IsCoprime s r\n\u22a2 IsCoprime (r ^ 2 + s ^ 2) r", "state_after": "r s : \u2124\nh2 : IsCoprime s r\n\u22a2 IsCoprime (r * r + s * s) r"}, {"tactic": "exact (IsCoprime.mul_left h2 h2).mul_add_left_left r", "annotated_tactic": ["exact (<a>IsCoprime.mul_left</a> h2 h2).<a>mul_add_left_left</a> r", [{"full_name": "IsCoprime.mul_left", "def_path": "Mathlib/RingTheory/Coprime/Basic.lean", "def_pos": [112, 9], "def_end_pos": [112, 27]}, {"full_name": "IsCoprime.mul_add_left_left", "def_path": "Mathlib/RingTheory/Coprime/Basic.lean", "def_pos": [313, 9], "def_end_pos": [313, 26]}]], "state_before": "r s : \u2124\nh2 : IsCoprime s r\n\u22a2 IsCoprime (r * r + s * s) r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/ContDiff.lean", "full_name": "iteratedFDeriv_add_apply'", "start": [1280, 1], "end": [1283, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/QuotientGroup.lean", "full_name": "QuotientGroup.homQuotientZPowOfHom_comp_of_rightInverse", "start": [532, 1], "end": [534, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "full_name": "SimpleGraph.Reachable.trans", "start": [1966, 11], "end": [1968, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/SMulWithZero.lean", "full_name": "zero_smul", "start": [70, 1], "end": [71, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Ring/Equiv.lean", "full_name": "RingEquiv.toMonoidHom_commutes", "start": [734, 1], "end": [736, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "full_name": "Rat.neg_mkRat", "start": [217, 1], "end": [218, 79], "traced_tactics": [{"tactic": "if z : d = 0 then simp [z] else simp [\u2190 normalize_eq_mkRat z, neg_normalize]", "annotated_tactic": ["if z : d = 0 then simp [z] else simp [\u2190 <a>normalize_eq_mkRat</a> z, <a>neg_normalize</a>]", [{"full_name": "Rat.normalize_eq_mkRat", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [87, 9], "def_end_pos": [87, 27]}, {"full_name": "Rat.neg_normalize", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [214, 9], "def_end_pos": [214, 22]}]], "state_before": "n : Int\nd : Nat\n\u22a2 -mkRat n d = mkRat (-n) d", "state_after": "no goals"}, {"tactic": "simp [z]", "annotated_tactic": ["simp [z]", []], "state_before": "n : Int\nd : Nat\nz : d = 0\n\u22a2 -mkRat n d = mkRat (-n) d", "state_after": "no goals"}, {"tactic": "simp [\u2190 normalize_eq_mkRat z, neg_normalize]", "annotated_tactic": ["simp [\u2190 <a>normalize_eq_mkRat</a> z, <a>neg_normalize</a>]", [{"full_name": "Rat.normalize_eq_mkRat", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [87, 9], "def_end_pos": [87, 27]}, {"full_name": "Rat.neg_normalize", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [214, 9], "def_end_pos": [214, 22]}]], "state_before": "n : Int\nd : Nat\nz : \u00acd = 0\n\u22a2 -mkRat n d = mkRat (-n) d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Field/Basic.lean", "full_name": "le_of_neg_of_one_div_le_one_div", "start": [872, 1], "end": [873, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/LocalProperties.lean", "full_name": "IsLocalization.lift_mem_adjoin_finsetIntegerMultiple", "start": [600, 1], "end": [610, 59], "traced_tactics": [{"tactic": "obtain \u27e8\u27e8_, a, ha, rfl\u27e9, e\u27e9 :=\n  IsLocalization.exists_smul_mem_of_mem_adjoin (M.map (algebraMap R S)) x s (Algebra.adjoin R _)\n    Algebra.subset_adjoin (by rintro _ \u27e8a, _, rfl\u27e9; exact Subalgebra.algebraMap_mem _ a) hx", "annotated_tactic": ["obtain \u27e8\u27e8_, a, ha, rfl\u27e9, e\u27e9 :=\n    <a>IsLocalization.exists_smul_mem_of_mem_adjoin</a> (M.map (<a>algebraMap</a> R S)) x s (<a>Algebra.adjoin</a> R _)\n      <a>Algebra.subset_adjoin</a> (by rintro _ \u27e8a, _, rfl\u27e9; exact <a>Subalgebra.algebraMap_mem</a> _ a) hx", [{"full_name": "IsLocalization.exists_smul_mem_of_mem_adjoin", "def_path": "Mathlib/RingTheory/LocalProperties.lean", "def_pos": [574, 9], "def_end_pos": [574, 53]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "Algebra.adjoin", "def_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "def_pos": [765, 5], "def_end_pos": [765, 11]}, {"full_name": "Algebra.subset_adjoin", "def_path": "Mathlib/RingTheory/Adjoin/Basic.lean", "def_pos": [44, 9], "def_end_pos": [44, 22]}, {"full_name": "Subalgebra.algebraMap_mem", "def_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "def_pos": [113, 19], "def_end_pos": [113, 33]}]], "state_before": "R S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\n\u22a2 \u2203 m, m \u2022 x \u2208 Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)", "state_after": "case intro.mk.intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\na : R\nha : a \u2208 \u2191M\ne :\n  { val := \u2191(algebraMap R S) a, property := (_ : \u2203 a_1, a_1 \u2208 \u2191M \u2227 \u2191(algebraMap R S) a_1 = \u2191(algebraMap R S) a) } \u2022 x \u2208\n    Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n\u22a2 \u2203 m, m \u2022 x \u2208 Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)"}, {"tactic": "refine' \u27e8\u27e8a, ha\u27e9, _\u27e9", "annotated_tactic": ["refine' \u27e8\u27e8a, ha\u27e9, _\u27e9", []], "state_before": "case intro.mk.intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\na : R\nha : a \u2208 \u2191M\ne :\n  { val := \u2191(algebraMap R S) a, property := (_ : \u2203 a_1, a_1 \u2208 \u2191M \u2227 \u2191(algebraMap R S) a_1 = \u2191(algebraMap R S) a) } \u2022 x \u2208\n    Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n\u22a2 \u2203 m, m \u2022 x \u2208 Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)", "state_after": "case intro.mk.intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\na : R\nha : a \u2208 \u2191M\ne :\n  { val := \u2191(algebraMap R S) a, property := (_ : \u2203 a_1, a_1 \u2208 \u2191M \u2227 \u2191(algebraMap R S) a_1 = \u2191(algebraMap R S) a) } \u2022 x \u2208\n    Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n\u22a2 { val := a, property := ha } \u2022 x \u2208 Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)"}, {"tactic": "simpa only [Submonoid.smul_def, algebraMap_smul] using e", "annotated_tactic": ["simpa only [<a>Submonoid.smul_def</a>, <a>algebraMap_smul</a>] using e", [{"full_name": "Submonoid.smul_def", "def_path": "Mathlib/GroupTheory/Submonoid/Operations.lean", "def_pos": [1529, 9], "def_end_pos": [1529, 17]}, {"full_name": "algebraMap_smul", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [835, 9], "def_end_pos": [835, 24]}]], "state_before": "case intro.mk.intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\na : R\nha : a \u2208 \u2191M\ne :\n  { val := \u2191(algebraMap R S) a, property := (_ : \u2203 a_1, a_1 \u2208 \u2191M \u2227 \u2191(algebraMap R S) a_1 = \u2191(algebraMap R S) a) } \u2022 x \u2208\n    Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)\n\u22a2 { val := a, property := ha } \u2022 x \u2208 Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)", "state_after": "no goals"}, {"tactic": "rintro _ \u27e8a, _, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8a, _, rfl\u27e9", []], "state_before": "R S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\n\u22a2 Submonoid.map (algebraMap R S) M \u2264\n    (Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)).toSubsemiring.toSubmonoid", "state_after": "case intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\na : R\nleft\u271d : a \u2208 \u2191M\n\u22a2 \u2191(algebraMap R S) a \u2208\n    (Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)).toSubsemiring.toSubmonoid"}, {"tactic": "exact Subalgebra.algebraMap_mem _ a", "annotated_tactic": ["exact <a>Subalgebra.algebraMap_mem</a> _ a", [{"full_name": "Subalgebra.algebraMap_mem", "def_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "def_pos": [113, 19], "def_end_pos": [113, 33]}]], "state_before": "case intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset S'\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\na : R\nleft\u271d : a \u2208 \u2191M\n\u22a2 \u2191(algebraMap R S) a \u2208\n    (Algebra.adjoin R \u2191(finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s)).toSubsemiring.toSubmonoid", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Quaternion.lean", "full_name": "QuaternionAlgebra.self_add_star", "start": [676, 1], "end": [676, 97], "traced_tactics": [{"tactic": "simp only [self_add_star', two_mul, coe_add]", "annotated_tactic": ["simp only [<a>self_add_star'</a>, <a>two_mul</a>, <a>coe_add</a>]", [{"full_name": "QuaternionAlgebra.self_add_star'", "def_path": "Mathlib/Algebra/Quaternion.lean", "def_pos": [673, 9], "def_end_pos": [673, 23]}, {"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}, {"full_name": "QuaternionAlgebra.coe_add", "def_path": "Mathlib/Algebra/Quaternion.lean", "def_pos": [237, 9], "def_end_pos": [237, 16]}]], "state_before": "S : Type u_1\nT : Type u_2\nR : Type u_3\ninst\u271d : CommRing R\nc\u2081 c\u2082 r x y z : R\na b c : \u210d[R,c\u2081,c\u2082]\n\u22a2 a + star a = 2 * \u2191a.re", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Limits/Final.lean", "full_name": "CategoryTheory.IsCofilteredOrEmpty.of_initial", "start": [862, 1], "end": [865, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/SpecificGroups/Dihedral.lean", "full_name": "DihedralGroup.exponent", "start": [194, 1], "end": [209, 32], "traced_tactics": [{"tactic": "rcases eq_zero_or_neZero n with (rfl | hn)", "annotated_tactic": ["rcases <a>eq_zero_or_neZero</a> n with (rfl | hn)", [{"full_name": "eq_zero_or_neZero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [45, 9], "def_end_pos": [45, 26]}]], "state_before": "n : \u2115\n\u22a2 Monoid.exponent (DihedralGroup n) = lcm n 2", "state_after": "case inl\n\n\u22a2 Monoid.exponent (DihedralGroup 0) = lcm 0 2\n\ncase inr\nn : \u2115\nhn : NeZero n\n\u22a2 Monoid.exponent (DihedralGroup n) = lcm n 2"}, {"tactic": "apply Nat.dvd_antisymm", "annotated_tactic": ["apply <a>Nat.dvd_antisymm</a>", [{"full_name": "Nat.dvd_antisymm", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [918, 19], "def_end_pos": [918, 31]}]], "state_before": "case inr\nn : \u2115\nhn : NeZero n\n\u22a2 Monoid.exponent (DihedralGroup n) = lcm n 2", "state_after": "case inr.a\nn : \u2115\nhn : NeZero n\n\u22a2 Monoid.exponent (DihedralGroup n) \u2223 lcm n 2\n\ncase inr.a\nn : \u2115\nhn : NeZero n\n\u22a2 lcm n 2 \u2223 Monoid.exponent (DihedralGroup n)"}, {"tactic": "exact Monoid.exponent_eq_zero_of_order_zero orderOf_r_one", "annotated_tactic": ["exact <a>Monoid.exponent_eq_zero_of_order_zero</a> <a>orderOf_r_one</a>", [{"full_name": "Monoid.exponent_eq_zero_of_order_zero", "def_path": "Mathlib/GroupTheory/Exponent.lean", "def_pos": [94, 9], "def_end_pos": [94, 39]}, {"full_name": "DihedralGroup.orderOf_r_one", "def_path": "Mathlib/GroupTheory/SpecificGroups/Dihedral.lean", "def_pos": [170, 9], "def_end_pos": [170, 22]}]], "state_before": "case inl\n\n\u22a2 Monoid.exponent (DihedralGroup 0) = lcm 0 2", "state_after": "no goals"}, {"tactic": "apply Monoid.exponent_dvd_of_forall_pow_eq_one", "annotated_tactic": ["apply <a>Monoid.exponent_dvd_of_forall_pow_eq_one</a>", [{"full_name": "Monoid.exponent_dvd_of_forall_pow_eq_one", "def_path": "Mathlib/GroupTheory/Exponent.lean", "def_pos": [163, 9], "def_end_pos": [163, 42]}]], "state_before": "case inr.a\nn : \u2115\nhn : NeZero n\n\u22a2 Monoid.exponent (DihedralGroup n) \u2223 lcm n 2", "state_after": "case inr.a.hG\nn : \u2115\nhn : NeZero n\n\u22a2 \u2200 (g : DihedralGroup n), g ^ lcm n 2 = 1"}, {"tactic": "rintro (m | m)", "annotated_tactic": ["rintro (m | m)", []], "state_before": "case inr.a.hG\nn : \u2115\nhn : NeZero n\n\u22a2 \u2200 (g : DihedralGroup n), g ^ lcm n 2 = 1", "state_after": "case inr.a.hG.r\nn : \u2115\nhn : NeZero n\nm : ZMod n\n\u22a2 r m ^ lcm n 2 = 1\n\ncase inr.a.hG.sr\nn : \u2115\nhn : NeZero n\nm : ZMod n\n\u22a2 sr m ^ lcm n 2 = 1"}, {"tactic": "rw [\u2190 orderOf_dvd_iff_pow_eq_one, orderOf_r]", "annotated_tactic": ["rw [\u2190 <a>orderOf_dvd_iff_pow_eq_one</a>, <a>orderOf_r</a>]", [{"full_name": "orderOf_dvd_iff_pow_eq_one", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [253, 9], "def_end_pos": [253, 35]}, {"full_name": "DihedralGroup.orderOf_r", "def_path": "Mathlib/GroupTheory/SpecificGroups/Dihedral.lean", "def_pos": [189, 9], "def_end_pos": [189, 18]}]], "state_before": "case inr.a.hG.r\nn : \u2115\nhn : NeZero n\nm : ZMod n\n\u22a2 r m ^ lcm n 2 = 1", "state_after": "case inr.a.hG.r\nn : \u2115\nhn : NeZero n\nm : ZMod n\n\u22a2 n / Nat.gcd n (ZMod.val m) \u2223 lcm n 2"}, {"tactic": "refine' Nat.dvd_trans \u27e8gcd n m.val, _\u27e9 (dvd_lcm_left n 2)", "annotated_tactic": ["refine' <a>Nat.dvd_trans</a> \u27e8<a>gcd</a> n m.val, _\u27e9 (<a>dvd_lcm_left</a> n 2)", [{"full_name": "Nat.dvd_trans", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [870, 19], "def_end_pos": [870, 28]}, {"full_name": "GCDMonoid.gcd", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [274, 3], "def_end_pos": [274, 6]}, {"full_name": "dvd_lcm_left", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [733, 9], "def_end_pos": [733, 21]}]], "state_before": "case inr.a.hG.r\nn : \u2115\nhn : NeZero n\nm : ZMod n\n\u22a2 n / Nat.gcd n (ZMod.val m) \u2223 lcm n 2", "state_after": "case inr.a.hG.r\nn : \u2115\nhn : NeZero n\nm : ZMod n\n\u22a2 n = n / Nat.gcd n (ZMod.val m) * gcd n (ZMod.val m)"}, {"tactic": "exact (Nat.div_mul_cancel (Nat.gcd_dvd_left n m.val)).symm", "annotated_tactic": ["exact (<a>Nat.div_mul_cancel</a> (<a>Nat.gcd_dvd_left</a> n m.val)).<a>symm</a>", [{"full_name": "Nat.div_mul_cancel", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [947, 19], "def_end_pos": [947, 33]}, {"full_name": "Nat.gcd_dvd_left", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [42, 9], "def_end_pos": [42, 21]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case inr.a.hG.r\nn : \u2115\nhn : NeZero n\nm : ZMod n\n\u22a2 n = n / Nat.gcd n (ZMod.val m) * gcd n (ZMod.val m)", "state_after": "no goals"}, {"tactic": "rw [\u2190 orderOf_dvd_iff_pow_eq_one, orderOf_sr]", "annotated_tactic": ["rw [\u2190 <a>orderOf_dvd_iff_pow_eq_one</a>, <a>orderOf_sr</a>]", [{"full_name": "orderOf_dvd_iff_pow_eq_one", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [253, 9], "def_end_pos": [253, 35]}, {"full_name": "DihedralGroup.orderOf_sr", "def_path": "Mathlib/GroupTheory/SpecificGroups/Dihedral.lean", "def_pos": [159, 9], "def_end_pos": [159, 19]}]], "state_before": "case inr.a.hG.sr\nn : \u2115\nhn : NeZero n\nm : ZMod n\n\u22a2 sr m ^ lcm n 2 = 1", "state_after": "case inr.a.hG.sr\nn : \u2115\nhn : NeZero n\nm : ZMod n\n\u22a2 2 \u2223 lcm n 2"}, {"tactic": "exact dvd_lcm_right n 2", "annotated_tactic": ["exact <a>dvd_lcm_right</a> n 2", [{"full_name": "dvd_lcm_right", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [737, 9], "def_end_pos": [737, 22]}]], "state_before": "case inr.a.hG.sr\nn : \u2115\nhn : NeZero n\nm : ZMod n\n\u22a2 2 \u2223 lcm n 2", "state_after": "no goals"}, {"tactic": "apply lcm_dvd", "annotated_tactic": ["apply <a>lcm_dvd</a>", [{"full_name": "lcm_dvd", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [741, 9], "def_end_pos": [741, 16]}]], "state_before": "case inr.a\nn : \u2115\nhn : NeZero n\n\u22a2 lcm n 2 \u2223 Monoid.exponent (DihedralGroup n)", "state_after": "case inr.a.hab\nn : \u2115\nhn : NeZero n\n\u22a2 n \u2223 Monoid.exponent (DihedralGroup n)\n\ncase inr.a.hcb\nn : \u2115\nhn : NeZero n\n\u22a2 2 \u2223 Monoid.exponent (DihedralGroup n)"}, {"tactic": "convert Monoid.order_dvd_exponent (r (1 : ZMod n))", "annotated_tactic": ["convert <a>Monoid.order_dvd_exponent</a> (<a>r</a> (1 : <a>ZMod</a> n))", [{"full_name": "Monoid.order_dvd_exponent", "def_path": "Mathlib/GroupTheory/Exponent.lean", "def_pos": [155, 9], "def_end_pos": [155, 27]}, {"full_name": "DihedralGroup.r", "def_path": "Mathlib/GroupTheory/SpecificGroups/Dihedral.lean", "def_pos": [27, 5], "def_end_pos": [27, 6]}, {"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [94, 5], "def_end_pos": [94, 9]}]], "state_before": "case inr.a.hab\nn : \u2115\nhn : NeZero n\n\u22a2 n \u2223 Monoid.exponent (DihedralGroup n)", "state_after": "case h.e'_3\nn : \u2115\nhn : NeZero n\n\u22a2 n = orderOf (r 1)"}, {"tactic": "exact orderOf_r_one.symm", "annotated_tactic": ["exact orderOf_r_one.symm", []], "state_before": "case h.e'_3\nn : \u2115\nhn : NeZero n\n\u22a2 n = orderOf (r 1)", "state_after": "no goals"}, {"tactic": "convert Monoid.order_dvd_exponent (sr (0 : ZMod n))", "annotated_tactic": ["convert <a>Monoid.order_dvd_exponent</a> (<a>sr</a> (0 : <a>ZMod</a> n))", [{"full_name": "Monoid.order_dvd_exponent", "def_path": "Mathlib/GroupTheory/Exponent.lean", "def_pos": [155, 9], "def_end_pos": [155, 27]}, {"full_name": "DihedralGroup.sr", "def_path": "Mathlib/GroupTheory/SpecificGroups/Dihedral.lean", "def_pos": [28, 5], "def_end_pos": [28, 7]}, {"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [94, 5], "def_end_pos": [94, 9]}]], "state_before": "case inr.a.hcb\nn : \u2115\nhn : NeZero n\n\u22a2 2 \u2223 Monoid.exponent (DihedralGroup n)", "state_after": "case h.e'_3\nn : \u2115\nhn : NeZero n\n\u22a2 2 = orderOf (sr 0)"}, {"tactic": "exact (orderOf_sr 0).symm", "annotated_tactic": ["exact (<a>orderOf_sr</a> 0).<a>symm</a>", [{"full_name": "DihedralGroup.orderOf_sr", "def_path": "Mathlib/GroupTheory/SpecificGroups/Dihedral.lean", "def_pos": [159, 9], "def_end_pos": [159, 19]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case h.e'_3\nn : \u2115\nhn : NeZero n\n\u22a2 2 = orderOf (sr 0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Normed/Group/AddCircle.lean", "full_name": "AddCircle.le_add_order_smul_norm_of_isOfFinAddOrder", "start": [261, 1], "end": [271, 77], "traced_tactics": [{"tactic": "obtain \u27e8n, hn\u27e9 := exists_norm_eq_of_isOfFinAddOrder hu", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 := <a>exists_norm_eq_of_isOfFinAddOrder</a> hu", [{"full_name": "AddCircle.exists_norm_eq_of_isOfFinAddOrder", "def_path": "Mathlib/Analysis/Normed/Group/AddCircle.lean", "def_pos": [252, 9], "def_end_pos": [252, 42]}]], "state_before": "p : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu : IsOfFinAddOrder u\nhu' : u \u2260 0\n\u22a2 p \u2264 addOrderOf u \u2022 \u2016u\u2016", "state_after": "case intro\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu : IsOfFinAddOrder u\nhu' : u \u2260 0\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\n\u22a2 p \u2264 addOrderOf u \u2022 \u2016u\u2016"}, {"tactic": "replace hu : (addOrderOf u : \u211d) \u2260 0", "annotated_tactic": ["replace hu : (<a>addOrderOf</a> u : \u211d) \u2260 0", [{"full_name": "addOrderOf", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [133, 3], "def_end_pos": [133, 14]}]], "state_before": "case intro\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu : IsOfFinAddOrder u\nhu' : u \u2260 0\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\n\u22a2 p \u2264 addOrderOf u \u2022 \u2016u\u2016", "state_after": "case hu\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu : IsOfFinAddOrder u\nhu' : u \u2260 0\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\n\u22a2 \u2191(addOrderOf u) \u2260 0\n\ncase intro\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu' : u \u2260 0\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\nhu : \u2191(addOrderOf u) \u2260 0\n\u22a2 p \u2264 addOrderOf u \u2022 \u2016u\u2016"}, {"tactic": "conv_lhs => rw [\u2190 mul_one p]", "annotated_tactic": ["conv_lhs => rw [\u2190 <a>mul_one</a> p]", [{"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "case intro\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu' : u \u2260 0\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\nhu : \u2191(addOrderOf u) \u2260 0\n\u22a2 p \u2264 addOrderOf u \u2022 \u2016u\u2016", "state_after": "case intro\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu' : u \u2260 0\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\nhu : \u2191(addOrderOf u) \u2260 0\n\u22a2 p * 1 \u2264 addOrderOf u \u2022 \u2016u\u2016"}, {"tactic": "rw [hn, nsmul_eq_mul, \u2190 mul_assoc, mul_comm _ p, mul_assoc, mul_div_cancel' _ hu,\n  mul_le_mul_left hp.out, Nat.one_le_cast, Nat.one_le_iff_ne_zero]", "annotated_tactic": ["rw [hn, <a>nsmul_eq_mul</a>, \u2190 <a>mul_assoc</a>, <a>mul_comm</a> _ p, <a>mul_assoc</a>, <a>mul_div_cancel'</a> _ hu,\n    <a>mul_le_mul_left</a> hp.out, <a>Nat.one_le_cast</a>, <a>Nat.one_le_iff_ne_zero</a>]", [{"full_name": "nsmul_eq_mul", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [509, 9], "def_end_pos": [509, 21]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_div_cancel'", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [173, 9], "def_end_pos": [173, 24]}, {"full_name": "mul_le_mul_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [209, 9], "def_end_pos": [209, 24]}, {"full_name": "Nat.one_le_cast", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [105, 9], "def_end_pos": [105, 20]}, {"full_name": "Nat.one_le_iff_ne_zero", "def_path": "Mathlib/Data/Nat/Order/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 27]}]], "state_before": "case intro\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu' : u \u2260 0\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\nhu : \u2191(addOrderOf u) \u2260 0\n\u22a2 p * 1 \u2264 addOrderOf u \u2022 \u2016u\u2016", "state_after": "case intro\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu' : u \u2260 0\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\nhu : \u2191(addOrderOf u) \u2260 0\n\u22a2 n \u2260 0"}, {"tactic": "contrapose! hu'", "annotated_tactic": ["contrapose! hu'", []], "state_before": "case intro\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu' : u \u2260 0\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\nhu : \u2191(addOrderOf u) \u2260 0\n\u22a2 n \u2260 0", "state_after": "case intro\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\nhu : \u2191(addOrderOf u) \u2260 0\nhu' : n = 0\n\u22a2 u = 0"}, {"tactic": "simpa only [hu', Nat.cast_zero, zero_div, mul_zero, norm_eq_zero] using hn", "annotated_tactic": ["simpa only [hu', <a>Nat.cast_zero</a>, <a>zero_div</a>, <a>mul_zero</a>, <a>norm_eq_zero</a>] using hn", [{"full_name": "Nat.cast_zero", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [114, 9], "def_end_pos": [114, 18]}, {"full_name": "zero_div", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [291, 9], "def_end_pos": [291, 17]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}, {"full_name": "norm_eq_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [2018, 30], "def_end_pos": [2018, 42]}]], "state_before": "case intro\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\nhu : \u2191(addOrderOf u) \u2260 0\nhu' : n = 0\n\u22a2 u = 0", "state_after": "no goals"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "case hu\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu : IsOfFinAddOrder u\nhu' : u \u2260 0\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\n\u22a2 \u2191(addOrderOf u) \u2260 0", "state_after": "case hu\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu : IsOfFinAddOrder u\nhu' : u \u2260 0\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\n\u22a2 \u00acaddOrderOf u = 0"}, {"tactic": "exact (addOrderOf_pos_iff.mpr hu).ne'", "annotated_tactic": ["exact (addOrderOf_pos_iff.mpr hu).<a>ne'</a>", [{"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "case hu\np : \u211d\nhp : Fact (0 < p)\nu : AddCircle p\nhu : IsOfFinAddOrder u\nhu' : u \u2260 0\nn : \u2115\nhn : \u2016u\u2016 = p * (\u2191n / \u2191(addOrderOf u))\n\u22a2 \u00acaddOrderOf u = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Variance.lean", "full_name": "ProbabilityTheory.variance_zero", "start": [180, 1], "end": [181, 60], "traced_tactics": [{"tactic": "simp only [variance, evariance_zero, ENNReal.zero_toReal]", "annotated_tactic": ["simp only [<a>variance</a>, <a>evariance_zero</a>, <a>ENNReal.zero_toReal</a>]", [{"full_name": "ProbabilityTheory.variance", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [61, 5], "def_end_pos": [61, 13]}, {"full_name": "ProbabilityTheory.evariance_zero", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [147, 9], "def_end_pos": [147, 23]}, {"full_name": "ENNReal.zero_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [242, 17], "def_end_pos": [242, 28]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc\u271d \u03bc : Measure \u03a9\n\u22a2 variance 0 \u03bc = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "MeasureTheory.Measure.addHaar_ball_mul", "start": [451, 1], "end": [456, 42], "traced_tactics": [{"tactic": "rcases hr.eq_or_lt with (rfl | h)", "annotated_tactic": ["rcases hr.eq_or_lt with (rfl | h)", []], "state_before": "E : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : MeasurableSpace E\ninst\u271d\u2076 : BorelSpace E\ninst\u271d\u2075 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2074 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ns\u271d : Set E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\nhr : 0 \u2264 r\ns : \u211d\n\u22a2 \u2191\u2191\u03bc (ball x (r * s)) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 s)", "state_after": "case inl\nE : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : MeasurableSpace E\ninst\u271d\u2076 : BorelSpace E\ninst\u271d\u2075 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2074 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ns\u271d : Set E\ninst\u271d : Nontrivial E\nx : E\ns : \u211d\nhr : 0 \u2264 0\n\u22a2 \u2191\u2191\u03bc (ball x (0 * s)) = ENNReal.ofReal (0 ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 s)\n\ncase inr\nE : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : MeasurableSpace E\ninst\u271d\u2076 : BorelSpace E\ninst\u271d\u2075 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2074 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ns\u271d : Set E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\nhr : 0 \u2264 r\ns : \u211d\nh : 0 < r\n\u22a2 \u2191\u2191\u03bc (ball x (r * s)) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 s)"}, {"tactic": "simp only [zero_pow (finrank_pos (K := \u211d) (V := E)), measure_empty, zero_mul,\n  ENNReal.ofReal_zero, ball_zero]", "annotated_tactic": ["simp only [<a>zero_pow</a> (<a>finrank_pos</a> (K := \u211d) (V := E)), <a>measure_empty</a>, <a>zero_mul</a>,\n      <a>ENNReal.ofReal_zero</a>, <a>ball_zero</a>]", [{"full_name": "zero_pow", "def_path": "Mathlib/Algebra/GroupPower/Ring.lean", "def_pos": [32, 9], "def_end_pos": [32, 17]}, {"full_name": "FiniteDimensional.finrank_pos", "def_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "def_pos": [334, 9], "def_end_pos": [334, 20]}, {"full_name": "MeasureTheory.measure_empty", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [185, 9], "def_end_pos": [185, 22]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "ENNReal.ofReal_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [245, 17], "def_end_pos": [245, 28]}, {"full_name": "Metric.ball_zero", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [440, 9], "def_end_pos": [440, 18]}]], "state_before": "case inl\nE : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : MeasurableSpace E\ninst\u271d\u2076 : BorelSpace E\ninst\u271d\u2075 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2074 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ns\u271d : Set E\ninst\u271d : Nontrivial E\nx : E\ns : \u211d\nhr : 0 \u2264 0\n\u22a2 \u2191\u2191\u03bc (ball x (0 * s)) = ENNReal.ofReal (0 ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 s)", "state_after": "no goals"}, {"tactic": "exact addHaar_ball_mul_of_pos \u03bc x h s", "annotated_tactic": ["exact <a>addHaar_ball_mul_of_pos</a> \u03bc x h s", [{"full_name": "MeasureTheory.Measure.addHaar_ball_mul_of_pos", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [439, 9], "def_end_pos": [439, 32]}]], "state_before": "case inr\nE : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : MeasurableSpace E\ninst\u271d\u2076 : BorelSpace E\ninst\u271d\u2075 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u2074 : IsAddHaarMeasure \u03bc\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ns\u271d : Set E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\nhr : 0 \u2264 r\ns : \u211d\nh : 0 < r\n\u22a2 \u2191\u2191\u03bc (ball x (r * s)) = ENNReal.ofReal (r ^ finrank \u211d E) * \u2191\u2191\u03bc (ball 0 s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Bounds/Basic.lean", "full_name": "IsLeast.isGLB", "start": [277, 1], "end": [278, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "full_name": "EMetric.diam_eq_zero_iff", "start": [1138, 1], "end": [1139, 94], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Fin/Tuple/Monotone.lean", "full_name": "Antitone.vecCons", "start": [81, 1], "end": [82, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/NAry.lean", "full_name": "Set.image3_congr'", "start": [255, 1], "end": [256, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "full_name": "LatticeOrderedGroup.m_neg_part_def", "start": [156, 1], "end": [157, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Basis.lean", "full_name": "Basis.coord_repr_symm", "start": [262, 1], "end": [263, 58], "traced_tactics": [{"tactic": "simp only [repr_symm_apply, coord_apply, repr_total]", "annotated_tactic": ["simp only [<a>repr_symm_apply</a>, <a>coord_apply</a>, <a>repr_total</a>]", [{"full_name": "Basis.repr_symm_apply", "def_path": "Mathlib/LinearAlgebra/Basis.lean", "def_pos": [157, 9], "def_end_pos": [157, 24]}, {"full_name": "Basis.coord_apply", "def_path": "Mathlib/LinearAlgebra/Basis.lean", "def_pos": [214, 3], "def_end_pos": [214, 9]}, {"full_name": "Basis.repr_total", "def_path": "Mathlib/LinearAlgebra/Basis.lean", "def_pos": [170, 9], "def_end_pos": [170, 19]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nR : Type u_3\nR\u2082 : Type u_4\nK : Type u_5\nM : Type u_6\nM' : Type u_7\nM'' : Type u_8\nV : Type u\nV' : Type u_9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nb\u271d b\u2081 : Basis \u03b9 R M\ni\u271d : \u03b9\nc : R\nx : M\nb : Basis \u03b9 R M\ni : \u03b9\nf : \u03b9 \u2192\u2080 R\n\u22a2 \u2191(coord b i) (\u2191(LinearEquiv.symm b.repr) f) = \u2191f i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.inv_apply", "start": [515, 1], "end": [516, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Basic.lean", "full_name": "Submodule.map_subtype_le", "start": [1548, 1], "end": [1549, 66], "traced_tactics": [{"tactic": "simpa using (map_le_range : map p.subtype p' \u2264 range p.subtype)", "annotated_tactic": ["simpa using (<a>map_le_range</a> : <a>map</a> p.subtype p' \u2264 <a>range</a> p.subtype)", [{"full_name": "LinearMap.map_le_range", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [1130, 9], "def_end_pos": [1130, 21]}, {"full_name": "Submodule.map", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [557, 5], "def_end_pos": [557, 8]}, {"full_name": "LinearMap.range", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [1080, 5], "def_end_pos": [1080, 10]}]], "state_before": "R : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : Semiring R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R\u2082 M\u2082\np p'\u271d : Submodule R M\nq : Submodule R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\np' : Submodule R { x // x \u2208 p }\n\u22a2 map (Submodule.subtype p) p' \u2264 p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Basic.lean", "full_name": "LinearMap.range_toAddSubmonoid", "start": [1088, 1], "end": [1090, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Determinant.lean", "full_name": "LinearMap.detAux_id", "start": [159, 1], "end": [160, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/MinMax.lean", "full_name": "min_lt_iff", "start": [63, 1], "end": [64, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Algebra/Operations.lean", "full_name": "Submodule.mem_div_iff_forall_mul_mem", "start": [687, 1], "end": [688, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/FieldTheory/Finite/Basic.lean", "full_name": "FiniteField.exists_nonsquare", "start": [582, 1], "end": [594, 10], "traced_tactics": [{"tactic": "let sq : F \u2192 F := fun x => x ^ 2", "annotated_tactic": ["let sq : F \u2192 F := fun x => x ^ 2", []], "state_before": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\n\u22a2 \u2203 a, \u00acIsSquare a", "state_after": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\n\u22a2 \u2203 a, \u00acIsSquare a"}, {"tactic": "have h : \u00acFunction.Injective sq := by\n  simp only [Function.Injective, not_forall, exists_prop]\n  refine' \u27e8-1, 1, _, Ring.neg_one_ne_one_of_char_ne_two hF\u27e9\n  simp only [one_pow, neg_one_sq]", "annotated_tactic": ["have h : \u00ac<a>Function.Injective</a> sq := by\n    simp only [<a>Function.Injective</a>, <a>not_forall</a>, <a>exists_prop</a>]\n    refine' \u27e8-1, 1, _, <a>Ring.neg_one_ne_one_of_char_ne_two</a> hF\u27e9\n    simp only [<a>one_pow</a>, <a>neg_one_sq</a>]", [{"full_name": "Function.Injective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [109, 5], "def_end_pos": [109, 14]}, {"full_name": "Function.Injective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [109, 5], "def_end_pos": [109, 14]}, {"full_name": "Classical.not_forall", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [686, 9], "def_end_pos": [686, 19]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "Ring.neg_one_ne_one_of_char_ne_two", "def_path": "Mathlib/Algebra/CharP/Basic.lean", "def_pos": [635, 9], "def_end_pos": [635, 43]}, {"full_name": "one_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [90, 9], "def_end_pos": [90, 16]}, {"full_name": "neg_one_sq", "def_path": "Mathlib/Algebra/GroupPower/Ring.lean", "def_pos": [223, 9], "def_end_pos": [223, 19]}]], "state_before": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\n\u22a2 \u2203 a, \u00acIsSquare a", "state_after": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\nh : \u00acFunction.Injective sq\n\u22a2 \u2203 a, \u00acIsSquare a"}, {"tactic": "rw [Finite.injective_iff_surjective] at h", "annotated_tactic": ["rw [<a>Finite.injective_iff_surjective</a>] at h", [{"full_name": "Finite.injective_iff_surjective", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [647, 9], "def_end_pos": [647, 33]}]], "state_before": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\nh : \u00acFunction.Injective sq\n\u22a2 \u2203 a, \u00acIsSquare a", "state_after": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\nh : \u00acFunction.Surjective sq\n\u22a2 \u2203 a, \u00acIsSquare a"}, {"tactic": "simp_rw [IsSquare, \u2190 pow_two, @eq_comm _ _ (_ ^ 2)]", "annotated_tactic": ["simp_rw [<a>IsSquare</a>, \u2190 <a>pow_two</a>, @<a>eq_comm</a> _ _ (_ ^ 2)]", [{"full_name": "IsSquare", "def_path": "Mathlib/Algebra/Parity.lean", "def_pos": [49, 5], "def_end_pos": [49, 13]}, {"full_name": "pow_two", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [102, 9], "def_end_pos": [102, 16]}, {"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\nh : \u00acFunction.Surjective sq\n\u22a2 \u2203 a, \u00acIsSquare a", "state_after": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\nh : \u00acFunction.Surjective sq\n\u22a2 \u2203 a, \u00ac\u2203 r, r ^ 2 = a"}, {"tactic": "unfold Function.Surjective at h", "annotated_tactic": ["unfold <a>Function.Surjective</a> at h", [{"full_name": "Function.Surjective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [119, 5], "def_end_pos": [119, 15]}]], "state_before": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\nh : \u00acFunction.Surjective sq\n\u22a2 \u2203 a, \u00ac\u2203 r, r ^ 2 = a", "state_after": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\nh : \u00ac\u2200 (b : F), \u2203 a, sq a = b\n\u22a2 \u2203 a, \u00ac\u2203 r, r ^ 2 = a"}, {"tactic": "push_neg at h \u22a2", "annotated_tactic": ["push_neg at h \u22a2", []], "state_before": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\nh : \u00ac\u2200 (b : F), \u2203 a, sq a = b\n\u22a2 \u2203 a, \u00ac\u2203 r, r ^ 2 = a", "state_after": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\nh : \u2203 b, \u2200 (a : F), sq a \u2260 b\n\u22a2 \u2203 a, \u2200 (r : F), r ^ 2 \u2260 a"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\nh : \u2203 b, \u2200 (a : F), sq a \u2260 b\n\u22a2 \u2203 a, \u2200 (r : F), r ^ 2 \u2260 a", "state_after": "no goals"}, {"tactic": "simp only [Function.Injective, not_forall, exists_prop]", "annotated_tactic": ["simp only [<a>Function.Injective</a>, <a>not_forall</a>, <a>exists_prop</a>]", [{"full_name": "Function.Injective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [109, 5], "def_end_pos": [109, 14]}, {"full_name": "Classical.not_forall", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [686, 9], "def_end_pos": [686, 19]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}]], "state_before": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\n\u22a2 \u00acFunction.Injective sq", "state_after": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\n\u22a2 \u2203 x x_1, x ^ 2 = x_1 ^ 2 \u2227 \u00acx = x_1"}, {"tactic": "refine' \u27e8-1, 1, _, Ring.neg_one_ne_one_of_char_ne_two hF\u27e9", "annotated_tactic": ["refine' \u27e8-1, 1, _, <a>Ring.neg_one_ne_one_of_char_ne_two</a> hF\u27e9", [{"full_name": "Ring.neg_one_ne_one_of_char_ne_two", "def_path": "Mathlib/Algebra/CharP/Basic.lean", "def_pos": [635, 9], "def_end_pos": [635, 43]}]], "state_before": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\n\u22a2 \u2203 x x_1, x ^ 2 = x_1 ^ 2 \u2227 \u00acx = x_1", "state_after": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\n\u22a2 (-1) ^ 2 = 1 ^ 2"}, {"tactic": "simp only [one_pow, neg_one_sq]", "annotated_tactic": ["simp only [<a>one_pow</a>, <a>neg_one_sq</a>]", [{"full_name": "one_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [90, 9], "def_end_pos": [90, 16]}, {"full_name": "neg_one_sq", "def_path": "Mathlib/Algebra/GroupPower/Ring.lean", "def_pos": [223, 9], "def_end_pos": [223, 19]}]], "state_before": "K : Type u_1\nR : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : Finite F\nhF : ringChar F \u2260 2\nsq : F \u2192 F := fun x => x ^ 2\n\u22a2 (-1) ^ 2 = 1 ^ 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.val_one_eq_one_mod", "start": [607, 1], "end": [608, 36], "traced_tactics": [{"tactic": "rw [\u2190 Nat.cast_one, val_nat_cast]", "annotated_tactic": ["rw [\u2190 <a>Nat.cast_one</a>, <a>val_nat_cast</a>]", [{"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [141, 9], "def_end_pos": [141, 17]}, {"full_name": "ZMod.val_nat_cast", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 21]}]], "state_before": "n : \u2115\n\u22a2 val 1 = 1 % n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/ContinuedFractions/Computation/ApproximationCorollaries.lean", "full_name": "GeneralizedContinuedFraction.of_convergence_epsilon", "start": [98, 1], "end": [147, 75], "traced_tactics": [{"tactic": "intro \u03b5 \u03b5_pos", "annotated_tactic": ["intro \u03b5 \u03b5_pos", []], "state_before": "K : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u22a2 \u2200 (\u03b5 : K), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 |v - convergents (of v) n| < \u03b5", "state_after": "K : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 |v - convergents (of v) n| < \u03b5"}, {"tactic": "rcases (exists_nat_gt (1 / \u03b5) : \u2203 N' : \u2115, 1 / \u03b5 < N') with \u27e8N', one_div_\u03b5_lt_N'\u27e9", "annotated_tactic": ["rcases (<a>exists_nat_gt</a> (1 / \u03b5) : \u2203 N' : \u2115, 1 / \u03b5 < N') with \u27e8N', one_div_\u03b5_lt_N'\u27e9", [{"full_name": "exists_nat_gt", "def_path": "Mathlib/Algebra/Order/Archimedean.lean", "def_pos": [118, 9], "def_end_pos": [118, 22]}]], "state_before": "K : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 |v - convergents (of v) n| < \u03b5", "state_after": "case intro\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 |v - convergents (of v) n| < \u03b5"}, {"tactic": "let N := max N' 5", "annotated_tactic": ["let N := <a>max</a> N' 5", [{"full_name": "Max.max", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1090, 3], "def_end_pos": [1090, 6]}]], "state_before": "case intro\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 |v - convergents (of v) n| < \u03b5", "state_after": "case intro\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 |v - convergents (of v) n| < \u03b5"}, {"tactic": "exists N", "annotated_tactic": ["exists N", []], "state_before": "case intro\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 |v - convergents (of v) n| < \u03b5", "state_after": "case intro\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\n\u22a2 \u2200 (n : \u2115), n \u2265 N \u2192 |v - convergents (of v) n| < \u03b5"}, {"tactic": "intro n n_ge_N", "annotated_tactic": ["intro n n_ge_N", []], "state_before": "case intro\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\n\u22a2 \u2200 (n : \u2115), n \u2265 N \u2192 |v - convergents (of v) n| < \u03b5", "state_after": "case intro\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\n\u22a2 |v - convergents (of v) n| < \u03b5"}, {"tactic": "let g := of v", "annotated_tactic": ["let g := <a>of</a> v", [{"full_name": "GeneralizedContinuedFraction.of", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean", "def_pos": [195, 15], "def_end_pos": [195, 17]}]], "state_before": "case intro\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\n\u22a2 |v - convergents (of v) n| < \u03b5", "state_after": "case intro\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\n\u22a2 |v - convergents (of v) n| < \u03b5"}, {"tactic": "cases' Decidable.em (g.TerminatedAt n) with terminated_at_n not_terminated_at_n", "annotated_tactic": ["cases' <a>Decidable.em</a> (g.TerminatedAt n) with terminated_at_n not_terminated_at_n", [{"full_name": "Decidable.em", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [734, 9], "def_end_pos": [734, 11]}]], "state_before": "case intro\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\n\u22a2 |v - convergents (of v) n| < \u03b5", "state_after": "case intro.inl\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nterminated_at_n : TerminatedAt g n\n\u22a2 |v - convergents (of v) n| < \u03b5\n\ncase intro.inr\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\n\u22a2 |v - convergents (of v) n| < \u03b5"}, {"tactic": "have : v = g.convergents n := of_correctness_of_terminatedAt terminated_at_n", "annotated_tactic": ["have : v = g.convergents n := <a>of_correctness_of_terminatedAt</a> terminated_at_n", [{"full_name": "GeneralizedContinuedFraction.of_correctness_of_terminatedAt", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean", "def_pos": [247, 9], "def_end_pos": [247, 39]}]], "state_before": "case intro.inl\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nterminated_at_n : TerminatedAt g n\n\u22a2 |v - convergents (of v) n| < \u03b5", "state_after": "case intro.inl\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nterminated_at_n : TerminatedAt g n\nthis : v = convergents g n\n\u22a2 |v - convergents (of v) n| < \u03b5"}, {"tactic": "have : v - g.convergents n = 0 := sub_eq_zero.mpr this", "annotated_tactic": ["have : v - g.convergents n = 0 := sub_eq_zero.mpr this", []], "state_before": "case intro.inl\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nterminated_at_n : TerminatedAt g n\nthis : v = convergents g n\n\u22a2 |v - convergents (of v) n| < \u03b5", "state_after": "case intro.inl\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nterminated_at_n : TerminatedAt g n\nthis\u271d : v = convergents g n\nthis : v - convergents g n = 0\n\u22a2 |v - convergents (of v) n| < \u03b5"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "case intro.inl\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nterminated_at_n : TerminatedAt g n\nthis\u271d : v = convergents g n\nthis : v - convergents g n = 0\n\u22a2 |v - convergents (of v) n| < \u03b5", "state_after": "case intro.inl\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nterminated_at_n : TerminatedAt g n\nthis\u271d : v = convergents g n\nthis : v - convergents g n = 0\n\u22a2 |0| < \u03b5"}, {"tactic": "exact_mod_cast \u03b5_pos", "annotated_tactic": ["exact_mod_cast \u03b5_pos", []], "state_before": "case intro.inl\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nterminated_at_n : TerminatedAt g n\nthis\u271d : v = convergents g n\nthis : v - convergents g n = 0\n\u22a2 |0| < \u03b5", "state_after": "no goals"}, {"tactic": "let B := g.denominators n", "annotated_tactic": ["let B := g.denominators n", []], "state_before": "case intro.inr\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\n\u22a2 |v - convergents (of v) n| < \u03b5", "state_after": "case intro.inr\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\n\u22a2 |v - convergents (of v) n| < \u03b5"}, {"tactic": "let nB := g.denominators (n + 1)", "annotated_tactic": ["let nB := g.denominators (n + 1)", []], "state_before": "case intro.inr\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\n\u22a2 |v - convergents (of v) n| < \u03b5", "state_after": "case intro.inr\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\n\u22a2 |v - convergents (of v) n| < \u03b5"}, {"tactic": "have abs_v_sub_conv_le : |v - g.convergents n| \u2264 1 / (B * nB) :=\n  abs_sub_convergents_le not_terminated_at_n", "annotated_tactic": ["have abs_v_sub_conv_le : |v - g.convergents n| \u2264 1 / (B * nB) :=\n      <a>abs_sub_convergents_le</a> not_terminated_at_n", [{"full_name": "GeneralizedContinuedFraction.abs_sub_convergents_le", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean", "def_pos": [454, 9], "def_end_pos": [454, 31]}]], "state_before": "case intro.inr\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\n\u22a2 |v - convergents (of v) n| < \u03b5", "state_after": "case intro.inr\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\n\u22a2 |v - convergents (of v) n| < \u03b5"}, {"tactic": "suffices : 1 / (B * nB) < \u03b5", "annotated_tactic": ["suffices : 1 / (B * nB) < \u03b5", []], "state_before": "case intro.inr\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\n\u22a2 |v - convergents (of v) n| < \u03b5", "state_after": "case intro.inr\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nthis : 1 / (B * nB) < \u03b5\n\u22a2 |v - convergents (of v) n| < \u03b5\n\ncase this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\n\u22a2 1 / (B * nB) < \u03b5"}, {"tactic": "exact lt_of_le_of_lt abs_v_sub_conv_le this", "annotated_tactic": ["exact <a>lt_of_le_of_lt</a> abs_v_sub_conv_le this", [{"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}]], "state_before": "case intro.inr\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nthis : 1 / (B * nB) < \u03b5\n\u22a2 |v - convergents (of v) n| < \u03b5\n\ncase this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\n\u22a2 1 / (B * nB) < \u03b5", "state_after": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\n\u22a2 1 / (B * nB) < \u03b5"}, {"tactic": "have nB_ineq : (fib (n + 2) : K) \u2264 nB :=\n  haveI : \u00acg.TerminatedAt (n + 1 - 1) := not_terminated_at_n\n  succ_nth_fib_le_of_nth_denom (Or.inr this)", "annotated_tactic": ["have nB_ineq : (<a>fib</a> (n + 2) : K) \u2264 nB :=\n      haveI : \u00acg.TerminatedAt (n + 1 - 1) := not_terminated_at_n\n      <a>succ_nth_fib_le_of_nth_denom</a> (<a>Or.inr</a> this)", [{"full_name": "Nat.fib", "def_path": "Mathlib/Data/Nat/Fib.lean", "def_pos": [70, 5], "def_end_pos": [70, 8]}, {"full_name": "GeneralizedContinuedFraction.succ_nth_fib_le_of_nth_denom", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean", "def_pos": [237, 9], "def_end_pos": [237, 37]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}]], "state_before": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\n\u22a2 1 / (B * nB) < \u03b5", "state_after": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\n\u22a2 1 / (B * nB) < \u03b5"}, {"tactic": "have B_ineq : (fib (n + 1) : K) \u2264 B :=\n  haveI : \u00acg.TerminatedAt (n - 1) := mt (terminated_stable n.pred_le) not_terminated_at_n\n  succ_nth_fib_le_of_nth_denom (Or.inr this)", "annotated_tactic": ["have B_ineq : (<a>fib</a> (n + 1) : K) \u2264 B :=\n      haveI : \u00acg.TerminatedAt (n - 1) := <a>mt</a> (<a>terminated_stable</a> n.pred_le) not_terminated_at_n\n      <a>succ_nth_fib_le_of_nth_denom</a> (<a>Or.inr</a> this)", [{"full_name": "Nat.fib", "def_path": "Mathlib/Data/Nat/Fib.lean", "def_pos": [70, 5], "def_end_pos": [70, 8]}, {"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}, {"full_name": "GeneralizedContinuedFraction.terminated_stable", "def_path": "Mathlib/Algebra/ContinuedFractions/TerminatedStable.lean", "def_pos": [24, 9], "def_end_pos": [24, 26]}, {"full_name": "GeneralizedContinuedFraction.succ_nth_fib_le_of_nth_denom", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean", "def_pos": [237, 9], "def_end_pos": [237, 37]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}]], "state_before": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\n\u22a2 1 / (B * nB) < \u03b5", "state_after": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\n\u22a2 1 / (B * nB) < \u03b5"}, {"tactic": "have zero_lt_B : 0 < B := B_ineq.trans_lt' $ by exact_mod_cast fib_pos.2 n.succ_pos", "annotated_tactic": ["have zero_lt_B : 0 < B := B_ineq.trans_lt' $ by exact_mod_cast <a>fib_pos</a>.2 n.succ_pos", [{"full_name": "Nat.fib_pos", "def_path": "Mathlib/Data/Nat/Fib.lean", "def_pos": [110, 15], "def_end_pos": [110, 22]}]], "state_before": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\n\u22a2 1 / (B * nB) < \u03b5", "state_after": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\n\u22a2 1 / (B * nB) < \u03b5"}, {"tactic": "have nB_pos : 0 < nB := nB_ineq.trans_lt' $ by exact_mod_cast fib_pos.2 $ succ_pos _", "annotated_tactic": ["have nB_pos : 0 < nB := nB_ineq.trans_lt' $ by exact_mod_cast <a>fib_pos</a>.2 $ <a>succ_pos</a> _", [{"full_name": "Nat.fib_pos", "def_path": "Mathlib/Data/Nat/Fib.lean", "def_pos": [110, 15], "def_end_pos": [110, 22]}, {"full_name": "Nat.succ_pos", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1608, 9], "def_end_pos": [1608, 21]}]], "state_before": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\n\u22a2 1 / (B * nB) < \u03b5", "state_after": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\n\u22a2 1 / (B * nB) < \u03b5"}, {"tactic": "have zero_lt_mul_conts : 0 < B * nB := by positivity", "annotated_tactic": ["have zero_lt_mul_conts : 0 < B * nB := by positivity", []], "state_before": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\n\u22a2 1 / (B * nB) < \u03b5", "state_after": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\n\u22a2 1 / (B * nB) < \u03b5"}, {"tactic": "suffices : 1 < \u03b5 * (B * nB)", "annotated_tactic": ["suffices : 1 < \u03b5 * (B * nB)", []], "state_before": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\n\u22a2 1 / (B * nB) < \u03b5", "state_after": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\nthis : 1 < \u03b5 * (B * nB)\n\u22a2 1 / (B * nB) < \u03b5\n\ncase this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\n\u22a2 1 < \u03b5 * (B * nB)"}, {"tactic": "exact (div_lt_iff zero_lt_mul_conts).mpr this", "annotated_tactic": ["exact (<a>div_lt_iff</a> zero_lt_mul_conts).<a>mpr</a> this", [{"full_name": "div_lt_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\nthis : 1 < \u03b5 * (B * nB)\n\u22a2 1 / (B * nB) < \u03b5\n\ncase this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\n\u22a2 1 < \u03b5 * (B * nB)", "state_after": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\n\u22a2 1 < \u03b5 * (B * nB)"}, {"tactic": "have one_lt_\u03b5_mul_N : 1 < \u03b5 * n := by\n  have one_lt_\u03b5_mul_N' : 1 < \u03b5 * (N' : K) := (div_lt_iff' \u03b5_pos).mp one_div_\u03b5_lt_N'\n  have : (N' : K) \u2264 N := by exact_mod_cast le_max_left _ _\n  have : \u03b5 * N' \u2264 \u03b5 * n :=\n    (mul_le_mul_left \u03b5_pos).mpr (le_trans this (by exact_mod_cast n_ge_N))\n  exact lt_of_lt_of_le one_lt_\u03b5_mul_N' this", "annotated_tactic": ["have one_lt_\u03b5_mul_N : 1 < \u03b5 * n := by\n      have one_lt_\u03b5_mul_N' : 1 < \u03b5 * (N' : K) := (<a>div_lt_iff'</a> \u03b5_pos).<a>mp</a> one_div_\u03b5_lt_N'\n      have : (N' : K) \u2264 N := by exact_mod_cast <a>le_max_left</a> _ _\n      have : \u03b5 * N' \u2264 \u03b5 * n :=\n        (<a>mul_le_mul_left</a> \u03b5_pos).<a>mpr</a> (<a>le_trans</a> this (by exact_mod_cast n_ge_N))\n      exact <a>lt_of_lt_of_le</a> one_lt_\u03b5_mul_N' this", [{"full_name": "div_lt_iff'", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [173, 9], "def_end_pos": [173, 20]}, {"full_name": "Iff.mp", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [90, 3], "def_end_pos": [90, 5]}, {"full_name": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}, {"full_name": "mul_le_mul_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [209, 9], "def_end_pos": [209, 24]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}]], "state_before": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\n\u22a2 1 < \u03b5 * (B * nB)", "state_after": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n\n\u22a2 1 < \u03b5 * (B * nB)"}, {"tactic": "suffices : \u03b5 * n \u2264 \u03b5 * (B * nB)", "annotated_tactic": ["suffices : \u03b5 * n \u2264 \u03b5 * (B * nB)", []], "state_before": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n\n\u22a2 1 < \u03b5 * (B * nB)", "state_after": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n\nthis : \u03b5 * \u2191n \u2264 \u03b5 * (B * nB)\n\u22a2 1 < \u03b5 * (B * nB)\n\ncase this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n\n\u22a2 \u03b5 * \u2191n \u2264 \u03b5 * (B * nB)"}, {"tactic": "exact lt_of_lt_of_le one_lt_\u03b5_mul_N this", "annotated_tactic": ["exact <a>lt_of_lt_of_le</a> one_lt_\u03b5_mul_N this", [{"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}]], "state_before": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n\nthis : \u03b5 * \u2191n \u2264 \u03b5 * (B * nB)\n\u22a2 1 < \u03b5 * (B * nB)\n\ncase this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n\n\u22a2 \u03b5 * \u2191n \u2264 \u03b5 * (B * nB)", "state_after": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n\n\u22a2 \u03b5 * \u2191n \u2264 \u03b5 * (B * nB)"}, {"tactic": "suffices : (n : K) \u2264 B * nB", "annotated_tactic": ["suffices : (n : K) \u2264 B * nB", []], "state_before": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n\n\u22a2 \u03b5 * \u2191n \u2264 \u03b5 * (B * nB)", "state_after": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n\nthis : \u2191n \u2264 B * nB\n\u22a2 \u03b5 * \u2191n \u2264 \u03b5 * (B * nB)\n\ncase this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n\n\u22a2 \u2191n \u2264 B * nB"}, {"tactic": "exact (mul_le_mul_left \u03b5_pos).mpr this", "annotated_tactic": ["exact (<a>mul_le_mul_left</a> \u03b5_pos).<a>mpr</a> this", [{"full_name": "mul_le_mul_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [209, 9], "def_end_pos": [209, 24]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n\nthis : \u2191n \u2264 B * nB\n\u22a2 \u03b5 * \u2191n \u2264 \u03b5 * (B * nB)\n\ncase this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n\n\u22a2 \u2191n \u2264 B * nB", "state_after": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n\n\u22a2 \u2191n \u2264 B * nB"}, {"tactic": "calc\n  (n : K) \u2264 fib n := by exact_mod_cast le_fib_self <| le_trans (le_max_right N' 5) n_ge_N\n  _ \u2264 fib (n + 1) := by exact_mod_cast fib_le_fib_succ\n  _ \u2264 fib (n + 1) * fib (n + 1) := by exact_mod_cast (fib (n + 1)).le_mul_self\n  _ \u2264 fib (n + 1) * fib (n + 2) :=\n        mul_le_mul_of_nonneg_left (by exact_mod_cast fib_le_fib_succ) (cast_nonneg _)\n  _ \u2264 B * nB := mul_le_mul B_ineq nB_ineq (cast_nonneg _) zero_lt_B.le", "annotated_tactic": ["calc\n      (n : K) \u2264 <a>fib</a> n := by exact_mod_cast <a>le_fib_self</a> <| <a>le_trans</a> (<a>le_max_right</a> N' 5) n_ge_N\n      _ \u2264 <a>fib</a> (n + 1) := by exact_mod_cast <a>fib_le_fib_succ</a>\n      _ \u2264 <a>fib</a> (n + 1) * <a>fib</a> (n + 1) := by exact_mod_cast (<a>fib</a> (n + 1)).<a>le_mul_self</a>\n      _ \u2264 <a>fib</a> (n + 1) * <a>fib</a> (n + 2) :=\n            <a>mul_le_mul_of_nonneg_left</a> (by exact_mod_cast <a>fib_le_fib_succ</a>) (<a>cast_nonneg</a> _)\n      _ \u2264 B * nB := <a>mul_le_mul</a> B_ineq nB_ineq (<a>cast_nonneg</a> _) zero_lt_B.le", [{"full_name": "Nat.fib", "def_path": "Mathlib/Data/Nat/Fib.lean", "def_pos": [70, 5], "def_end_pos": [70, 8]}, {"full_name": "Nat.le_fib_self", "def_path": "Mathlib/Data/Nat/Fib.lean", "def_pos": [138, 9], "def_end_pos": [138, 20]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}, {"full_name": "Nat.fib", "def_path": "Mathlib/Data/Nat/Fib.lean", "def_pos": [70, 5], "def_end_pos": [70, 8]}, {"full_name": "Nat.fib_le_fib_succ", "def_path": "Mathlib/Data/Nat/Fib.lean", "def_pos": [97, 9], "def_end_pos": [97, 24]}, {"full_name": "Nat.fib", "def_path": "Mathlib/Data/Nat/Fib.lean", "def_pos": [70, 5], "def_end_pos": [70, 8]}, {"full_name": "Nat.fib", "def_path": "Mathlib/Data/Nat/Fib.lean", "def_pos": [70, 5], "def_end_pos": [70, 8]}, {"full_name": "Nat.fib", "def_path": "Mathlib/Data/Nat/Fib.lean", "def_pos": [70, 5], "def_end_pos": [70, 8]}, {"full_name": "Nat.le_mul_self", "def_path": "Mathlib/Data/Nat/Order/Basic.lean", "def_pos": [288, 9], "def_end_pos": [288, 20]}, {"full_name": "Nat.fib", "def_path": "Mathlib/Data/Nat/Fib.lean", "def_pos": [70, 5], "def_end_pos": [70, 8]}, {"full_name": "Nat.fib", "def_path": "Mathlib/Data/Nat/Fib.lean", "def_pos": [70, 5], "def_end_pos": [70, 8]}, {"full_name": "mul_le_mul_of_nonneg_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [152, 9], "def_end_pos": [152, 34]}, {"full_name": "Nat.fib_le_fib_succ", "def_path": "Mathlib/Data/Nat/Fib.lean", "def_pos": [97, 9], "def_end_pos": [97, 24]}, {"full_name": "Nat.cast_nonneg", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [44, 9], "def_end_pos": [44, 20]}, {"full_name": "mul_le_mul", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [414, 9], "def_end_pos": [414, 19]}, {"full_name": "Nat.cast_nonneg", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [44, 9], "def_end_pos": [44, 20]}]], "state_before": "case this\nK : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n\n\u22a2 \u2191n \u2264 B * nB", "state_after": "no goals"}, {"tactic": "exact_mod_cast fib_pos.2 n.succ_pos", "annotated_tactic": ["exact_mod_cast <a>fib_pos</a>.2 n.succ_pos", [{"full_name": "Nat.fib_pos", "def_path": "Mathlib/Data/Nat/Fib.lean", "def_pos": [110, 15], "def_end_pos": [110, 22]}]], "state_before": "K : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\n\u22a2 0 < \u2191(fib (n + 1))", "state_after": "no goals"}, {"tactic": "exact_mod_cast fib_pos.2 $ succ_pos _", "annotated_tactic": ["exact_mod_cast <a>fib_pos</a>.2 $ <a>succ_pos</a> _", [{"full_name": "Nat.fib_pos", "def_path": "Mathlib/Data/Nat/Fib.lean", "def_pos": [110, 15], "def_end_pos": [110, 22]}, {"full_name": "Nat.succ_pos", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1608, 9], "def_end_pos": [1608, 21]}]], "state_before": "K : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\n\u22a2 0 < \u2191(fib (n + 2))", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "K : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\n\u22a2 0 < B * nB", "state_after": "no goals"}, {"tactic": "have one_lt_\u03b5_mul_N' : 1 < \u03b5 * (N' : K) := (div_lt_iff' \u03b5_pos).mp one_div_\u03b5_lt_N'", "annotated_tactic": ["have one_lt_\u03b5_mul_N' : 1 < \u03b5 * (N' : K) := (<a>div_lt_iff'</a> \u03b5_pos).<a>mp</a> one_div_\u03b5_lt_N'", [{"full_name": "div_lt_iff'", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [173, 9], "def_end_pos": [173, 20]}, {"full_name": "Iff.mp", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [90, 3], "def_end_pos": [90, 5]}]], "state_before": "K : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\n\u22a2 1 < \u03b5 * \u2191n", "state_after": "K : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N' : 1 < \u03b5 * \u2191N'\n\u22a2 1 < \u03b5 * \u2191n"}, {"tactic": "have : (N' : K) \u2264 N := by exact_mod_cast le_max_left _ _", "annotated_tactic": ["have : (N' : K) \u2264 N := by exact_mod_cast <a>le_max_left</a> _ _", [{"full_name": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}]], "state_before": "K : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N' : 1 < \u03b5 * \u2191N'\n\u22a2 1 < \u03b5 * \u2191n", "state_after": "K : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N' : 1 < \u03b5 * \u2191N'\nthis : \u2191N' \u2264 \u2191N\n\u22a2 1 < \u03b5 * \u2191n"}, {"tactic": "have : \u03b5 * N' \u2264 \u03b5 * n :=\n  (mul_le_mul_left \u03b5_pos).mpr (le_trans this (by exact_mod_cast n_ge_N))", "annotated_tactic": ["have : \u03b5 * N' \u2264 \u03b5 * n :=\n        (<a>mul_le_mul_left</a> \u03b5_pos).<a>mpr</a> (<a>le_trans</a> this (by exact_mod_cast n_ge_N))", [{"full_name": "mul_le_mul_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [209, 9], "def_end_pos": [209, 24]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "state_before": "K : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N' : 1 < \u03b5 * \u2191N'\nthis : \u2191N' \u2264 \u2191N\n\u22a2 1 < \u03b5 * \u2191n", "state_after": "K : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N' : 1 < \u03b5 * \u2191N'\nthis\u271d : \u2191N' \u2264 \u2191N\nthis : \u03b5 * \u2191N' \u2264 \u03b5 * \u2191n\n\u22a2 1 < \u03b5 * \u2191n"}, {"tactic": "exact lt_of_lt_of_le one_lt_\u03b5_mul_N' this", "annotated_tactic": ["exact <a>lt_of_lt_of_le</a> one_lt_\u03b5_mul_N' this", [{"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}]], "state_before": "K : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N' : 1 < \u03b5 * \u2191N'\nthis\u271d : \u2191N' \u2264 \u2191N\nthis : \u03b5 * \u2191N' \u2264 \u03b5 * \u2191n\n\u22a2 1 < \u03b5 * \u2191n", "state_after": "no goals"}, {"tactic": "exact_mod_cast le_max_left _ _", "annotated_tactic": ["exact_mod_cast <a>le_max_left</a> _ _", [{"full_name": "le_max_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}]], "state_before": "K : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N' : 1 < \u03b5 * \u2191N'\n\u22a2 \u2191N' \u2264 \u2191N", "state_after": "no goals"}, {"tactic": "exact_mod_cast n_ge_N", "annotated_tactic": ["exact_mod_cast n_ge_N", []], "state_before": "K : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N' : 1 < \u03b5 * \u2191N'\nthis : \u2191N' \u2264 \u2191N\n\u22a2 \u2191N \u2264 \u2191n", "state_after": "no goals"}, {"tactic": "exact_mod_cast le_fib_self <| le_trans (le_max_right N' 5) n_ge_N", "annotated_tactic": ["exact_mod_cast <a>le_fib_self</a> <| <a>le_trans</a> (<a>le_max_right</a> N' 5) n_ge_N", [{"full_name": "Nat.le_fib_self", "def_path": "Mathlib/Data/Nat/Fib.lean", "def_pos": [138, 9], "def_end_pos": [138, 20]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_max_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}]], "state_before": "K : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n\n\u22a2 \u2191n \u2264 \u2191(fib n)", "state_after": "no goals"}, {"tactic": "exact_mod_cast fib_le_fib_succ", "annotated_tactic": ["exact_mod_cast <a>fib_le_fib_succ</a>", [{"full_name": "Nat.fib_le_fib_succ", 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[{"full_name": "Nat.fib", "def_path": "Mathlib/Data/Nat/Fib.lean", "def_pos": [70, 5], "def_end_pos": [70, 8]}, {"full_name": "Nat.le_mul_self", "def_path": "Mathlib/Data/Nat/Order/Basic.lean", "def_pos": [288, 9], "def_end_pos": [288, 20]}]], "state_before": "K : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n\n\u22a2 \u2191(fib (n + 1)) \u2264 \u2191(fib (n + 1)) * \u2191(fib (n + 1))", "state_after": "no goals"}, {"tactic": "exact_mod_cast fib_le_fib_succ", "annotated_tactic": ["exact_mod_cast <a>fib_le_fib_succ</a>", [{"full_name": "Nat.fib_le_fib_succ", "def_path": "Mathlib/Data/Nat/Fib.lean", "def_pos": [97, 9], "def_end_pos": [97, 24]}]], "state_before": "K : Type u_1\nv : K\ninst\u271d\u00b2 : LinearOrderedField K\ninst\u271d\u00b9 : FloorRing K\ninst\u271d : Archimedean K\n\u03b5 : K\n\u03b5_pos : \u03b5 > 0\nN' : \u2115\none_div_\u03b5_lt_N' : 1 / \u03b5 < \u2191N'\nN : \u2115 := max N' 5\nn : \u2115\nn_ge_N : n \u2265 N\ng : GeneralizedContinuedFraction K := of v\nnot_terminated_at_n : \u00acTerminatedAt g n\nB : K := denominators g n\nnB : K := denominators g (n + 1)\nabs_v_sub_conv_le : |v - convergents g n| \u2264 1 / (B * nB)\nnB_ineq : \u2191(fib (n + 2)) \u2264 nB\nB_ineq : \u2191(fib (n + 1)) \u2264 B\nzero_lt_B : 0 < B\nnB_pos : 0 < nB\nzero_lt_mul_conts : 0 < B * nB\none_lt_\u03b5_mul_N : 1 < \u03b5 * \u2191n\n\u22a2 \u2191(fib (n + 1)) \u2264 \u2191(fib (n + 2))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "full_name": "WithTop.image_coe_Ici", "start": [93, 1], "end": [94, 83], "traced_tactics": [{"tactic": "rw [\u2190 preimage_coe_Ici, image_preimage_eq_inter_range, range_coe, Ici_inter_Iio]", "annotated_tactic": ["rw [\u2190 <a>preimage_coe_Ici</a>, <a>image_preimage_eq_inter_range</a>, <a>range_coe</a>, <a>Ici_inter_Iio</a>]", [{"full_name": "WithTop.preimage_coe_Ici", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [44, 9], "def_end_pos": [44, 25]}, {"full_name": "Set.image_preimage_eq_inter_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [796, 9], "def_end_pos": [796, 38]}, {"full_name": "WithTop.range_coe", "def_path": "Mathlib/Data/Set/Intervals/WithBotTop.lean", "def_pos": [32, 9], "def_end_pos": [32, 18]}, {"full_name": "Set.Ici_inter_Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [622, 9], "def_end_pos": [622, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\na b : \u03b1\n\u22a2 some '' Ici a = Ico \u2191a \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Nat.floor_eq_zero", "start": [232, 1], "end": [234, 34], "traced_tactics": [{"tactic": "rw [\u2190 lt_one_iff, \u2190 @cast_one \u03b1]", "annotated_tactic": ["rw [\u2190 <a>lt_one_iff</a>, \u2190 @<a>cast_one</a> \u03b1]", [{"full_name": "Nat.lt_one_iff", "def_path": "Mathlib/Data/Nat/Order/Basic.lean", "def_pos": [159, 9], "def_end_pos": [159, 19]}, {"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [141, 9], "def_end_pos": [141, 17]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\n\u22a2 \u230aa\u230b\u208a = 0 \u2194 a < 1", "state_after": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\n\u22a2 \u230aa\u230b\u208a < 1 \u2194 a < \u21911"}, {"tactic": "exact floor_lt' Nat.one_ne_zero", "annotated_tactic": ["exact <a>floor_lt'</a> <a>Nat.one_ne_zero</a>", [{"full_name": "Nat.floor_lt'", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [206, 9], "def_end_pos": [206, 18]}, {"full_name": "Nat.one_ne_zero", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [426, 19], "def_end_pos": [426, 30]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\n\u22a2 \u230aa\u230b\u208a < 1 \u2194 a < \u21911", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Localization/Construction.lean", "full_name": "CategoryTheory.Localization.Construction.morphismProperty_is_top", "start": [232, 1], "end": [256, 18], "traced_tactics": [{"tactic": "funext X Y f", "annotated_tactic": ["funext X Y f", []], "state_before": "C : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\n\u22a2 P = \u22a4", "state_after": "case h.h.h\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\n\u22a2 P f = \u22a4 f"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "case h.h.h\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\n\u22a2 P f = \u22a4 f", "state_after": "case h.h.h.a\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\n\u22a2 P f \u2194 \u22a4 f"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h.h.h.a\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\n\u22a2 P f \u2194 \u22a4 f", "state_after": "case h.h.h.a.mp\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\n\u22a2 P f \u2192 \u22a4 f\n\ncase h.h.h.a.mpr\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\n\u22a2 \u22a4 f \u2192 P f"}, {"tactic": "intro", "annotated_tactic": ["intro", []], "state_before": "case h.h.h.a.mp\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\n\u22a2 P f \u2192 \u22a4 f", "state_after": "case h.h.h.a.mp\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : P f\n\u22a2 \u22a4 f"}, {"tactic": "apply MorphismProperty.top_apply", "annotated_tactic": ["apply <a>MorphismProperty.top_apply</a>", [{"full_name": "CategoryTheory.MorphismProperty.top_apply", "def_path": "Mathlib/CategoryTheory/MorphismProperty.lean", "def_pos": [58, 7], "def_end_pos": [58, 16]}]], "state_before": "case h.h.h.a.mp\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : P f\n\u22a2 \u22a4 f", "state_after": "no goals"}, {"tactic": "intro", "annotated_tactic": ["intro", []], "state_before": "case h.h.h.a.mpr\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\n\u22a2 \u22a4 f \u2192 P f", "state_after": "case h.h.h.a.mpr\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\n\u22a2 P f"}, {"tactic": "let G : _ \u2964 W.Localization := Quotient.functor _", "annotated_tactic": ["let G : _ \u2964 W.Localization := <a>Quotient.functor</a> _", [{"full_name": "CategoryTheory.Quotient.functor", "def_path": "Mathlib/CategoryTheory/Quotient.lean", "def_pos": [113, 5], "def_end_pos": [113, 12]}]], "state_before": "case h.h.h.a.mpr\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\n\u22a2 P f", "state_after": "case h.h.h.a.mpr\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\n\u22a2 P f"}, {"tactic": "haveI : Full G := Quotient.fullFunctor _", "annotated_tactic": ["haveI : <a>Full</a> G := <a>Quotient.fullFunctor</a> _", [{"full_name": "CategoryTheory.Full", "def_path": "Mathlib/CategoryTheory/Functor/FullyFaithful.lean", "def_pos": [45, 7], "def_end_pos": [45, 11]}, {"full_name": "CategoryTheory.Quotient.fullFunctor", "def_path": "Mathlib/CategoryTheory/Quotient.lean", "def_pos": [118, 24], "def_end_pos": [118, 35]}]], "state_before": "case h.h.h.a.mpr\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\n\u22a2 P f", "state_after": "case h.h.h.a.mpr\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis : Full G\n\u22a2 P f"}, {"tactic": "suffices \u2200 (X\u2081 X\u2082 : Paths (LocQuiver W)) (f : X\u2081 \u27f6 X\u2082), P (G.map f) by\n  rcases X with \u27e8\u27e8X\u27e9\u27e9\n  rcases Y with \u27e8\u27e8Y\u27e9\u27e9\n  simpa only [Functor.image_preimage] using this _ _ (G.preimage f)", "annotated_tactic": ["suffices \u2200 (X\u2081 X\u2082 : <a>Paths</a> (<a>LocQuiver</a> W)) (f : X\u2081 \u27f6 X\u2082), P (G.map f) by\n      rcases X with \u27e8\u27e8X\u27e9\u27e9\n      rcases Y with \u27e8\u27e8Y\u27e9\u27e9\n      simpa only [<a>Functor.image_preimage</a>] using this _ _ (G.preimage f)", [{"full_name": "CategoryTheory.Paths", "def_path": "Mathlib/CategoryTheory/PathCategory.lean", "def_pos": [32, 5], "def_end_pos": [32, 10]}, {"full_name": "CategoryTheory.Localization.Construction.LocQuiver", "def_path": "Mathlib/CategoryTheory/Localization/Construction.lean", "def_pos": [58, 11], "def_end_pos": [58, 20]}, {"full_name": "CategoryTheory.Functor.image_preimage", "def_path": "Mathlib/CategoryTheory/Functor/FullyFaithful.lean", "def_pos": [90, 9], "def_end_pos": [90, 23]}]], "state_before": "case h.h.h.a.mpr\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis : Full G\n\u22a2 P f", "state_after": "case h.h.h.a.mpr\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis : Full G\n\u22a2 \u2200 (X\u2081 X\u2082 : Paths (LocQuiver W)) (f : X\u2081 \u27f6 X\u2082), P (G.map f)"}, {"tactic": "intros X\u2081 X\u2082 p", "annotated_tactic": ["intros X\u2081 X\u2082 p", []], "state_before": "case h.h.h.a.mpr\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis : Full G\n\u22a2 \u2200 (X\u2081 X\u2082 : Paths (LocQuiver W)) (f : X\u2081 \u27f6 X\u2082), P (G.map f)", "state_after": "case h.h.h.a.mpr\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis : Full G\nX\u2081 X\u2082 : Paths (LocQuiver W)\np : X\u2081 \u27f6 X\u2082\n\u22a2 P (G.map p)"}, {"tactic": "induction' p with X\u2082 X\u2083 p g hp", "annotated_tactic": ["induction' p with X\u2082 X\u2083 p g hp", []], "state_before": "case h.h.h.a.mpr\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis : Full G\nX\u2081 X\u2082 : Paths (LocQuiver W)\np : X\u2081 \u27f6 X\u2082\n\u22a2 P (G.map p)", "state_after": "case h.h.h.a.mpr.nil\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis : Full G\nX\u2081 X\u2082 : Paths (LocQuiver W)\n\u22a2 P (G.map Quiver.Path.nil)\n\ncase h.h.h.a.mpr.cons\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis : Full G\nX\u2081 X\u2082\u271d X\u2082 X\u2083 : Paths (LocQuiver W)\np : Quiver.Path X\u2081 X\u2082\ng : X\u2082 \u27f6 X\u2083\nhp : P (G.map p)\n\u22a2 P (G.map (Quiver.Path.cons p g))"}, {"tactic": "rcases X with \u27e8\u27e8X\u27e9\u27e9", "annotated_tactic": ["rcases X with \u27e8\u27e8X\u27e9\u27e9", []], "state_before": "C : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis\u271d : Full G\nthis : \u2200 (X\u2081 X\u2082 : Paths (LocQuiver W)) (f : X\u2081 \u27f6 X\u2082), P (G.map f)\n\u22a2 P f", "state_after": "case mk.mk\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nY : MorphismProperty.Localization W\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis\u271d : Full G\nthis : \u2200 (X\u2081 X\u2082 : Paths (LocQuiver W)) (f : X\u2081 \u27f6 X\u2082), P (G.map f)\nX : C\nf : { as := { obj := X } } \u27f6 Y\na\u271d : \u22a4 f\n\u22a2 P f"}, {"tactic": "rcases Y with \u27e8\u27e8Y\u27e9\u27e9", "annotated_tactic": ["rcases Y with \u27e8\u27e8Y\u27e9\u27e9", []], "state_before": "case mk.mk\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nY : MorphismProperty.Localization W\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis\u271d : Full G\nthis : \u2200 (X\u2081 X\u2082 : Paths (LocQuiver W)) (f : X\u2081 \u27f6 X\u2082), P (G.map f)\nX : C\nf : { as := { obj := X } } \u27f6 Y\na\u271d : \u22a4 f\n\u22a2 P f", "state_after": "case mk.mk.mk.mk\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis\u271d : Full G\nthis : \u2200 (X\u2081 X\u2082 : Paths (LocQuiver W)) (f : X\u2081 \u27f6 X\u2082), P (G.map f)\nX Y : C\nf : { as := { obj := X } } \u27f6 { as := { obj := Y } }\na\u271d : \u22a4 f\n\u22a2 P f"}, {"tactic": "simpa only [Functor.image_preimage] using this _ _ (G.preimage f)", "annotated_tactic": ["simpa only [<a>Functor.image_preimage</a>] using this _ _ (G.preimage f)", [{"full_name": "CategoryTheory.Functor.image_preimage", "def_path": "Mathlib/CategoryTheory/Functor/FullyFaithful.lean", "def_pos": [90, 9], "def_end_pos": [90, 23]}]], "state_before": "case mk.mk.mk.mk\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis\u271d : Full G\nthis : \u2200 (X\u2081 X\u2082 : Paths (LocQuiver W)) (f : X\u2081 \u27f6 X\u2082), P (G.map f)\nX Y : C\nf : { as := { obj := X } } \u27f6 { as := { obj := Y } }\na\u271d : \u22a4 f\n\u22a2 P f", "state_after": "no goals"}, {"tactic": "simpa only [Functor.map_id] using hP\u2081 (\ud835\udfd9 X\u2081.obj)", "annotated_tactic": ["simpa only [<a>Functor.map_id</a>] using hP\u2081 (\ud835\udfd9 X\u2081.obj)", [{"full_name": "CategoryTheory.Functor.map_id", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [41, 3], "def_end_pos": [41, 9]}]], "state_before": "case h.h.h.a.mpr.nil\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis : Full G\nX\u2081 X\u2082 : Paths (LocQuiver W)\n\u22a2 P (G.map Quiver.Path.nil)", "state_after": "no goals"}, {"tactic": "let p' : X\u2081 \u27f6X\u2082 := p", "annotated_tactic": ["let p' : X\u2081 \u27f6X\u2082 := p", []], "state_before": "case h.h.h.a.mpr.cons\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis : Full G\nX\u2081 X\u2082\u271d X\u2082 X\u2083 : Paths (LocQuiver W)\np : Quiver.Path X\u2081 X\u2082\ng : X\u2082 \u27f6 X\u2083\nhp : P (G.map p)\n\u22a2 P (G.map (Quiver.Path.cons p g))", "state_after": "case h.h.h.a.mpr.cons\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis : Full G\nX\u2081 X\u2082\u271d X\u2082 X\u2083 : Paths (LocQuiver W)\np : Quiver.Path X\u2081 X\u2082\ng : X\u2082 \u27f6 X\u2083\nhp : P (G.map p)\np' : X\u2081 \u27f6 X\u2082 := p\n\u22a2 P (G.map (Quiver.Path.cons p g))"}, {"tactic": "rw [show p'.cons g = p' \u226b Quiver.Hom.toPath g by rfl, G.map_comp]", "annotated_tactic": ["rw [show p'.cons g = p' \u226b <a>Quiver.Hom.toPath</a> g by rfl, G.map_comp]", [{"full_name": "Quiver.Hom.toPath", "def_path": "Mathlib/Combinatorics/Quiver/Path.lean", "def_pos": [34, 5], "def_end_pos": [34, 15]}]], "state_before": "case h.h.h.a.mpr.cons\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis : Full G\nX\u2081 X\u2082\u271d X\u2082 X\u2083 : Paths (LocQuiver W)\np : Quiver.Path X\u2081 X\u2082\ng : X\u2082 \u27f6 X\u2083\nhp : P (G.map p)\np' : X\u2081 \u27f6 X\u2082 := p\n\u22a2 P (G.map (Quiver.Path.cons p g))", "state_after": "case h.h.h.a.mpr.cons\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis : Full G\nX\u2081 X\u2082\u271d X\u2082 X\u2083 : Paths (LocQuiver W)\np : Quiver.Path X\u2081 X\u2082\ng : X\u2082 \u27f6 X\u2083\nhp : P (G.map p)\np' : X\u2081 \u27f6 X\u2082 := p\n\u22a2 P (G.map p' \u226b G.map (Quiver.Hom.toPath g))"}, {"tactic": "refine' hP\u2083 _ _ hp _", "annotated_tactic": ["refine' hP\u2083 _ _ hp _", []], "state_before": "case h.h.h.a.mpr.cons\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis : Full G\nX\u2081 X\u2082\u271d X\u2082 X\u2083 : Paths (LocQuiver W)\np : Quiver.Path X\u2081 X\u2082\ng : X\u2082 \u27f6 X\u2083\nhp : P (G.map p)\np' : X\u2081 \u27f6 X\u2082 := p\n\u22a2 P (G.map p' \u226b G.map (Quiver.Hom.toPath g))", "state_after": "case h.h.h.a.mpr.cons\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis : Full G\nX\u2081 X\u2082\u271d X\u2082 X\u2083 : Paths (LocQuiver W)\np : Quiver.Path X\u2081 X\u2082\ng : X\u2082 \u27f6 X\u2083\nhp : P (G.map p)\np' : X\u2081 \u27f6 X\u2082 := p\n\u22a2 P (G.map (Quiver.Hom.toPath g))"}, {"tactic": "rcases g with (g | \u27e8g, hg\u27e9)", "annotated_tactic": ["rcases g with (g | \u27e8g, hg\u27e9)", []], "state_before": "case h.h.h.a.mpr.cons\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis : Full G\nX\u2081 X\u2082\u271d X\u2082 X\u2083 : Paths (LocQuiver W)\np : Quiver.Path X\u2081 X\u2082\ng : X\u2082 \u27f6 X\u2083\nhp : P (G.map p)\np' : X\u2081 \u27f6 X\u2082 := p\n\u22a2 P (G.map (Quiver.Hom.toPath g))", "state_after": "case h.h.h.a.mpr.cons.inl\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis : Full G\nX\u2081 X\u2082\u271d X\u2082 X\u2083 : Paths (LocQuiver W)\np : Quiver.Path X\u2081 X\u2082\nhp : P (G.map p)\np' : X\u2081 \u27f6 X\u2082 := p\ng : X\u2082.obj \u27f6 X\u2083.obj\n\u22a2 P (G.map (Quiver.Hom.toPath (Sum.inl g)))\n\ncase h.h.h.a.mpr.cons.inr.mk\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis : Full G\nX\u2081 X\u2082\u271d X\u2082 X\u2083 : Paths (LocQuiver W)\np : Quiver.Path X\u2081 X\u2082\nhp : P (G.map p)\np' : X\u2081 \u27f6 X\u2082 := p\ng : X\u2083.obj \u27f6 X\u2082.obj\nhg : W g\n\u22a2 P (G.map (Quiver.Hom.toPath (Sum.inr { val := g, property := hg })))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "C : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis : Full G\nX\u2081 X\u2082\u271d X\u2082 X\u2083 : Paths (LocQuiver W)\np : Quiver.Path X\u2081 X\u2082\ng : X\u2082 \u27f6 X\u2083\nhp : P (G.map p)\np' : X\u2081 \u27f6 X\u2082 := p\n\u22a2 Quiver.Path.cons p' g = p' \u226b Quiver.Hom.toPath g", "state_after": "no goals"}, {"tactic": "apply hP\u2081", "annotated_tactic": ["apply hP\u2081", []], "state_before": "case h.h.h.a.mpr.cons.inl\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis : Full G\nX\u2081 X\u2082\u271d X\u2082 X\u2083 : Paths (LocQuiver W)\np : Quiver.Path X\u2081 X\u2082\nhp : P (G.map p)\np' : X\u2081 \u27f6 X\u2082 := p\ng : X\u2082.obj \u27f6 X\u2083.obj\n\u22a2 P (G.map (Quiver.Hom.toPath (Sum.inl g)))", "state_after": "no goals"}, {"tactic": "apply hP\u2082", "annotated_tactic": ["apply hP\u2082", []], "state_before": "case h.h.h.a.mpr.cons.inr.mk\nC : Type uC\ninst\u271d\u00b9 : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst\u271d : Category.{uD', uD} D\nG\u271d : C \u2964 D\nhG : MorphismProperty.IsInvertedBy W G\u271d\nP : MorphismProperty (MorphismProperty.Localization W)\nhP\u2081 : \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), P ((MorphismProperty.Q W).map f)\nhP\u2082 : \u2200 \u2983X Y : C\u2984 (w : X \u27f6 Y) (hw : W w), P (winv w hw)\nhP\u2083 : MorphismProperty.StableUnderComposition P\nX Y : MorphismProperty.Localization W\nf : X \u27f6 Y\na\u271d : \u22a4 f\nG : Paths (LocQuiver W) \u2964 MorphismProperty.Localization W := Quotient.functor (relations W)\nthis : Full G\nX\u2081 X\u2082\u271d X\u2082 X\u2083 : Paths (LocQuiver W)\np : Quiver.Path X\u2081 X\u2082\nhp : P (G.map p)\np' : X\u2081 \u27f6 X\u2082 := p\ng : X\u2083.obj \u27f6 X\u2082.obj\nhg : W g\n\u22a2 P (G.map (Quiver.Hom.toPath (Sum.inr { val := g, property := hg })))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean", "full_name": "Matrix.mul_inv_eq_iff_eq_mul_of_invertible", "start": [330, 1], "end": [333, 59], "traced_tactics": [{"tactic": "rw [\u2190 h, inv_mul_cancel_right_of_invertible]", "annotated_tactic": ["rw [\u2190 h, <a>inv_mul_cancel_right_of_invertible</a>]", [{"full_name": "Matrix.inv_mul_cancel_right_of_invertible", "def_path": "Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean", "def_pos": [315, 9], "def_end_pos": [315, 43]}]], "state_before": "l : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA\u271d B\u271d A B C : Matrix n n \u03b1\ninst\u271d : Invertible A\nh : B * A\u207b\u00b9 = C\n\u22a2 B = C * A", "state_after": "no goals"}, {"tactic": "rw [h, mul_inv_cancel_right_of_invertible]", "annotated_tactic": ["rw [h, <a>mul_inv_cancel_right_of_invertible</a>]", [{"full_name": "Matrix.mul_inv_cancel_right_of_invertible", "def_path": "Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean", "def_pos": [305, 9], "def_end_pos": [305, 43]}]], "state_before": "l : Type u_1\nm : Type u\nn : Type u'\n\u03b1 : Type v\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA\u271d B\u271d A B C : Matrix n n \u03b1\ninst\u271d : Invertible A\nh : B = C * A\n\u22a2 B * A\u207b\u00b9 = C", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Category/ModuleCat/Monoidal/Closed.lean", "full_name": "ModuleCat.monoidalClosed_curry", "start": [71, 1], "end": [74, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/DFinsupp/Basic.lean", "full_name": "DFinsupp.comapDomain'_single", "start": [1428, 1], "end": [1435, 79], "traced_tactics": [{"tactic": "ext i", "annotated_tactic": ["ext i", []], "state_before": "\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : DecidableEq \u03ba\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nh' : \u03b9 \u2192 \u03ba\nhh' : Function.LeftInverse h' h\nk : \u03ba\nx : \u03b2 (h k)\n\u22a2 comapDomain' h hh' (single (h k) x) = single k x", "state_after": "case h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : DecidableEq \u03ba\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nh' : \u03b9 \u2192 \u03ba\nhh' : Function.LeftInverse h' h\nk : \u03ba\nx : \u03b2 (h k)\ni : \u03ba\n\u22a2 \u2191(comapDomain' h hh' (single (h k) x)) i = \u2191(single k x) i"}, {"tactic": "rw [comapDomain'_apply]", "annotated_tactic": ["rw [<a>comapDomain'_apply</a>]", [{"full_name": "DFinsupp.comapDomain'_apply", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [1399, 9], "def_end_pos": [1399, 27]}]], "state_before": "case h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : DecidableEq \u03ba\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nh' : \u03b9 \u2192 \u03ba\nhh' : Function.LeftInverse h' h\nk : \u03ba\nx : \u03b2 (h k)\ni : \u03ba\n\u22a2 \u2191(comapDomain' h hh' (single (h k) x)) i = \u2191(single k x) i", "state_after": "case h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : DecidableEq \u03ba\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nh' : \u03b9 \u2192 \u03ba\nhh' : Function.LeftInverse h' h\nk : \u03ba\nx : \u03b2 (h k)\ni : \u03ba\n\u22a2 \u2191(single (h k) x) (h i) = \u2191(single k x) i"}, {"tactic": "obtain rfl | hik := Decidable.eq_or_ne i k", "annotated_tactic": ["obtain rfl | hik := <a>Decidable.eq_or_ne</a> i k", [{"full_name": "Decidable.eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [203, 9], "def_end_pos": [203, 27]}]], "state_before": "case h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : DecidableEq \u03ba\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nh' : \u03b9 \u2192 \u03ba\nhh' : Function.LeftInverse h' h\nk : \u03ba\nx : \u03b2 (h k)\ni : \u03ba\n\u22a2 \u2191(single (h k) x) (h i) = \u2191(single k x) i", "state_after": "case h.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : DecidableEq \u03ba\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nh' : \u03b9 \u2192 \u03ba\nhh' : Function.LeftInverse h' h\ni : \u03ba\nx : \u03b2 (h i)\n\u22a2 \u2191(single (h i) x) (h i) = \u2191(single i x) i\n\ncase h.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : DecidableEq \u03ba\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nh' : \u03b9 \u2192 \u03ba\nhh' : Function.LeftInverse h' h\nk : \u03ba\nx : \u03b2 (h k)\ni : \u03ba\nhik : i \u2260 k\n\u22a2 \u2191(single (h k) x) (h i) = \u2191(single k x) i"}, {"tactic": "rw [single_eq_same, single_eq_same]", "annotated_tactic": ["rw [<a>single_eq_same</a>, <a>single_eq_same</a>]", [{"full_name": "DFinsupp.single_eq_same", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [661, 9], "def_end_pos": [661, 23]}, {"full_name": "DFinsupp.single_eq_same", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [661, 9], "def_end_pos": [661, 23]}]], "state_before": "case h.inl\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : DecidableEq \u03ba\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nh' : \u03b9 \u2192 \u03ba\nhh' : Function.LeftInverse h' h\ni : \u03ba\nx : \u03b2 (h i)\n\u22a2 \u2191(single (h i) x) (h i) = \u2191(single i x) i", "state_after": "no goals"}, {"tactic": "rw [single_eq_of_ne hik.symm, single_eq_of_ne (hh'.injective.ne hik.symm)]", "annotated_tactic": ["rw [<a>single_eq_of_ne</a> hik.symm, <a>single_eq_of_ne</a> (hh'.injective.ne hik.symm)]", [{"full_name": "DFinsupp.single_eq_of_ne", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [665, 9], "def_end_pos": [665, 24]}, {"full_name": "DFinsupp.single_eq_of_ne", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [665, 9], "def_end_pos": [665, 24]}]], "state_before": "case h.inr\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : DecidableEq \u03ba\ninst\u271d : (i : \u03b9) \u2192 Zero (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nh' : \u03b9 \u2192 \u03ba\nhh' : Function.LeftInverse h' h\nk : \u03ba\nx : \u03b2 (h k)\ni : \u03ba\nhik : i \u2260 k\n\u22a2 \u2191(single (h k) x) (h i) = \u2191(single k x) i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/Forall2.lean", "full_name": "List.Forall\u2082.mp", "start": [43, 1], "end": [47, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Monoidal/End.lean", "full_name": "CategoryTheory.\u03bc_inv_naturality", "start": [154, 1], "end": [157, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/ModelTheory/LanguageMap.lean", "full_name": "FirstOrder.Language.card_withConstants", "start": [458, 1], "end": [460, 49], "traced_tactics": [{"tactic": "rw [withConstants, card_sum, card_constantsOn]", "annotated_tactic": ["rw [<a>withConstants</a>, <a>card_sum</a>, <a>card_constantsOn</a>]", [{"full_name": "FirstOrder.Language.withConstants", "def_path": "Mathlib/ModelTheory/LanguageMap.lean", "def_pos": [449, 5], "def_end_pos": [449, 18]}, {"full_name": "FirstOrder.Language.card_sum", "def_path": "Mathlib/ModelTheory/Basic.lean", "def_pos": [266, 9], "def_end_pos": [266, 17]}, {"full_name": "FirstOrder.Language.card_constantsOn", "def_path": "Mathlib/ModelTheory/LanguageMap.lean", "def_pos": [413, 9], "def_end_pos": [413, 25]}]], "state_before": "L : Language\nL' : Language\nM : Type w\ninst\u271d : Structure L M\n\u03b1 : Type w'\n\u22a2 card (L[[\u03b1]]) = lift.{w', max u v} (card L) + lift.{max u v, w'} #\u03b1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "full_name": "Asymptotics.IsLittleO.add", "start": [1110, 1], "end": [1113, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "iInf_pair", "start": [1530, 1], "end": [1531, 35], "traced_tactics": [{"tactic": "rw [iInf_insert, iInf_singleton]", "annotated_tactic": ["rw [<a>iInf_insert</a>, <a>iInf_singleton</a>]", [{"full_name": "iInf_insert", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1515, 9], "def_end_pos": [1515, 20]}, {"full_name": "iInf_singleton", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1523, 9], "def_end_pos": [1523, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na\u271d b\u271d : \u03b1\nf : \u03b2 \u2192 \u03b1\na b : \u03b2\n\u22a2 \u2a05 x \u2208 {a, b}, f x = f a \u2293 f b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Bicategory/Free.lean", "full_name": "CategoryTheory.FreeBicategory.mk_id", "start": [271, 1], "end": [272, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.stoppedProcess_eq_of_mem_finset", "start": [914, 1], "end": [935, 22], "traced_tactics": [{"tactic": "ext \u03c9", "annotated_tactic": ["ext \u03c9", []], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u22a2 stoppedProcess u \u03c4 n =\n    Set.indicator {a | n \u2264 \u03c4 a} (u n) + \u2211 i in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = i} (u i)", "state_after": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\n\u22a2 stoppedProcess u \u03c4 n \u03c9 =\n    (Set.indicator {a | n \u2264 \u03c4 a} (u n) + \u2211 i in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = i} (u i)) \u03c9"}, {"tactic": "rw [Pi.add_apply, Finset.sum_apply]", "annotated_tactic": ["rw [<a>Pi.add_apply</a>, <a>Finset.sum_apply</a>]", [{"full_name": "Pi.add_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "Finset.sum_apply", "def_path": "Mathlib/Algebra/BigOperators/Pi.lean", "def_pos": [41, 3], "def_end_pos": [41, 14]}]], "state_before": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\n\u22a2 stoppedProcess u \u03c4 n \u03c9 =\n    (Set.indicator {a | n \u2264 \u03c4 a} (u n) + \u2211 i in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = i} (u i)) \u03c9", "state_after": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\n\u22a2 stoppedProcess u \u03c4 n \u03c9 =\n    Set.indicator {a | n \u2264 \u03c4 a} (u n) \u03c9 + \u2211 c in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = c} (u c) \u03c9"}, {"tactic": "cases' le_or_lt n (\u03c4 \u03c9) with h h", "annotated_tactic": ["cases' <a>le_or_lt</a> n (\u03c4 \u03c9) with h h", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [340, 9], "def_end_pos": [340, 17]}]], "state_before": "case h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\n\u22a2 stoppedProcess u \u03c4 n \u03c9 =\n    Set.indicator {a | n \u2264 \u03c4 a} (u n) \u03c9 + \u2211 c in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = c} (u c) \u03c9", "state_after": "case h.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\n\u22a2 stoppedProcess u \u03c4 n \u03c9 =\n    Set.indicator {a | n \u2264 \u03c4 a} (u n) \u03c9 + \u2211 c in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = c} (u c) \u03c9\n\ncase h.inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 stoppedProcess u \u03c4 n \u03c9 =\n    Set.indicator {a | n \u2264 \u03c4 a} (u n) \u03c9 + \u2211 c in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = c} (u c) \u03c9"}, {"tactic": "rw [stoppedProcess_eq_of_le h, Set.indicator_of_mem, Finset.sum_eq_zero, add_zero]", "annotated_tactic": ["rw [<a>stoppedProcess_eq_of_le</a> h, <a>Set.indicator_of_mem</a>, <a>Finset.sum_eq_zero</a>, <a>add_zero</a>]", [{"full_name": "MeasureTheory.stoppedProcess_eq_of_le", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [794, 9], "def_end_pos": [794, 32]}, {"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}, {"full_name": "Finset.sum_eq_zero", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [728, 3], "def_end_pos": [728, 14]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "case h.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\n\u22a2 stoppedProcess u \u03c4 n \u03c9 =\n    Set.indicator {a | n \u2264 \u03c4 a} (u n) \u03c9 + \u2211 c in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = c} (u c) \u03c9", "state_after": "case h.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\n\u22a2 \u2200 (x : \u03b9), x \u2208 Finset.filter (fun x => x < n) s \u2192 Set.indicator {\u03c9 | \u03c4 \u03c9 = x} (u x) \u03c9 = 0\n\ncase h.inl.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\n\u22a2 \u03c9 \u2208 {a | n \u2264 \u03c4 a}"}, {"tactic": "intro m hm", "annotated_tactic": ["intro m hm", []], "state_before": "case h.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\n\u22a2 \u2200 (x : \u03b9), x \u2208 Finset.filter (fun x => x < n) s \u2192 Set.indicator {\u03c9 | \u03c4 \u03c9 = x} (u x) \u03c9 = 0", "state_after": "case h.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\nm : \u03b9\nhm : m \u2208 Finset.filter (fun x => x < n) s\n\u22a2 Set.indicator {\u03c9 | \u03c4 \u03c9 = m} (u m) \u03c9 = 0"}, {"tactic": "refine' Set.indicator_of_not_mem _ _", "annotated_tactic": ["refine' <a>Set.indicator_of_not_mem</a> _ _", [{"full_name": "Set.indicator_of_not_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [73, 3], "def_end_pos": [73, 14]}]], "state_before": "case h.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\nm : \u03b9\nhm : m \u2208 Finset.filter (fun x => x < n) s\n\u22a2 Set.indicator {\u03c9 | \u03c4 \u03c9 = m} (u m) \u03c9 = 0", "state_after": "case h.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\nm : \u03b9\nhm : m \u2208 Finset.filter (fun x => x < n) s\n\u22a2 \u00ac\u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 = m}"}, {"tactic": "rw [Finset.mem_filter] at hm", "annotated_tactic": ["rw [<a>Finset.mem_filter</a>] at hm", [{"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2708, 9], "def_end_pos": [2708, 19]}]], "state_before": "case h.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\nm : \u03b9\nhm : m \u2208 Finset.filter (fun x => x < n) s\n\u22a2 \u00ac\u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 = m}", "state_after": "case h.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\nm : \u03b9\nhm : m \u2208 s \u2227 m < n\n\u22a2 \u00ac\u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 = m}"}, {"tactic": "exact (hm.2.trans_le h).ne'", "annotated_tactic": ["exact (hm.2.<a>trans_le</a> h).<a>ne'</a>", [{"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [148, 7], "def_end_pos": [148, 21]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "case h.inl\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u271d : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\nm : \u03b9\nhm : m \u2208 s \u2227 m < n\n\u22a2 \u00ac\u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 = m}", "state_after": "no goals"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "case h.inl.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : n \u2264 \u03c4 \u03c9\n\u22a2 \u03c9 \u2208 {a | n \u2264 \u03c4 a}", "state_after": "no goals"}, {"tactic": "rw [stoppedProcess_eq_of_ge (le_of_lt h), Finset.sum_eq_single_of_mem (\u03c4 \u03c9)]", "annotated_tactic": ["rw [<a>stoppedProcess_eq_of_ge</a> (<a>le_of_lt</a> h), <a>Finset.sum_eq_single_of_mem</a> (\u03c4 \u03c9)]", [{"full_name": "MeasureTheory.stoppedProcess_eq_of_ge", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [798, 9], "def_end_pos": [798, 32]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "Finset.sum_eq_single_of_mem", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [784, 3], "def_end_pos": [784, 14]}]], "state_before": "case h.inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 stoppedProcess u \u03c4 n \u03c9 =\n    Set.indicator {a | n \u2264 \u03c4 a} (u n) \u03c9 + \u2211 c in Finset.filter (fun x => x < n) s, Set.indicator {\u03c9 | \u03c4 \u03c9 = c} (u c) \u03c9", "state_after": "case h.inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 u (\u03c4 \u03c9) \u03c9 = Set.indicator {a | n \u2264 \u03c4 a} (u n) \u03c9 + Set.indicator {\u03c9_1 | \u03c4 \u03c9_1 = \u03c4 \u03c9} (u (\u03c4 \u03c9)) \u03c9\n\ncase h.inr.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 \u03c4 \u03c9 \u2208 Finset.filter (fun x => x < n) s\n\ncase h.inr.h\u2080\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 \u2200 (b : \u03b9), b \u2208 Finset.filter (fun x => x < n) s \u2192 b \u2260 \u03c4 \u03c9 \u2192 Set.indicator {\u03c9 | \u03c4 \u03c9 = b} (u b) \u03c9 = 0"}, {"tactic": "rw [Set.indicator_of_not_mem, zero_add, Set.indicator_of_mem]", "annotated_tactic": ["rw [<a>Set.indicator_of_not_mem</a>, <a>zero_add</a>, <a>Set.indicator_of_mem</a>]", [{"full_name": "Set.indicator_of_not_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [73, 3], "def_end_pos": [73, 14]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}]], "state_before": "case h.inr\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 u (\u03c4 \u03c9) \u03c9 = Set.indicator {a | n \u2264 \u03c4 a} (u n) \u03c9 + Set.indicator {\u03c9_1 | \u03c4 \u03c9_1 = \u03c4 \u03c9} (u (\u03c4 \u03c9)) \u03c9", "state_after": "case h.inr.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 \u03c9 \u2208 {\u03c9_1 | \u03c4 \u03c9_1 = \u03c4 \u03c9}\n\ncase h.inr.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 \u00ac\u03c9 \u2208 {a | n \u2264 \u03c4 a}"}, {"tactic": "exact rfl", "annotated_tactic": ["exact <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case h.inr.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 \u03c9 \u2208 {\u03c9_1 | \u03c4 \u03c9_1 = \u03c4 \u03c9}", "state_after": "no goals"}, {"tactic": "exact not_le.2 h", "annotated_tactic": ["exact <a>not_le</a>.2 h", [{"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "case h.inr.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 \u00ac\u03c9 \u2208 {a | n \u2264 \u03c4 a}", "state_after": "no goals"}, {"tactic": "rw [Finset.mem_filter]", "annotated_tactic": ["rw [<a>Finset.mem_filter</a>]", [{"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2708, 9], "def_end_pos": [2708, 19]}]], "state_before": "case h.inr.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 \u03c4 \u03c9 \u2208 Finset.filter (fun x => x < n) s", "state_after": "case h.inr.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 \u03c4 \u03c9 \u2208 s \u2227 \u03c4 \u03c9 < n"}, {"tactic": "exact \u27e8hbdd \u03c9 h, h\u27e9", "annotated_tactic": ["exact \u27e8hbdd \u03c9 h, h\u27e9", []], "state_before": "case h.inr.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 \u03c4 \u03c9 \u2208 s \u2227 \u03c4 \u03c9 < n", "state_after": "no goals"}, {"tactic": "intro b _ hneq", "annotated_tactic": ["intro b _ hneq", []], "state_before": "case h.inr.h\u2080\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\n\u22a2 \u2200 (b : \u03b9), b \u2208 Finset.filter (fun x => x < n) s \u2192 b \u2260 \u03c4 \u03c9 \u2192 Set.indicator {\u03c9 | \u03c4 \u03c9 = b} (u b) \u03c9 = 0", "state_after": "case h.inr.h\u2080\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\nb : \u03b9\na\u271d : b \u2208 Finset.filter (fun x => x < n) s\nhneq : b \u2260 \u03c4 \u03c9\n\u22a2 Set.indicator {\u03c9 | \u03c4 \u03c9 = b} (u b) \u03c9 = 0"}, {"tactic": "rw [Set.indicator_of_not_mem]", "annotated_tactic": ["rw [<a>Set.indicator_of_not_mem</a>]", [{"full_name": "Set.indicator_of_not_mem", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [73, 3], "def_end_pos": [73, 14]}]], "state_before": "case h.inr.h\u2080\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\nb : \u03b9\na\u271d : b \u2208 Finset.filter (fun x => x < n) s\nhneq : b \u2260 \u03c4 \u03c9\n\u22a2 Set.indicator {\u03c9 | \u03c4 \u03c9 = b} (u b) \u03c9 = 0", "state_after": "case h.inr.h\u2080.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\nb : \u03b9\na\u271d : b \u2208 Finset.filter (fun x => x < n) s\nhneq : b \u2260 \u03c4 \u03c9\n\u22a2 \u00ac\u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 = b}"}, {"tactic": "exact hneq.symm", "annotated_tactic": ["exact hneq.symm", []], "state_before": "case h.inr.h\u2080.h\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u03c4 \u03c3 : \u03a9 \u2192 \u03b9\nE : Type u_4\np : \u211d\u22650\u221e\nu : \u03b9 \u2192 \u03a9 \u2192 E\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : AddCommMonoid E\ns : Finset \u03b9\nn : \u03b9\nhbdd : \u2200 (\u03c9 : \u03a9), \u03c4 \u03c9 < n \u2192 \u03c4 \u03c9 \u2208 s\n\u03c9 : \u03a9\nh : \u03c4 \u03c9 < n\nb : \u03b9\na\u271d : b \u2208 Finset.filter (fun x => x < n) s\nhneq : b \u2260 \u03c4 \u03c9\n\u22a2 \u00ac\u03c9 \u2208 {\u03c9 | \u03c4 \u03c9 = b}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "full_name": "Int.not_lt", "start": [667, 11], "end": [668, 42], "traced_tactics": [{"tactic": "rw [\u2190 Int.not_le, Decidable.not_not]", "annotated_tactic": ["rw [\u2190 <a>Int.not_le</a>, <a>Decidable.not_not</a>]", [{"full_name": "Int.not_le", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [663, 19], "def_end_pos": [663, 25]}, {"full_name": "Decidable.not_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [513, 9], "def_end_pos": [513, 26]}]], "state_before": "a b : Int\n\u22a2 \u00aca < b \u2194 b \u2264 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Tactic/Group.lean", "full_name": "Mathlib.Tactic.Group.zpow_trick", "start": [37, 1], "end": [38, 73], "traced_tactics": [{"tactic": "rw [mul_assoc, \u2190 zpow_add]", "annotated_tactic": ["rw [<a>mul_assoc</a>, \u2190 <a>zpow_add</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "zpow_add", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [215, 9], "def_end_pos": [215, 17]}]], "state_before": "G : Type u_1\ninst\u271d : Group G\na b : G\nn m : \u2124\n\u22a2 a * b ^ n * b ^ m = a * b ^ (n + m)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.diff_univ", "start": [1935, 1], "end": [1936, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Module/Multilinear.lean", "full_name": "ContinuousMultilinearMap.coe_restrictScalars", "start": [443, 1], "end": [444, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/UniformSpace/Basic.lean", "full_name": "prod_mk_mem_compRel", "start": [175, 1], "end": [177, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "full_name": "Real.Angle.cos_coe", "start": [323, 1], "end": [324, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Instances/ENNReal.lean", "full_name": "nhds_eq_nhds_emetric_ball", "start": [1399, 1], "end": [1401, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/WittVector/Isocrystal.lean", "full_name": "WittVector.StandardOneDimIsocrystal.frobenius_apply", "start": [199, 1], "end": [204, 6], "traced_tactics": [{"tactic": "erw [smul_eq_mul]", "annotated_tactic": ["erw [<a>smul_eq_mul</a>]", [{"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}]], "state_before": "p : \u2115\ninst\u271d\u2074 : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d\u00b3 : CommRing k\ninst\u271d\u00b2 : IsDomain k\ninst\u271d\u00b9 : CharP k p\ninst\u271d : PerfectRing k p\nm : \u2124\nx : StandardOneDimIsocrystal p k m\n\u22a2 \u2191\u03a6(p, k) x = \u2191p ^ m \u2022 \u2191\u03c6(p, k) x", "state_after": "p : \u2115\ninst\u271d\u2074 : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d\u00b3 : CommRing k\ninst\u271d\u00b2 : IsDomain k\ninst\u271d\u00b9 : CharP k p\ninst\u271d : PerfectRing k p\nm : \u2124\nx : StandardOneDimIsocrystal p k m\n\u22a2 \u2191\u03a6(p, k) x = \u2191p ^ m * \u2191\u03c6(p, k) x"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "p : \u2115\ninst\u271d\u2074 : Fact (Nat.Prime p)\nk : Type u_1\ninst\u271d\u00b3 : CommRing k\ninst\u271d\u00b2 : IsDomain k\ninst\u271d\u00b9 : CharP k p\ninst\u271d : PerfectRing k p\nm : \u2124\nx : StandardOneDimIsocrystal p k m\n\u22a2 \u2191\u03a6(p, k) x = \u2191p ^ m * \u2191\u03c6(p, k) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBNode.Stream.toList_nil", "start": [403, 9], "end": [403, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/OrdConnected.lean", "full_name": "Set.ordConnected_biInter", "start": [116, 1], "end": [118, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Subsemiring/Basic.lean", "full_name": "Subsemiring.mem_iSup_of_directed", "start": [1110, 1], "end": [1119, 55], "traced_tactics": [{"tactic": "refine' \u27e8_, fun \u27e8i, hi\u27e9 => (SetLike.le_def.1 <| le_iSup S i) hi\u27e9", "annotated_tactic": ["refine' \u27e8_, fun \u27e8i, hi\u27e9 => (<a>SetLike.le_def</a>.1 <| <a>le_iSup</a> S i) hi\u27e9", [{"full_name": "SetLike.le_def", "def_path": "Mathlib/Data/SetLike/Basic.lean", "def_pos": [191, 9], "def_end_pos": [191, 15]}, {"full_name": "le_iSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [810, 9], "def_end_pos": [810, 16]}]], "state_before": "R : Type u\nS\u271d : Type v\nT : Type w\ninst\u271d\u00b2 : NonAssocSemiring R\nM : Submonoid R\ninst\u271d\u00b9 : NonAssocSemiring S\u271d\ninst\u271d : NonAssocSemiring T\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subsemiring R\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : R\n\u22a2 x \u2208 \u2a06 i, S i \u2194 \u2203 i, x \u2208 S i", "state_after": "R : Type u\nS\u271d : Type v\nT : Type w\ninst\u271d\u00b2 : NonAssocSemiring R\nM : Submonoid R\ninst\u271d\u00b9 : NonAssocSemiring S\u271d\ninst\u271d : NonAssocSemiring T\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subsemiring R\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : R\n\u22a2 x \u2208 \u2a06 i, S i \u2192 \u2203 i, x \u2208 S i"}, {"tactic": "let U : Subsemiring R :=\n  Subsemiring.mk' (\u22c3 i, (S i : Set R)) (\u2a06 i, (S i).toSubmonoid)\n    (Submonoid.coe_iSup_of_directed <| hS.mono_comp _ fun _ _ => id) (\u2a06 i, (S i).toAddSubmonoid)\n    (AddSubmonoid.coe_iSup_of_directed <| hS.mono_comp _ fun _ _ => id)", "annotated_tactic": ["let U : <a>Subsemiring</a> R :=\n    <a>Subsemiring.mk'</a> (\u22c3 i, (S i : <a>Set</a> R)) (\u2a06 i, (S i).<a>toSubmonoid</a>)\n      (<a>Submonoid.coe_iSup_of_directed</a> <| hS.mono_comp _ fun _ _ => <a>id</a>) (\u2a06 i, (S i).<a>toAddSubmonoid</a>)\n      (<a>AddSubmonoid.coe_iSup_of_directed</a> <| hS.mono_comp _ fun _ _ => <a>id</a>)", [{"full_name": "Subsemiring", "def_path": "Mathlib/RingTheory/Subsemiring/Basic.lean", "def_pos": [188, 11], "def_end_pos": [188, 22]}, {"full_name": "Subsemiring.mk'", "def_path": "Mathlib/RingTheory/Subsemiring/Basic.lean", "def_pos": [273, 15], "def_end_pos": [273, 18]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Subsemiring.toSubmonoid", "def_path": "Mathlib/RingTheory/Subsemiring/Basic.lean", "def_pos": [192, 14], "def_end_pos": [192, 37]}, {"full_name": "Submonoid.coe_iSup_of_directed", "def_path": "Mathlib/GroupTheory/Submonoid/Membership.lean", "def_pos": [212, 9], "def_end_pos": [212, 29]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"full_name": "Subsemiring.toAddSubmonoid", "def_path": "Mathlib/RingTheory/Subsemiring/Basic.lean", "def_pos": [195, 14], "def_end_pos": [195, 40]}, {"full_name": "AddSubmonoid.coe_iSup_of_directed", "def_path": "Mathlib/GroupTheory/Submonoid/Membership.lean", "def_pos": [211, 3], "def_end_pos": [211, 14]}, {"full_name": "id", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "state_before": "R : Type u\nS\u271d : Type v\nT : Type w\ninst\u271d\u00b2 : NonAssocSemiring R\nM : Submonoid R\ninst\u271d\u00b9 : NonAssocSemiring S\u271d\ninst\u271d : NonAssocSemiring T\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subsemiring R\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : R\n\u22a2 x \u2208 \u2a06 i, S i \u2192 \u2203 i, x \u2208 S i", "state_after": "R : Type u\nS\u271d : Type v\nT : Type w\ninst\u271d\u00b2 : NonAssocSemiring R\nM : Submonoid R\ninst\u271d\u00b9 : NonAssocSemiring S\u271d\ninst\u271d : NonAssocSemiring T\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subsemiring R\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : R\nU : Subsemiring R :=\n  Subsemiring.mk' (\u22c3 i, \u2191(S i)) (\u2a06 i, (S i).toSubmonoid) (_ : \u2191(\u2a06 i, (S i).toSubmonoid) = \u22c3 i, \u2191(S i).toSubmonoid)\n    (\u2a06 i, toAddSubmonoid (S i)) (_ : \u2191(\u2a06 i, toAddSubmonoid (S i)) = \u22c3 i, \u2191(toAddSubmonoid (S i)))\n\u22a2 x \u2208 \u2a06 i, S i \u2192 \u2203 i, x \u2208 S i"}, {"tactic": "suffices h : \u2a06 i, S i \u2264 U by simpa using @h x", "annotated_tactic": ["suffices h : \u2a06 i, S i \u2264 U by simpa using @h x", []], "state_before": "R : Type u\nS\u271d : Type v\nT : Type w\ninst\u271d\u00b2 : NonAssocSemiring R\nM : Submonoid R\ninst\u271d\u00b9 : NonAssocSemiring S\u271d\ninst\u271d : NonAssocSemiring T\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subsemiring R\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : R\nU : Subsemiring R :=\n  Subsemiring.mk' (\u22c3 i, \u2191(S i)) (\u2a06 i, (S i).toSubmonoid) (_ : \u2191(\u2a06 i, (S i).toSubmonoid) = \u22c3 i, \u2191(S i).toSubmonoid)\n    (\u2a06 i, toAddSubmonoid (S i)) (_ : \u2191(\u2a06 i, toAddSubmonoid (S i)) = \u22c3 i, \u2191(toAddSubmonoid (S i)))\n\u22a2 x \u2208 \u2a06 i, S i \u2192 \u2203 i, x \u2208 S i", "state_after": "R : Type u\nS\u271d : Type v\nT : Type w\ninst\u271d\u00b2 : NonAssocSemiring R\nM : Submonoid R\ninst\u271d\u00b9 : NonAssocSemiring S\u271d\ninst\u271d : NonAssocSemiring T\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subsemiring R\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : R\nU : Subsemiring R :=\n  Subsemiring.mk' (\u22c3 i, \u2191(S i)) (\u2a06 i, (S i).toSubmonoid) (_ : \u2191(\u2a06 i, (S i).toSubmonoid) = \u22c3 i, \u2191(S i).toSubmonoid)\n    (\u2a06 i, toAddSubmonoid (S i)) (_ : \u2191(\u2a06 i, toAddSubmonoid (S i)) = \u22c3 i, \u2191(toAddSubmonoid (S i)))\n\u22a2 \u2a06 i, S i \u2264 U"}, {"tactic": "exact iSup_le fun i x hx => Set.mem_iUnion.2 \u27e8i, hx\u27e9", "annotated_tactic": ["exact <a>iSup_le</a> fun i x hx => <a>Set.mem_iUnion</a>.2 \u27e8i, hx\u27e9", [{"full_name": "iSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [875, 9], "def_end_pos": [875, 16]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}]], "state_before": "R : Type u\nS\u271d : Type v\nT : Type w\ninst\u271d\u00b2 : NonAssocSemiring R\nM : Submonoid R\ninst\u271d\u00b9 : NonAssocSemiring S\u271d\ninst\u271d : NonAssocSemiring T\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subsemiring R\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : R\nU : Subsemiring R :=\n  Subsemiring.mk' (\u22c3 i, \u2191(S i)) (\u2a06 i, (S i).toSubmonoid) (_ : \u2191(\u2a06 i, (S i).toSubmonoid) = \u22c3 i, \u2191(S i).toSubmonoid)\n    (\u2a06 i, toAddSubmonoid (S i)) (_ : \u2191(\u2a06 i, toAddSubmonoid (S i)) = \u22c3 i, \u2191(toAddSubmonoid (S i)))\n\u22a2 \u2a06 i, S i \u2264 U", "state_after": "no goals"}, {"tactic": "simpa using @h x", "annotated_tactic": ["simpa using @h x", []], "state_before": "R : Type u\nS\u271d : Type v\nT : Type w\ninst\u271d\u00b2 : NonAssocSemiring R\nM : Submonoid R\ninst\u271d\u00b9 : NonAssocSemiring S\u271d\ninst\u271d : NonAssocSemiring T\n\u03b9 : Sort u_1\nh\u03b9 : Nonempty \u03b9\nS : \u03b9 \u2192 Subsemiring R\nhS : Directed (fun x x_1 => x \u2264 x_1) S\nx : R\nU : Subsemiring R :=\n  Subsemiring.mk' (\u22c3 i, \u2191(S i)) (\u2a06 i, (S i).toSubmonoid) (_ : \u2191(\u2a06 i, (S i).toSubmonoid) = \u22c3 i, \u2191(S i).toSubmonoid)\n    (\u2a06 i, toAddSubmonoid (S i)) (_ : \u2191(\u2a06 i, toAddSubmonoid (S i)) = \u22c3 i, \u2191(toAddSubmonoid (S i)))\nh : \u2a06 i, S i \u2264 U\n\u22a2 x \u2208 \u2a06 i, S i \u2192 \u2203 i, x \u2208 S i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean", "full_name": "MeasureTheory.integral_abs_condexp_le", "start": [93, 1], "end": [114, 73], "traced_tactics": [{"tactic": "by_cases hm : m \u2264 m0", "annotated_tactic": ["by_cases hm : m \u2264 m0", []], "state_before": "\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\n\u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\n\u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : \u00acm \u2264 m0\n\u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\n\u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : \u00acm \u2264 m0\n\u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : \u00acm \u2264 m0\n\u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc\n\ncase pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\n\u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc"}, {"tactic": "by_cases hfint : Integrable f \u03bc", "annotated_tactic": ["by_cases hfint : <a>Integrable</a> f \u03bc", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\n\u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : \u00acIntegrable f\n\u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : \u00acIntegrable f\n\u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : \u00acIntegrable f\n\u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc\n\ncase pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc"}, {"tactic": "rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae]", "annotated_tactic": ["rw [<a>integral_eq_lintegral_of_nonneg_ae</a>, <a>integral_eq_lintegral_of_nonneg_ae</a>]", [{"full_name": "MeasureTheory.integral_eq_lintegral_of_nonneg_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1158, 9], "def_end_pos": [1158, 43]}, {"full_name": "MeasureTheory.integral_eq_lintegral_of_nonneg_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1158, 9], "def_end_pos": [1158, 43]}]], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal |(\u03bc[f|m]) a| \u2202\u03bc) \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal |f a| \u2202\u03bc)\n\ncase pos.hf\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 0 \u2264\u1d50[\u03bc] fun x => |f x|\n\ncase pos.hfm\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 AEStronglyMeasurable (fun x => |f x|) \u03bc\n\ncase pos.hf\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 0 \u2264\u1d50[\u03bc] fun x => |(\u03bc[f|m]) x|\n\ncase pos.hfm\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 AEStronglyMeasurable (fun x => |(\u03bc[f|m]) x|) \u03bc"}, {"tactic": "simp_rw [condexp_of_not_le hm, Pi.zero_apply, abs_zero, integral_zero]", "annotated_tactic": ["simp_rw [<a>condexp_of_not_le</a> hm, <a>Pi.zero_apply</a>, <a>abs_zero</a>, <a>integral_zero</a>]", [{"full_name": "MeasureTheory.condexp_of_not_le", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [106, 9], "def_end_pos": [106, 26]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "abs_zero", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [128, 9], "def_end_pos": [128, 17]}, {"full_name": "MeasureTheory.integral_zero", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [850, 9], "def_end_pos": [850, 22]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : \u00acm \u2264 m0\n\u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : \u00acm \u2264 m0\n\u22a2 0 \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc"}, {"tactic": "exact integral_nonneg fun x => abs_nonneg _", "annotated_tactic": ["exact <a>integral_nonneg</a> fun x => <a>abs_nonneg</a> _", [{"full_name": "MeasureTheory.integral_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 24]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : \u00acm \u2264 m0\n\u22a2 0 \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc", "state_after": "no goals"}, {"tactic": "simp only [condexp_undef hfint, Pi.zero_apply, abs_zero, integral_const, Algebra.id.smul_eq_mul,\n  mul_zero]", "annotated_tactic": ["simp only [<a>condexp_undef</a> hfint, <a>Pi.zero_apply</a>, <a>abs_zero</a>, <a>integral_const</a>, <a>Algebra.id.smul_eq_mul</a>,\n      <a>mul_zero</a>]", [{"full_name": "MeasureTheory.condexp_undef", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [159, 9], "def_end_pos": [159, 22]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "abs_zero", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [128, 9], "def_end_pos": [128, 17]}, {"full_name": "MeasureTheory.integral_const", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 23]}, {"full_name": "Algebra.id.smul_eq_mul", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [453, 9], "def_end_pos": [453, 20]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : \u00acIntegrable f\n\u22a2 \u222b (x : \u03b1), |(\u03bc[f|m]) x| \u2202\u03bc \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : \u00acIntegrable f\n\u22a2 0 \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc"}, {"tactic": "exact integral_nonneg fun x => abs_nonneg _", "annotated_tactic": ["exact <a>integral_nonneg</a> fun x => <a>abs_nonneg</a> _", [{"full_name": "MeasureTheory.integral_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 24]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : \u00acIntegrable f\n\u22a2 0 \u2264 \u222b (x : \u03b1), |f x| \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [ENNReal.toReal_le_toReal] <;> simp_rw [\u2190 Real.norm_eq_abs, ofReal_norm_eq_coe_nnnorm]", "annotated_tactic": ["rw [<a>ENNReal.toReal_le_toReal</a>] <;> simp_rw [\u2190 <a>Real.norm_eq_abs</a>, <a>ofReal_norm_eq_coe_nnnorm</a>]", [{"full_name": "ENNReal.toReal_le_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2036, 9], "def_end_pos": [2036, 25]}, {"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 20]}, {"full_name": "ofReal_norm_eq_coe_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [999, 15], "def_end_pos": [999, 40]}]], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal |(\u03bc[f|m]) a| \u2202\u03bc) \u2264 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal |f a| \u2202\u03bc)", "state_after": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016(\u03bc[f|m]) a\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc\n\ncase pos.ha\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016(\u03bc[f|m]) a\u2016\u208a \u2202\u03bc \u2260 \u22a4\n\ncase pos.hb\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc \u2260 \u22a4"}, {"tactic": "rw [\u2190 snorm_one_eq_lintegral_nnnorm, \u2190 snorm_one_eq_lintegral_nnnorm]", "annotated_tactic": ["rw [\u2190 <a>snorm_one_eq_lintegral_nnnorm</a>, \u2190 <a>snorm_one_eq_lintegral_nnnorm</a>]", [{"full_name": "MeasureTheory.snorm_one_eq_lintegral_nnnorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [97, 9], "def_end_pos": [97, 38]}, {"full_name": "MeasureTheory.snorm_one_eq_lintegral_nnnorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [97, 9], "def_end_pos": [97, 38]}]], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016(\u03bc[f|m]) a\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 snorm (fun a => (\u03bc[f|m]) a) 1 \u03bc \u2264 snorm (fun a => f a) 1 \u03bc"}, {"tactic": "exact snorm_one_condexp_le_snorm _", "annotated_tactic": ["exact <a>snorm_one_condexp_le_snorm</a> _", [{"full_name": "MeasureTheory.snorm_one_condexp_le_snorm", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean", "def_pos": [60, 9], "def_end_pos": [60, 35]}]], "state_before": "case pos\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 snorm (fun a => (\u03bc[f|m]) a) 1 \u03bc \u2264 snorm (fun a => f a) 1 \u03bc", "state_after": "no goals"}, {"tactic": "exact ne_of_lt integrable_condexp.2", "annotated_tactic": ["exact <a>ne_of_lt</a> <a>integrable_condexp</a>.2", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "MeasureTheory.integrable_condexp", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [210, 9], "def_end_pos": [210, 27]}]], "state_before": "case pos.ha\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016(\u03bc[f|m]) a\u2016\u208a \u2202\u03bc \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "exact ne_of_lt hfint.2", "annotated_tactic": ["exact <a>ne_of_lt</a> hfint.2", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}]], "state_before": "case pos.hb\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "exact eventually_of_forall fun x => abs_nonneg _", "annotated_tactic": ["exact <a>eventually_of_forall</a> fun x => <a>abs_nonneg</a> _", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}]], "state_before": "case pos.hf\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 0 \u2264\u1d50[\u03bc] fun x => |f x|", "state_after": "no goals"}, {"tactic": "simp_rw [\u2190 Real.norm_eq_abs]", "annotated_tactic": ["simp_rw [\u2190 <a>Real.norm_eq_abs</a>]", [{"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 20]}]], "state_before": "case pos.hfm\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 AEStronglyMeasurable (fun x => |f x|) \u03bc", "state_after": "case pos.hfm\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 AEStronglyMeasurable (fun x => \u2016f x\u2016) \u03bc"}, {"tactic": "exact hfint.1.norm", "annotated_tactic": ["exact hfint.1.<a>norm</a>", [{"full_name": "MeasureTheory.AEStronglyMeasurable.norm", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1479, 19], "def_end_pos": [1479, 23]}]], "state_before": "case pos.hfm\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 AEStronglyMeasurable (fun x => \u2016f x\u2016) \u03bc", "state_after": "no goals"}, {"tactic": "exact eventually_of_forall fun x => abs_nonneg _", "annotated_tactic": ["exact <a>eventually_of_forall</a> fun x => <a>abs_nonneg</a> _", [{"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 29]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}]], "state_before": "case pos.hf\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 0 \u2264\u1d50[\u03bc] fun x => |(\u03bc[f|m]) x|", "state_after": "no goals"}, {"tactic": "simp_rw [\u2190 Real.norm_eq_abs]", "annotated_tactic": ["simp_rw [\u2190 <a>Real.norm_eq_abs</a>]", [{"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1761, 9], "def_end_pos": [1761, 20]}]], "state_before": "case pos.hfm\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 AEStronglyMeasurable (fun x => |(\u03bc[f|m]) x|) \u03bc", "state_after": "case pos.hfm\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 AEStronglyMeasurable (fun x => \u2016(\u03bc[f|m]) x\u2016) \u03bc"}, {"tactic": "exact (stronglyMeasurable_condexp.mono hm).aestronglyMeasurable.norm", "annotated_tactic": ["exact (stronglyMeasurable_condexp.mono hm).aestronglyMeasurable.norm", []], "state_before": "case pos.hfm\n\u03b1 : Type u_1\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\nhm : m \u2264 m0\nhfint : Integrable f\n\u22a2 AEStronglyMeasurable (fun x => \u2016(\u03bc[f|m]) x\u2016) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/Padics/PadicIntegers.lean", "full_name": "PadicInt.p_nonnunit", "start": [587, 1], "end": [589, 41], "traced_tactics": [{"tactic": "have : (p : \u211d)\u207b\u00b9 < 1 := inv_lt_one <| by exact_mod_cast hp.1.one_lt", "annotated_tactic": ["have : (p : \u211d)\u207b\u00b9 < 1 := <a>inv_lt_one</a> <| by exact_mod_cast hp.1.<a>one_lt</a>", [{"full_name": "inv_lt_one", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [297, 9], "def_end_pos": [297, 19]}, {"full_name": "Nat.Prime.one_lt", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [76, 9], "def_end_pos": [76, 21]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 \u2191p \u2208 nonunits \u2124_[p]", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nthis : (\u2191p)\u207b\u00b9 < 1\n\u22a2 \u2191p \u2208 nonunits \u2124_[p]"}, {"tactic": "rwa [\u2190 norm_p, \u2190 mem_nonunits] at this", "annotated_tactic": ["rwa [\u2190 <a>norm_p</a>, \u2190 <a>mem_nonunits</a>] at this", [{"full_name": "PadicInt.norm_p", "def_path": "Mathlib/NumberTheory/Padics/PadicIntegers.lean", "def_pos": [305, 9], "def_end_pos": [305, 15]}, {"full_name": "PadicInt.mem_nonunits", "def_path": "Mathlib/NumberTheory/Padics/PadicIntegers.lean", "def_pos": [460, 9], "def_end_pos": [460, 21]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nthis : (\u2191p)\u207b\u00b9 < 1\n\u22a2 \u2191p \u2208 nonunits \u2124_[p]", "state_after": "no goals"}, {"tactic": "exact_mod_cast hp.1.one_lt", "annotated_tactic": ["exact_mod_cast hp.1.<a>one_lt</a>", [{"full_name": "Nat.Prime.one_lt", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [76, 9], "def_end_pos": [76, 21]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 1 < \u2191p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/VectorBundle/Basic.lean", "full_name": "VectorBundleCore.mem_localTriv_source", "start": [698, 1], "end": [699, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "full_name": "MeasureTheory.Measure.measure_Ioo_eq_zero", "start": [182, 1], "end": [183, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "full_name": "mul_left_cancel''", "start": [293, 1], "end": [295, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Uniform.lean", "full_name": "PMF.toMeasure_uniformOfFintype_apply", "start": [142, 1], "end": [145, 6], "traced_tactics": [{"tactic": "simp [uniformOfFintype, hs]", "annotated_tactic": ["simp [<a>uniformOfFintype</a>, hs]", [{"full_name": "PMF.uniformOfFintype", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Uniform.lean", "def_pos": [111, 5], "def_end_pos": [111, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ns : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(toMeasure (uniformOfFintype \u03b1)) s = \u2191(Fintype.card \u2191s) / \u2191(Fintype.card \u03b1)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ns : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191(Finset.card (Finset.filter (fun x => x \u2208 s) Finset.univ)) / \u2191(Finset.card Finset.univ) =\n    \u2191(Finset.card (Finset.filter (fun x => x \u2208 s) Finset.univ)) / \u2191(Fintype.card \u03b1)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ns : Set \u03b1\ninst\u271d : MeasurableSpace \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191(Finset.card (Finset.filter (fun x => x \u2208 s) Finset.univ)) / \u2191(Finset.card Finset.univ) =\n    \u2191(Finset.card (Finset.filter (fun x => x \u2208 s) Finset.univ)) / \u2191(Fintype.card \u03b1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Ordinal/Notation.lean", "full_name": "ONote.NFBelow.lt", "start": [252, 1], "end": [253, 50], "traced_tactics": [{"tactic": "(cases' h with _ _ _ _ eb _ h\u2081 h\u2082 h\u2083; exact h\u2083)", "annotated_tactic": ["(cases' h with _ _ _ _ eb _ h\u2081 h\u2082 h\u2083; exact h\u2083)", []], "state_before": "e : ONote\nn : \u2115+\na : ONote\nb : Ordinal.{0}\nh : NFBelow (ONote.oadd e n a) b\n\u22a2 repr e < b", "state_after": "no goals"}, {"tactic": "cases' h with _ _ _ _ eb _ h\u2081 h\u2082 h\u2083", "annotated_tactic": ["cases' h with _ _ _ _ eb _ h\u2081 h\u2082 h\u2083", []], "state_before": "e : ONote\nn : \u2115+\na : ONote\nb : Ordinal.{0}\nh : NFBelow (ONote.oadd e n a) b\n\u22a2 repr e < b", "state_after": "case oadd'\ne : ONote\nn : \u2115+\na : ONote\nb eb : Ordinal.{0}\nh\u2081 : NFBelow e eb\nh\u2082 : NFBelow a (repr e)\nh\u2083 : repr e < b\n\u22a2 repr e < b"}, {"tactic": "exact h\u2083", "annotated_tactic": ["exact h\u2083", []], "state_before": "case oadd'\ne : ONote\nn : \u2115+\na : ONote\nb eb : Ordinal.{0}\nh\u2081 : NFBelow e eb\nh\u2082 : NFBelow a (repr e)\nh\u2083 : repr e < b\n\u22a2 repr e < b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/UniformSpace/AbstractCompletion.lean", "full_name": "AbstractCompletion.continuous_map", "start": [195, 1], "end": [196, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/AlgebraicCard.lean", "full_name": "Algebraic.cardinal_mk_le_mul", "start": [93, 1], "end": [95, 36], "traced_tactics": [{"tactic": "rw [\u2190 lift_id #_, \u2190 lift_id #R[X]]", "annotated_tactic": ["rw [\u2190 <a>lift_id</a> #_, \u2190 <a>lift_id</a> #R[X]]", [{"full_name": "Cardinal.lift_id", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [224, 9], "def_end_pos": [224, 16]}, {"full_name": "Cardinal.lift_id", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [224, 9], "def_end_pos": [224, 16]}]], "state_before": "R A : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\n\u22a2 #{ x // IsAlgebraic R x } \u2264 #R[X] * \u2135\u2080", "state_after": "R A : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\n\u22a2 lift.{u, u} #{ x // IsAlgebraic R x } \u2264 lift.{u, u} #R[X] * \u2135\u2080"}, {"tactic": "exact cardinal_mk_lift_le_mul R A", "annotated_tactic": ["exact <a>cardinal_mk_lift_le_mul</a> R A", [{"full_name": "Algebraic.cardinal_mk_lift_le_mul", "def_path": "Mathlib/Algebra/AlgebraicCard.lean", "def_pos": [45, 9], "def_end_pos": [45, 32]}]], "state_before": "R A : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : IsDomain A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroSMulDivisors R A\n\u22a2 lift.{u, u} #{ x // IsAlgebraic R x } \u2264 lift.{u, u} #R[X] * \u2135\u2080", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean", "full_name": "IsCoercive.ker_eq_bot", "start": [79, 1], "end": [82, 32], "traced_tactics": [{"tactic": "rw [LinearMapClass.ker_eq_bot]", "annotated_tactic": ["rw [<a>LinearMapClass.ker_eq_bot</a>]", [{"full_name": "LinearMapClass.ker_eq_bot", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [1415, 9], "def_end_pos": [1415, 41]}]], "state_before": "V : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\ncoercive : IsCoercive B\n\u22a2 ker (continuousLinearMapOfBilin B) = \u22a5", "state_after": "V : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\ncoercive : IsCoercive B\n\u22a2 Function.Injective \u2191(continuousLinearMapOfBilin B)"}, {"tactic": "rcases coercive.antilipschitz with \u27e8_, _, antilipschitz\u27e9", "annotated_tactic": ["rcases coercive.antilipschitz with \u27e8_, _, antilipschitz\u27e9", []], "state_before": "V : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\ncoercive : IsCoercive B\n\u22a2 Function.Injective \u2191(continuousLinearMapOfBilin B)", "state_after": "case intro.intro\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\ncoercive : IsCoercive B\nw\u271d : \u211d\u22650\nleft\u271d : 0 < w\u271d\nantilipschitz : AntilipschitzWith w\u271d \u2191(continuousLinearMapOfBilin B)\n\u22a2 Function.Injective \u2191(continuousLinearMapOfBilin B)"}, {"tactic": "exact antilipschitz.injective", "annotated_tactic": ["exact antilipschitz.injective", []], "state_before": "case intro.intro\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\ncoercive : IsCoercive B\nw\u271d : \u211d\u22650\nleft\u271d : 0 < w\u271d\nantilipschitz : AntilipschitzWith w\u271d \u2191(continuousLinearMapOfBilin B)\n\u22a2 Function.Injective \u2191(continuousLinearMapOfBilin B)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Accumulate.lean", "full_name": "Set.mem_accumulate", "start": [31, 1], "end": [32, 53], "traced_tactics": [{"tactic": "simp_rw [accumulate_def, mem_iUnion\u2082, exists_prop]", "annotated_tactic": ["simp_rw [<a>accumulate_def</a>, <a>mem_iUnion\u2082</a>, <a>exists_prop</a>]", [{"full_name": "Set.accumulate_def", "def_path": "Mathlib/Data/Set/Accumulate.lean", "def_pos": [26, 9], "def_end_pos": [26, 23]}, {"full_name": "Set.mem_iUnion\u2082", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [212, 9], "def_end_pos": [212, 20]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ns : \u03b1 \u2192 Set \u03b2\nt : \u03b1 \u2192 Set \u03b3\ninst\u271d : LE \u03b1\nx : \u03b1\nz : \u03b2\n\u22a2 z \u2208 Accumulate s x \u2194 \u2203 y, y \u2264 x \u2227 z \u2208 s y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Seq/WSeq.lean", "full_name": "Stream'.WSeq.toSeq_ofSeq", "start": [1384, 1], "end": [1387, 33], "traced_tactics": [{"tactic": "apply Subtype.eq", "annotated_tactic": ["apply <a>Subtype.eq</a>", [{"full_name": "Subtype.eq", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [960, 19], "def_end_pos": [960, 21]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\n\u22a2 toSeq \u2191s = s", "state_after": "case a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\n\u22a2 \u2191(toSeq \u2191s) = \u2191s"}, {"tactic": "funext n", "annotated_tactic": ["funext n", []], "state_before": "case a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\n\u22a2 \u2191(toSeq \u2191s) = \u2191s", "state_after": "case a.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nn : \u2115\n\u22a2 \u2191(toSeq \u2191s) n = \u2191s n"}, {"tactic": "dsimp [toSeq]", "annotated_tactic": ["dsimp [<a>toSeq</a>]", [{"full_name": "Stream'.WSeq.toSeq", "def_path": "Mathlib/Data/Seq/WSeq.lean", "def_pos": [879, 5], "def_end_pos": [879, 10]}]], "state_before": "case a.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nn : \u2115\n\u22a2 \u2191(toSeq \u2191s) n = \u2191s n", "state_after": "case a.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nn : \u2115\n\u22a2 Computation.get (get? (\u2191s) n) = \u2191s n"}, {"tactic": "apply get_eq_of_mem", "annotated_tactic": ["apply <a>get_eq_of_mem</a>", [{"full_name": "Computation.get_eq_of_mem", "def_path": "Mathlib/Data/Seq/Computation.lean", "def_pos": [463, 9], "def_end_pos": [463, 22]}]], "state_before": "case a.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nn : \u2115\n\u22a2 Computation.get (get? (\u2191s) n) = \u2191s n", "state_after": "case a.h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nn : \u2115\n\u22a2 \u2191s n \u2208 get? (\u2191s) n"}, {"tactic": "rw [get?_ofSeq]", "annotated_tactic": ["rw [<a>get?_ofSeq</a>]", [{"full_name": "Stream'.WSeq.get?_ofSeq", "def_path": "Mathlib/Data/Seq/WSeq.lean", "def_pos": [1376, 9], "def_end_pos": [1376, 19]}]], "state_before": "case a.h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nn : \u2115\n\u22a2 \u2191s n \u2208 get? (\u2191s) n", "state_after": "case a.h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nn : \u2115\n\u22a2 \u2191s n \u2208 Computation.pure (Seq.get? s n)"}, {"tactic": "apply ret_mem", "annotated_tactic": ["apply <a>ret_mem</a>", [{"full_name": "Computation.ret_mem", "def_path": "Mathlib/Data/Seq/Computation.lean", "def_pos": [369, 9], "def_end_pos": [369, 16]}]], "state_before": "case a.h.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns : Seq \u03b1\nn : \u2115\n\u22a2 \u2191s n \u2208 Computation.pure (Seq.get? s n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Lipschitz.lean", "full_name": "LipschitzOnWith.edist_le_mul_of_le", "start": [520, 1], "end": [523, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Order/Hom/Basic.lean", "full_name": "ContinuousOrderHom.coe_copy", "start": [128, 1], "end": [129, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_iInf_ae", "start": [982, 1], "end": [1010, 74], "traced_tactics": [{"tactic": "induction' n with n ih", "annotated_tactic": ["induction' n with n ih", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh_meas : \u2200 (n : \u2115), Measurable (f n)\nh_mono\u271d : \u2200 (n : \u2115), f (Nat.succ n) \u2264\u1d50[\u03bc] f n\nh_fin : \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc \u2260 \u22a4\nfn_le_f0 : \u222b\u207b (a : \u03b1), \u2a05 n, f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nfn_le_f0' : \u2a05 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\nn : \u2115\na : \u03b1\nh : \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\n\u22a2 f n a \u2264 f 0 a", "state_after": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh_meas : \u2200 (n : \u2115), Measurable (f n)\nh_mono\u271d : \u2200 (n : \u2115), f (Nat.succ n) \u2264\u1d50[\u03bc] f n\nh_fin : \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc \u2260 \u22a4\nfn_le_f0 : \u222b\u207b (a : \u03b1), \u2a05 n, f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nfn_le_f0' : \u2a05 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\na : \u03b1\nh : \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\n\u22a2 f Nat.zero a \u2264 f 0 a\n\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh_meas : \u2200 (n : \u2115), Measurable (f n)\nh_mono\u271d : \u2200 (n : \u2115), f (Nat.succ n) \u2264\u1d50[\u03bc] f n\nh_fin : \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc \u2260 \u22a4\nfn_le_f0 : \u222b\u207b (a : \u03b1), \u2a05 n, f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nfn_le_f0' : \u2a05 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\na : \u03b1\nh : \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\nn : \u2115\nih : f n a \u2264 f 0 a\n\u22a2 f (Nat.succ n) a \u2264 f 0 a"}, {"tactic": "exact le_rfl", "annotated_tactic": ["exact <a>le_rfl</a>", [{"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh_meas : \u2200 (n : \u2115), Measurable (f n)\nh_mono\u271d : \u2200 (n : \u2115), f (Nat.succ n) \u2264\u1d50[\u03bc] f n\nh_fin : \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc \u2260 \u22a4\nfn_le_f0 : \u222b\u207b (a : \u03b1), \u2a05 n, f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nfn_le_f0' : \u2a05 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\na : \u03b1\nh : \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\n\u22a2 f Nat.zero a \u2264 f 0 a", "state_after": "no goals"}, {"tactic": "exact le_trans (h n) ih", "annotated_tactic": ["exact <a>le_trans</a> (h n) ih", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "state_before": "case succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh_meas : \u2200 (n : \u2115), Measurable (f n)\nh_mono\u271d : \u2200 (n : \u2115), f (Nat.succ n) \u2264\u1d50[\u03bc] f n\nh_fin : \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc \u2260 \u22a4\nfn_le_f0 : \u222b\u207b (a : \u03b1), \u2a05 n, f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nfn_le_f0' : \u2a05 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f 0 a \u2202\u03bc\nh_mono : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\na : \u03b1\nh : \u2200 (n : \u2115), f (Nat.succ n) a \u2264 f n a\nn : \u2115\nih : f n a \u2264 f 0 a\n\u22a2 f (Nat.succ n) a \u2264 f 0 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/DiffContOnCl.lean", "full_name": "DiffContOnCl.const_smul", "start": [118, 1], "end": [120, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean", "full_name": "CliffordAlgebra.forall_mul_self_eq_iff", "start": [248, 1], "end": [257, 75], "traced_tactics": [{"tactic": "simp_rw [FunLike.ext_iff]", "annotated_tactic": ["simp_rw [<a>FunLike.ext_iff</a>]", [{"full_name": "FunLike.ext_iff", "def_path": "Mathlib/Data/FunLike/Basic.lean", "def_pos": [181, 9], "def_end_pos": [181, 16]}]], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA\u271d : Type u_3\ninst\u271d\u00b3 : Semiring A\u271d\ninst\u271d\u00b2 : Algebra R A\u271d\nA : Type u_4\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nh2 : IsUnit 2\nf : M \u2192\u2097[R] A\n\u22a2 (\u2200 (x : M), \u2191f x * \u2191f x = \u2191(algebraMap R A) (\u2191Q x)) \u2194\n    LinearMap.comp (LinearMap.compl\u2082 (LinearMap.mul R A) f) f +\n        LinearMap.comp (LinearMap.compl\u2082 (LinearMap.flip (LinearMap.mul R A)) f) f =\n      LinearMap.compr\u2082 (\u2191BilinForm.toLin (QuadraticForm.polarBilin Q)) (Algebra.linearMap R A)", "state_after": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA\u271d : Type u_3\ninst\u271d\u00b3 : Semiring A\u271d\ninst\u271d\u00b2 : Algebra R A\u271d\nA : Type u_4\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nh2 : IsUnit 2\nf : M \u2192\u2097[R] A\n\u22a2 (\u2200 (x : M), \u2191f x * \u2191f x = \u2191(algebraMap R A) (\u2191Q x)) \u2194\n    \u2200 (x x_1 : M),\n      \u2191(\u2191(LinearMap.comp (LinearMap.compl\u2082 (LinearMap.mul R A) f) f +\n                  LinearMap.comp (LinearMap.compl\u2082 (LinearMap.flip (LinearMap.mul R A)) f) f)\n              x)\n          x_1 =\n        \u2191(\u2191(LinearMap.compr\u2082 (\u2191BilinForm.toLin (QuadraticForm.polarBilin Q)) (Algebra.linearMap R A)) x) x_1"}, {"tactic": "refine \u27e8mul_add_swap_eq_polar_of_forall_mul_self_eq _, fun h x => ?_\u27e9", "annotated_tactic": ["refine \u27e8<a>mul_add_swap_eq_polar_of_forall_mul_self_eq</a> _, fun h x => ?_\u27e9", [{"full_name": "CliffordAlgebra.mul_add_swap_eq_polar_of_forall_mul_self_eq", "def_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean", "def_pos": [233, 9], "def_end_pos": [233, 52]}]], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA\u271d : Type u_3\ninst\u271d\u00b3 : Semiring A\u271d\ninst\u271d\u00b2 : Algebra R A\u271d\nA : Type u_4\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nh2 : IsUnit 2\nf : M \u2192\u2097[R] A\n\u22a2 (\u2200 (x : M), \u2191f x * \u2191f x = \u2191(algebraMap R A) (\u2191Q x)) \u2194\n    \u2200 (x x_1 : M),\n      \u2191(\u2191(LinearMap.comp (LinearMap.compl\u2082 (LinearMap.mul R A) f) f +\n                  LinearMap.comp (LinearMap.compl\u2082 (LinearMap.flip (LinearMap.mul R A)) f) f)\n              x)\n          x_1 =\n        \u2191(\u2191(LinearMap.compr\u2082 (\u2191BilinForm.toLin (QuadraticForm.polarBilin Q)) (Algebra.linearMap R A)) x) x_1", "state_after": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA\u271d : Type u_3\ninst\u271d\u00b3 : Semiring A\u271d\ninst\u271d\u00b2 : Algebra R A\u271d\nA : Type u_4\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nh2 : IsUnit 2\nf : M \u2192\u2097[R] A\nh :\n  \u2200 (x x_1 : M),\n    \u2191(\u2191(LinearMap.comp (LinearMap.compl\u2082 (LinearMap.mul R A) f) f +\n                LinearMap.comp (LinearMap.compl\u2082 (LinearMap.flip (LinearMap.mul R A)) f) f)\n            x)\n        x_1 =\n      \u2191(\u2191(LinearMap.compr\u2082 (\u2191BilinForm.toLin (QuadraticForm.polarBilin Q)) (Algebra.linearMap R A)) x) x_1\nx : M\n\u22a2 \u2191f x * \u2191f x = \u2191(algebraMap R A) (\u2191Q x)"}, {"tactic": "change \u2200 x y : M, f x * f y + f y * f x = algebraMap R A (QuadraticForm.polar Q x y) at h", "annotated_tactic": ["change \u2200 x y : M, f x * f y + f y * f x = <a>algebraMap</a> R A (<a>QuadraticForm.polar</a> Q x y) at h", [{"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 15]}, {"full_name": "QuadraticForm.polar", "def_path": "Mathlib/LinearAlgebra/QuadraticForm/Basic.lean", "def_pos": [94, 5], "def_end_pos": [94, 10]}]], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA\u271d : Type u_3\ninst\u271d\u00b3 : Semiring A\u271d\ninst\u271d\u00b2 : Algebra R A\u271d\nA : Type u_4\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nh2 : IsUnit 2\nf : M \u2192\u2097[R] A\nh :\n  \u2200 (x x_1 : M),\n    \u2191(\u2191(LinearMap.comp (LinearMap.compl\u2082 (LinearMap.mul R A) f) f +\n                LinearMap.comp (LinearMap.compl\u2082 (LinearMap.flip (LinearMap.mul R A)) f) f)\n            x)\n        x_1 =\n      \u2191(\u2191(LinearMap.compr\u2082 (\u2191BilinForm.toLin (QuadraticForm.polarBilin Q)) (Algebra.linearMap R A)) x) x_1\nx : M\n\u22a2 \u2191f x * \u2191f x = \u2191(algebraMap R A) (\u2191Q x)", "state_after": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA\u271d : Type u_3\ninst\u271d\u00b3 : Semiring A\u271d\ninst\u271d\u00b2 : Algebra R A\u271d\nA : Type u_4\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nh2 : IsUnit 2\nf : M \u2192\u2097[R] A\nx : M\nh : \u2200 (x y : M), \u2191f x * \u2191f y + \u2191f y * \u2191f x = \u2191(algebraMap R A) (QuadraticForm.polar (\u2191Q) x y)\n\u22a2 \u2191f x * \u2191f x = \u2191(algebraMap R A) (\u2191Q x)"}, {"tactic": "apply h2.mul_left_cancel", "annotated_tactic": ["apply h2.mul_left_cancel", []], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA\u271d : Type u_3\ninst\u271d\u00b3 : Semiring A\u271d\ninst\u271d\u00b2 : Algebra R A\u271d\nA : Type u_4\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nh2 : IsUnit 2\nf : M \u2192\u2097[R] A\nx : M\nh : \u2200 (x y : M), \u2191f x * \u2191f y + \u2191f y * \u2191f x = \u2191(algebraMap R A) (QuadraticForm.polar (\u2191Q) x y)\n\u22a2 \u2191f x * \u2191f x = \u2191(algebraMap R A) (\u2191Q x)", "state_after": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA\u271d : Type u_3\ninst\u271d\u00b3 : Semiring A\u271d\ninst\u271d\u00b2 : Algebra R A\u271d\nA : Type u_4\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nh2 : IsUnit 2\nf : M \u2192\u2097[R] A\nx : M\nh : \u2200 (x y : M), \u2191f x * \u2191f y + \u2191f y * \u2191f x = \u2191(algebraMap R A) (QuadraticForm.polar (\u2191Q) x y)\n\u22a2 2 * (\u2191f x * \u2191f x) = 2 * \u2191(algebraMap R A) (\u2191Q x)"}, {"tactic": "rw [two_mul, two_mul, h x x, QuadraticForm.polar_self, two_mul, map_add]", "annotated_tactic": ["rw [<a>two_mul</a>, <a>two_mul</a>, h x x, <a>QuadraticForm.polar_self</a>, <a>two_mul</a>, <a>map_add</a>]", [{"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}, {"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}, {"full_name": "QuadraticForm.polar_self", "def_path": "Mathlib/LinearAlgebra/QuadraticForm/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 19]}, {"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}, {"full_name": "map_add", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [298, 3], "def_end_pos": [298, 14]}]], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nQ : QuadraticForm R M\nn : \u2115\nA\u271d : Type u_3\ninst\u271d\u00b3 : Semiring A\u271d\ninst\u271d\u00b2 : Algebra R A\u271d\nA : Type u_4\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nh2 : IsUnit 2\nf : M \u2192\u2097[R] A\nx : M\nh : \u2200 (x y : M), \u2191f x * \u2191f y + \u2191f y * \u2191f x = \u2191(algebraMap R A) (QuadraticForm.polar (\u2191Q) x y)\n\u22a2 2 * (\u2191f x * \u2191f x) = 2 * \u2191(algebraMap R A) (\u2191Q x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Ideal/Operations.lean", "full_name": "Ideal.map_isPrime_of_surjective", "start": [2213, 1], "end": [2227, 98], "traced_tactics": [{"tactic": "refine' \u27e8fun h => H.ne_top (eq_top_iff.2 _), fun {x y} => _\u27e9", "annotated_tactic": ["refine' \u27e8fun h => H.ne_top (<a>eq_top_iff</a>.2 _), fun {x y} => _\u27e9", [{"full_name": "eq_top_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [165, 9], "def_end_pos": [165, 19]}]], "state_before": "R : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\n\u22a2 IsPrime (map f I)", "state_after": "case refine'_1\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nh : map f I = \u22a4\n\u22a2 \u22a4 \u2264 I\n\ncase refine'_2\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\n\u22a2 x * y \u2208 map f I \u2192 x \u2208 map f I \u2228 y \u2208 map f I"}, {"tactic": "replace h := congr_arg (comap f) h", "annotated_tactic": ["replace h := <a>congr_arg</a> (<a>comap</a> f) h", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "Ideal.comap", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [1372, 5], "def_end_pos": [1372, 10]}]], "state_before": "case refine'_1\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nh : map f I = \u22a4\n\u22a2 \u22a4 \u2264 I", "state_after": "case refine'_1\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nh : comap f (map f I) = comap f \u22a4\n\u22a2 \u22a4 \u2264 I"}, {"tactic": "rw [comap_map_of_surjective _ hf, comap_top] at h", "annotated_tactic": ["rw [<a>comap_map_of_surjective</a> _ hf, <a>comap_top</a>] at h", [{"full_name": "Ideal.comap_map_of_surjective", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [1716, 9], "def_end_pos": [1716, 32]}, {"full_name": "Ideal.comap_top", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [1507, 9], "def_end_pos": [1507, 18]}]], "state_before": "case refine'_1\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nh : comap f (map f I) = comap f \u22a4\n\u22a2 \u22a4 \u2264 I", "state_after": "case refine'_1\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nh : I \u2294 comap f \u22a5 = \u22a4\n\u22a2 \u22a4 \u2264 I"}, {"tactic": "exact h \u25b8 sup_le (le_of_eq rfl) hk", "annotated_tactic": ["exact h \u25b8 <a>sup_le</a> (<a>le_of_eq</a> <a>rfl</a>) hk", [{"full_name": "sup_le", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [167, 9], "def_end_pos": [167, 15]}, {"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case refine'_1\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nh : I \u2294 comap f \u22a5 = \u22a4\n\u22a2 \u22a4 \u2264 I", "state_after": "no goals"}, {"tactic": "refine' fun hxy => (hf x).recOn fun a ha => (hf y).recOn fun b hb => _", "annotated_tactic": ["refine' fun hxy => (hf x).<a>recOn</a> fun a ha => (hf y).<a>recOn</a> fun b hb => _", [{"full_name": "Exists.recOn", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [196, 11], "def_end_pos": [196, 17]}, {"full_name": "Exists.recOn", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [196, 11], "def_end_pos": [196, 17]}]], "state_before": "case refine'_2\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\n\u22a2 x * y \u2208 map f I \u2192 x \u2208 map f I \u2228 y \u2208 map f I", "state_after": "case refine'_2\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\nhxy : x * y \u2208 map f I\na : R\nha : \u2191f a = x\nb : R\nhb : \u2191f b = y\n\u22a2 x \u2208 map f I \u2228 y \u2208 map f I"}, {"tactic": "rw [\u2190 ha, \u2190 hb, \u2190 _root_.map_mul f, mem_map_iff_of_surjective _ hf] at hxy", "annotated_tactic": ["rw [\u2190 ha, \u2190 hb, \u2190 <a>_root_.map_mul</a> f, <a>mem_map_iff_of_surjective</a> _ hf] at hxy", [{"full_name": "map_mul", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [299, 9], "def_end_pos": [299, 16]}, {"full_name": "Ideal.mem_map_iff_of_surjective", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [1672, 9], "def_end_pos": [1672, 34]}]], "state_before": "case refine'_2\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\nhxy : x * y \u2208 map f I\na : R\nha : \u2191f a = x\nb : R\nhb : \u2191f b = y\n\u22a2 x \u2208 map f I \u2228 y \u2208 map f I", "state_after": "case refine'_2\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\na : R\nha : \u2191f a = x\nb : R\nhxy : \u2203 x, x \u2208 I \u2227 \u2191f x = \u2191f (a * b)\nhb : \u2191f b = y\n\u22a2 x \u2208 map f I \u2228 y \u2208 map f I"}, {"tactic": "rcases hxy with \u27e8c, hc, hc'\u27e9", "annotated_tactic": ["rcases hxy with \u27e8c, hc, hc'\u27e9", []], "state_before": "case refine'_2\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\na : R\nha : \u2191f a = x\nb : R\nhxy : \u2203 x, x \u2208 I \u2227 \u2191f x = \u2191f (a * b)\nhb : \u2191f b = y\n\u22a2 x \u2208 map f I \u2228 y \u2208 map f I", "state_after": "case refine'_2.intro.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\na : R\nha : \u2191f a = x\nb : R\nhb : \u2191f b = y\nc : R\nhc : c \u2208 I\nhc' : \u2191f c = \u2191f (a * b)\n\u22a2 x \u2208 map f I \u2228 y \u2208 map f I"}, {"tactic": "rw [\u2190 sub_eq_zero, \u2190 map_sub] at hc'", "annotated_tactic": ["rw [\u2190 <a>sub_eq_zero</a>, \u2190 <a>map_sub</a>] at hc'", [{"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [801, 3], "def_end_pos": [801, 14]}, {"full_name": "map_sub", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [428, 3], "def_end_pos": [428, 14]}]], "state_before": "case refine'_2.intro.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\na : R\nha : \u2191f a = x\nb : R\nhb : \u2191f b = y\nc : R\nhc : c \u2208 I\nhc' : \u2191f c = \u2191f (a * b)\n\u22a2 x \u2208 map f I \u2228 y \u2208 map f I", "state_after": "case refine'_2.intro.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\na : R\nha : \u2191f a = x\nb : R\nhb : \u2191f b = y\nc : R\nhc : c \u2208 I\nhc'\u271d : \u2191f c = \u2191f (a * b)\nhc' : \u2191f (c - a * b) = 0\n\u22a2 x \u2208 map f I \u2228 y \u2208 map f I"}, {"tactic": "have : a * b \u2208 I := by\n  convert I.sub_mem hc (hk (hc' : c - a * b \u2208 RingHom.ker f)) using 1\n  abel", "annotated_tactic": ["have : a * b \u2208 I := by\n      convert I.sub_mem hc (hk (hc' : c - a * b \u2208 <a>RingHom.ker</a> f)) using 1\n      abel", [{"full_name": "RingHom.ker", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [2069, 5], "def_end_pos": [2069, 8]}]], "state_before": "case refine'_2.intro.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\na : R\nha : \u2191f a = x\nb : R\nhb : \u2191f b = y\nc : R\nhc : c \u2208 I\nhc'\u271d : \u2191f c = \u2191f (a * b)\nhc' : \u2191f (c - a * b) = 0\n\u22a2 x \u2208 map f I \u2228 y \u2208 map f I", "state_after": "case refine'_2.intro.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\na : R\nha : \u2191f a = x\nb : R\nhb : \u2191f b = y\nc : R\nhc : c \u2208 I\nhc'\u271d : \u2191f c = \u2191f (a * b)\nhc' : \u2191f (c - a * b) = 0\nthis : a * b \u2208 I\n\u22a2 x \u2208 map f I \u2228 y \u2208 map f I"}, {"tactic": "exact\n  (H.mem_or_mem this).imp (fun h => ha \u25b8 mem_map_of_mem f h) fun h => hb \u25b8 mem_map_of_mem f h", "annotated_tactic": ["exact\n      (H.mem_or_mem this).<a>imp</a> (fun h => ha \u25b8 <a>mem_map_of_mem</a> f h) fun h => hb \u25b8 <a>mem_map_of_mem</a> f h", [{"full_name": "Or.imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [249, 9], "def_end_pos": [249, 15]}, {"full_name": "Ideal.mem_map_of_mem", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [1393, 9], "def_end_pos": [1393, 23]}, {"full_name": "Ideal.mem_map_of_mem", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [1393, 9], "def_end_pos": [1393, 23]}]], "state_before": "case refine'_2.intro.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\na : R\nha : \u2191f a = x\nb : R\nhb : \u2191f b = y\nc : R\nhc : c \u2208 I\nhc'\u271d : \u2191f c = \u2191f (a * b)\nhc' : \u2191f (c - a * b) = 0\nthis : a * b \u2208 I\n\u22a2 x \u2208 map f I \u2228 y \u2208 map f I", "state_after": "no goals"}, {"tactic": "convert I.sub_mem hc (hk (hc' : c - a * b \u2208 RingHom.ker f)) using 1", "annotated_tactic": ["convert I.sub_mem hc (hk (hc' : c - a * b \u2208 <a>RingHom.ker</a> f)) using 1", [{"full_name": "RingHom.ker", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [2069, 5], "def_end_pos": [2069, 8]}]], "state_before": "R : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\na : R\nha : \u2191f a = x\nb : R\nhb : \u2191f b = y\nc : R\nhc : c \u2208 I\nhc'\u271d : \u2191f c = \u2191f (a * b)\nhc' : \u2191f (c - a * b) = 0\n\u22a2 a * b \u2208 I", "state_after": "case h.e'_4\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\na : R\nha : \u2191f a = x\nb : R\nhb : \u2191f b = y\nc : R\nhc : c \u2208 I\nhc'\u271d : \u2191f c = \u2191f (a * b)\nhc' : \u2191f (c - a * b) = 0\n\u22a2 a * b = c - (c - a * b)"}, {"tactic": "abel", "annotated_tactic": ["abel", []], "state_before": "case h.e'_4\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\na : R\nha : \u2191f a = x\nb : R\nhb : \u2191f b = y\nc : R\nhc : c \u2208 I\nhc'\u271d : \u2191f c = \u2191f (a * b)\nhc' : \u2191f (c - a * b) = 0\n\u22a2 a * b = c - (c - a * b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Subobject/Limits.lean", "full_name": "CategoryTheory.Limits.kernelSubobject_factors_iff", "start": [120, 1], "end": [125, 33], "traced_tactics": [{"tactic": "rw [\u2190 Subobject.factorThru_arrow _ _ w, Category.assoc, kernelSubobject_arrow_comp,\n  comp_zero]", "annotated_tactic": ["rw [\u2190 <a>Subobject.factorThru_arrow</a> _ _ w, <a>Category.assoc</a>, <a>kernelSubobject_arrow_comp</a>,\n      <a>comp_zero</a>]", [{"full_name": "CategoryTheory.Subobject.factorThru_arrow", "def_path": "Mathlib/CategoryTheory/Subobject/FactorThru.lean", "def_pos": [121, 9], "def_end_pos": [121, 25]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [159, 3], "def_end_pos": [159, 8]}, {"full_name": "CategoryTheory.Limits.kernelSubobject_arrow_comp", "def_path": "Mathlib/CategoryTheory/Subobject/Limits.lean", "def_pos": [110, 9], "def_end_pos": [110, 35]}, {"full_name": "CategoryTheory.Limits.comp_zero", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean", "def_pos": [67, 9], "def_end_pos": [67, 18]}]], "state_before": "C : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX Y Z : C\ninst\u271d\u00b9 : HasZeroMorphisms C\nf : X \u27f6 Y\ninst\u271d : HasKernel f\nW : C\nh : W \u27f6 X\nw : Factors (kernelSubobject f) h\n\u22a2 h \u226b f = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.piecewise_compl", "start": [241, 1], "end": [243, 67], "traced_tactics": [{"tactic": "simp [hs]", "annotated_tactic": ["simp [hs]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\u1d9c\nf g : \u03b1 \u2192\u209b \u03b2\n\u22a2 \u2191(piecewise s\u1d9c hs f g) = \u2191(piecewise s (_ : MeasurableSet s) g f)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\u1d9c\nf g : \u03b1 \u2192\u209b \u03b2\n\u22a2 Set.piecewise s\u1d9c \u2191f \u2191g = Set.piecewise s \u2191g \u2191f"}, {"tactic": "convert Set.piecewise_compl s f g", "annotated_tactic": ["convert <a>Set.piecewise_compl</a> s f g", [{"full_name": "Set.piecewise_compl", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1469, 9], "def_end_pos": [1469, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\u1d9c\nf g : \u03b1 \u2192\u209b \u03b2\n\u22a2 Set.piecewise s\u1d9c \u2191f \u2191g = Set.piecewise s \u2191g \u2191f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean", "full_name": "Asymptotics.isLittleO_sum_range_of_tendsto_zero", "start": [139, 1], "end": [144, 42], "traced_tactics": [{"tactic": "have := ((isLittleO_one_iff \u211d).2 h).sum_range fun i => zero_le_one", "annotated_tactic": ["have := ((<a>isLittleO_one_iff</a> \u211d).2 h).<a>sum_range</a> fun i => <a>zero_le_one</a>", [{"full_name": "Asymptotics.isLittleO_one_iff", "def_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "def_pos": [1363, 9], "def_end_pos": [1363, 26]}, {"full_name": "Asymptotics.IsLittleO.sum_range", "def_path": "Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean", "def_pos": [106, 9], "def_end_pos": [106, 40]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nh : Tendsto f atTop (\ud835\udcdd 0)\n\u22a2 (fun n => \u2211 i in range n, f i) =o[atTop] fun n => \u2191n", "state_after": "\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nh : Tendsto f atTop (\ud835\udcdd 0)\nthis :\n  Tendsto (fun n => \u2211 i in range n, 1) atTop atTop \u2192 (fun n => \u2211 i in range n, f i) =o[atTop] fun n => \u2211 i in range n, 1\n\u22a2 (fun n => \u2211 i in range n, f i) =o[atTop] fun n => \u2191n"}, {"tactic": "simp only [sum_const, card_range, Nat.smul_one_eq_coe] at this", "annotated_tactic": ["simp only [<a>sum_const</a>, <a>card_range</a>, <a>Nat.smul_one_eq_coe</a>] at this", [{"full_name": "Finset.sum_const", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1440, 3], "def_end_pos": [1440, 14]}, {"full_name": "Finset.card_range", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [177, 9], "def_end_pos": [177, 19]}, {"full_name": "Nat.smul_one_eq_coe", "def_path": "Mathlib/Algebra/Module/Basic.lean", "def_pos": [758, 9], "def_end_pos": [758, 28]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nh : Tendsto f atTop (\ud835\udcdd 0)\nthis :\n  Tendsto (fun n => \u2211 i in range n, 1) atTop atTop \u2192 (fun n => \u2211 i in range n, f i) =o[atTop] fun n => \u2211 i in range n, 1\n\u22a2 (fun n => \u2211 i in range n, f i) =o[atTop] fun n => \u2191n", "state_after": "\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nh : Tendsto f atTop (\ud835\udcdd 0)\nthis : Tendsto (fun n => \u2191n) atTop atTop \u2192 (fun n => \u2211 i in range n, f i) =o[atTop] fun n => \u2191n\n\u22a2 (fun n => \u2211 i in range n, f i) =o[atTop] fun n => \u2191n"}, {"tactic": "exact this tendsto_nat_cast_atTop_atTop", "annotated_tactic": ["exact this <a>tendsto_nat_cast_atTop_atTop</a>", [{"full_name": "tendsto_nat_cast_atTop_atTop", "def_path": "Mathlib/Order/Filter/Archimedean.lean", "def_pos": [37, 9], "def_end_pos": [37, 37]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : NormedAddCommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nh : Tendsto f atTop (\ud835\udcdd 0)\nthis : Tendsto (fun n => \u2191n) atTop atTop \u2192 (fun n => \u2211 i in range n, f i) =o[atTop] fun n => \u2191n\n\u22a2 (fun n => \u2211 i in range n, f i) =o[atTop] fun n => \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean", "full_name": "Complex.log_im_le_pi", "start": [42, 1], "end": [42, 82], "traced_tactics": [{"tactic": "simp only [log_im, arg_le_pi]", "annotated_tactic": ["simp only [<a>log_im</a>, <a>arg_le_pi</a>]", [{"full_name": "Complex.log_im", "def_path": "Mathlib/Analysis/SpecialFunctions/Complex/Log.lean", "def_pos": 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S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : Module S M\ninst\u271d\u2076 : IsScalarTower S R M\nR' : Type u_4\nM' : Type u_5\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : Module R' M'\ninst\u271d\u00b2 : Algebra R R'\ninst\u271d\u00b9 : Module R M'\ninst\u271d : IsScalarTower R R' M'\np p' : R[X]\nq : PolynomialModule R M\n\u22a2 \u2191(comp p) (p' \u2022 q) = Polynomial.comp p' p \u2022 \u2191(comp p) q", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\nI : Ideal R\nS : Type u_3\ninst\u271d\u2079 : CommSemiring S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : Module S M\ninst\u271d\u2076 : IsScalarTower S R M\nR' : Type u_4\nM' : Type u_5\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : Module R' M'\ninst\u271d\u00b2 : Algebra R R'\ninst\u271d\u00b9 : Module R M'\ninst\u271d : IsScalarTower R R' M'\np p' : R[X]\nq : PolynomialModule R M\n\u22a2 eval\u2082 (algebraMap R R[X]) p p' \u2022 \u2191(eval p) (\u2191(map R[X] (lsingle R 0)) q) =\n    eval\u2082 C p p' \u2022 \u2191(eval p) (\u2191(map R[X] (lsingle R 0)) q)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\nI : Ideal R\nS : Type u_3\ninst\u271d\u2079 : CommSemiring S\ninst\u271d\u2078 : Algebra S R\ninst\u271d\u2077 : Module S M\ninst\u271d\u2076 : IsScalarTower S R M\nR' : Type u_4\nM' : Type u_5\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : Module R' M'\ninst\u271d\u00b2 : Algebra R R'\ninst\u271d\u00b9 : Module R M'\ninst\u271d : IsScalarTower R R' M'\np p' : R[X]\nq : PolynomialModule R M\n\u22a2 eval\u2082 (algebraMap R R[X]) p p' \u2022 \u2191(eval p) (\u2191(map R[X] (lsingle R 0)) q) =\n    eval\u2082 C p p' \u2022 \u2191(eval p) (\u2191(map R[X] (lsingle R 0)) q)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "full_name": "mul_le_of_one_le_left", "start": [399, 1], "end": [400, 61], "traced_tactics": [{"tactic": "simpa only [one_mul] using mul_le_mul_of_nonpos_right h hb", "annotated_tactic": ["simpa only [<a>one_mul</a>] using <a>mul_le_mul_of_nonpos_right</a> h hb", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "mul_le_mul_of_nonpos_right", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [355, 9], "def_end_pos": [355, 35]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : OrderedRing \u03b1\na b c d : \u03b1\nhb : b \u2264 0\nh : 1 \u2264 a\n\u22a2 a * b \u2264 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Group/Units.lean", "full_name": "Units.inv_mul_cancel_right", "start": [302, 1], "end": [303, 35], "traced_tactics": [{"tactic": "rw [mul_assoc, inv_mul, mul_one]", "annotated_tactic": ["rw [<a>mul_assoc</a>, <a>inv_mul</a>, <a>mul_one</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "Units.inv_mul", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [262, 9], "def_end_pos": [262, 16]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na\u271d b\u271d c u : \u03b1\u02e3\na : \u03b1\nb : \u03b1\u02e3\n\u22a2 a * \u2191b\u207b\u00b9 * \u2191b = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PNat/Factors.lean", "full_name": "PrimeMultiset.coeNat_injective", "start": [81, 1], "end": [82, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/MinMax.lean", "full_name": "List.argmin_singleton", "start": [125, 1], "end": [126, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Subsemiring/Basic.lean", "full_name": "Subsemiring.center_eq_top", "start": [746, 1], "end": [747, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/AlgebraMap.lean", "full_name": "Polynomial.eval\u2082_algebraMap_X", "start": [127, 1], "end": [132, 36], "traced_tactics": [{"tactic": "conv_rhs => rw [\u2190 Polynomial.sum_C_mul_X_pow_eq p]", "annotated_tactic": ["conv_rhs => rw [\u2190 <a>Polynomial.sum_C_mul_X_pow_eq</a> p]", [{"full_name": "Polynomial.sum_C_mul_X_pow_eq", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [1046, 9], "def_end_pos": [1046, 27]}]], "state_before": "R\u271d : Type u\nS : Type v\nT : Type w\nA\u271d : Type z\nA' : Type u_1\nB' : Type u_2\na b : R\u271d\nn : \u2115\ninst\u271d\u2077 : CommSemiring A'\ninst\u271d\u2076 : Semiring B'\ninst\u271d\u2075 : CommSemiring R\u271d\np\u271d q r : R\u271d[X]\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\np : R[X]\nf : R[X] \u2192\u2090[R] A\n\u22a2 eval\u2082 (algebraMap R A) (\u2191f X) p = \u2191f p", "state_after": "R\u271d : Type u\nS : Type v\nT : Type w\nA\u271d : Type z\nA' : Type u_1\nB' : Type u_2\na b : R\u271d\nn : \u2115\ninst\u271d\u2077 : CommSemiring A'\ninst\u271d\u2076 : Semiring B'\ninst\u271d\u2075 : CommSemiring R\u271d\np\u271d q r : R\u271d[X]\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\np : R[X]\nf : R[X] \u2192\u2090[R] A\n\u22a2 eval\u2082 (algebraMap R A) (\u2191f X) p = \u2191f (sum p fun n a => \u2191C a * X ^ n)"}, {"tactic": "simp only [eval\u2082_eq_sum, sum_def]", "annotated_tactic": ["simp only [<a>eval\u2082_eq_sum</a>, <a>sum_def</a>]", [{"full_name": "Polynomial.eval\u2082_eq_sum", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [49, 9], "def_end_pos": [49, 21]}, {"full_name": "Polynomial.sum_def", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [977, 9], "def_end_pos": [977, 16]}]], "state_before": "R\u271d : Type u\nS : Type v\nT : Type w\nA\u271d : Type z\nA' : Type u_1\nB' : Type u_2\na b : R\u271d\nn : \u2115\ninst\u271d\u2077 : CommSemiring A'\ninst\u271d\u2076 : Semiring B'\ninst\u271d\u2075 : CommSemiring R\u271d\np\u271d q r : R\u271d[X]\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\np : R[X]\nf : R[X] \u2192\u2090[R] A\n\u22a2 eval\u2082 (algebraMap R A) (\u2191f X) p = \u2191f (sum p fun n a => \u2191C a * X ^ n)", "state_after": "R\u271d : Type u\nS : Type v\nT : Type w\nA\u271d : Type z\nA' : Type u_1\nB' : Type u_2\na b : R\u271d\nn : \u2115\ninst\u271d\u2077 : CommSemiring A'\ninst\u271d\u2076 : Semiring B'\ninst\u271d\u2075 : CommSemiring R\u271d\np\u271d q r : R\u271d[X]\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\np : R[X]\nf : R[X] \u2192\u2090[R] A\n\u22a2 \u2211 n in support p, \u2191(algebraMap R A) (coeff p n) * \u2191f X ^ n = \u2191f (\u2211 n in support p, \u2191C (coeff p n) * X ^ n)"}, {"tactic": "simp only [f.map_sum, f.map_mul, f.map_pow, eq_intCast, map_intCast]", "annotated_tactic": ["simp only [f.map_sum, f.map_mul, f.map_pow, <a>eq_intCast</a>, <a>map_intCast</a>]", [{"full_name": "eq_intCast", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [306, 9], "def_end_pos": [306, 19]}, {"full_name": "map_intCast", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [311, 9], "def_end_pos": [311, 20]}]], "state_before": "R\u271d : Type u\nS : Type v\nT : Type w\nA\u271d : Type z\nA' : Type u_1\nB' : Type u_2\na b : R\u271d\nn : \u2115\ninst\u271d\u2077 : CommSemiring A'\ninst\u271d\u2076 : Semiring B'\ninst\u271d\u2075 : CommSemiring R\u271d\np\u271d q r : R\u271d[X]\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\np : R[X]\nf : R[X] \u2192\u2090[R] A\n\u22a2 \u2211 n in support p, \u2191(algebraMap R A) (coeff p n) * \u2191f X ^ n = \u2191f (\u2211 n in support p, \u2191C (coeff p n) * X ^ n)", "state_after": "R\u271d : Type u\nS : Type v\nT : Type w\nA\u271d : Type z\nA' : Type u_1\nB' : Type u_2\na b : R\u271d\nn : \u2115\ninst\u271d\u2077 : CommSemiring A'\ninst\u271d\u2076 : Semiring B'\ninst\u271d\u2075 : CommSemiring R\u271d\np\u271d q r : R\u271d[X]\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\np : R[X]\nf : R[X] \u2192\u2090[R] A\n\u22a2 \u2211 n in support p, \u2191(algebraMap R A) (coeff p n) * \u2191f X ^ n = \u2211 x in support p, \u2191f (\u2191C (coeff p x)) * \u2191f X ^ x"}, {"tactic": "simp [Polynomial.C_eq_algebraMap]", "annotated_tactic": ["simp [<a>Polynomial.C_eq_algebraMap</a>]", [{"full_name": "Polynomial.C_eq_algebraMap", "def_path": "Mathlib/Data/Polynomial/AlgebraMap.lean", "def_pos": [76, 9], "def_end_pos": [76, 24]}]], "state_before": "R\u271d : Type u\nS : Type v\nT : Type w\nA\u271d : Type z\nA' : Type u_1\nB' : Type u_2\na b : R\u271d\nn : \u2115\ninst\u271d\u2077 : CommSemiring A'\ninst\u271d\u2076 : Semiring B'\ninst\u271d\u2075 : CommSemiring R\u271d\np\u271d q r : R\u271d[X]\ninst\u271d\u2074 : Semiring A\u271d\ninst\u271d\u00b3 : Algebra R\u271d A\u271d\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\np : R[X]\nf : R[X] \u2192\u2090[R] A\n\u22a2 \u2211 n in support p, \u2191(algebraMap R A) (coeff p n) * \u2191f X ^ n = \u2211 x in support p, \u2191f (\u2191C (coeff p x)) * \u2191f X ^ x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Subring/Basic.lean", "full_name": "Subring.mem_map", "start": [601, 1], "end": [602, 40], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "R : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : Ring T\ns\u271d : Subring R\nf : R \u2192+* S\ns : Subring R\ny : S\n\u22a2 (\u2203 x x_1, \u2191f x = y) \u2194 \u2203 x, x \u2208 s \u2227 \u2191f x = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Group/Bounds.lean", "full_name": "IsLUB.exists_between_sub_self", "start": [30, 1], "end": [31, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "full_name": "WithLp.equiv_symm_single", "start": [258, 1], "end": [260, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/DirectSum/TensorProduct.lean", "full_name": "TensorProduct.directSum_symm_lof_tmul", "start": [161, 1], "end": [165, 57], "traced_tactics": [{"tactic": "rw [LinearEquiv.symm_apply_eq, directSum_lof_tmul_lof]", "annotated_tactic": ["rw [<a>LinearEquiv.symm_apply_eq</a>, <a>directSum_lof_tmul_lof</a>]", [{"full_name": "LinearEquiv.symm_apply_eq", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [403, 9], "def_end_pos": [403, 22]}, {"full_name": "TensorProduct.directSum_lof_tmul_lof", "def_path": "Mathlib/LinearAlgebra/DirectSum/TensorProduct.lean", "def_pos": [154, 9], "def_end_pos": [154, 31]}]], "state_before": "R : Type u\ninst\u271d\u00b9\u2070 : CommRing R\n\u03b9\u2081 : Type v\u2081\n\u03b9\u2082 : Type v\u2082\ninst\u271d\u2079 : DecidableEq \u03b9\u2081\ninst\u271d\u2078 : DecidableEq \u03b9\u2082\nM\u2081 : \u03b9\u2081 \u2192 Type w\u2081\nM\u2081' : Type w\u2081'\nM\u2082 : \u03b9\u2082 \u2192 Type w\u2082\nM\u2082' : Type w\u2082'\ninst\u271d\u2077 : (i\u2081 : \u03b9\u2081) \u2192 AddCommGroup (M\u2081 i\u2081)\ninst\u271d\u2076 : AddCommGroup M\u2081'\ninst\u271d\u2075 : (i\u2082 : \u03b9\u2082) \u2192 AddCommGroup (M\u2082 i\u2082)\ninst\u271d\u2074 : AddCommGroup M\u2082'\ninst\u271d\u00b3 : (i\u2081 : \u03b9\u2081) \u2192 Module R (M\u2081 i\u2081)\ninst\u271d\u00b2 : Module R M\u2081'\ninst\u271d\u00b9 : (i\u2082 : \u03b9\u2082) \u2192 Module R (M\u2082 i\u2082)\ninst\u271d : Module R M\u2082'\ni\u2081 : \u03b9\u2081\nm\u2081 : M\u2081 i\u2081\ni\u2082 : \u03b9\u2082\nm\u2082 : M\u2082 i\u2082\n\u22a2 \u2191(LinearEquiv.symm (TensorProduct.directSum R M\u2081 M\u2082))\n      (\u2191(lof R (\u03b9\u2081 \u00d7 \u03b9\u2082) (fun i => M\u2081 i.1 \u2297[R] M\u2082 i.2) (i\u2081, i\u2082)) (m\u2081 \u2297\u209c[R] m\u2082)) =\n    \u2191(lof R \u03b9\u2081 M\u2081 i\u2081) m\u2081 \u2297\u209c[R] \u2191(lof R \u03b9\u2082 M\u2082 i\u2082) m\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Convex/Basic.lean", "full_name": "AntitoneOn.convex_le", "start": [395, 1], "end": [397, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Constructions.lean", "full_name": "inducing_subtype_val", "start": [982, 1], "end": [982, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "full_name": "Subgroup.mem_iSup_of_mem", "start": [1052, 1], "end": [1054, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Star/NonUnitalSubalgebra.lean", "full_name": "NonUnitalStarSubalgebra.coe_center", "start": [1020, 1], "end": [1021, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/CondCount.lean", "full_name": "ProbabilityTheory.condCount_add_compl_eq", "start": [193, 1], "end": [202, 33], "traced_tactics": [{"tactic": "have : condCount s t = (condCount (s \u2229 u) t * condCount (s \u2229 u \u222a s \u2229 u\u1d9c) (s \u2229 u) +\n    condCount (s \u2229 u\u1d9c) t * condCount (s \u2229 u \u222a s \u2229 u\u1d9c) (s \u2229 u\u1d9c)) := by\n  rw [condCount_disjoint_union (hs.inter_of_left _) (hs.inter_of_left _)\n    (disjoint_compl_right.mono inf_le_right inf_le_right), Set.inter_union_compl]", "annotated_tactic": ["have : <a>condCount</a> s t = (<a>condCount</a> (s \u2229 u) t * <a>condCount</a> (s \u2229 u \u222a s \u2229 u\u1d9c) (s \u2229 u) +\n      <a>condCount</a> (s \u2229 u\u1d9c) t * <a>condCount</a> (s \u2229 u \u222a s \u2229 u\u1d9c) (s \u2229 u\u1d9c)) := by\n    rw [<a>condCount_disjoint_union</a> (hs.inter_of_left _) (hs.inter_of_left _)\n      (disjoint_compl_right.mono <a>inf_le_right</a> <a>inf_le_right</a>), <a>Set.inter_union_compl</a>]", [{"full_name": "ProbabilityTheory.condCount", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [54, 5], "def_end_pos": [54, 14]}, {"full_name": "ProbabilityTheory.condCount", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [54, 5], "def_end_pos": [54, 14]}, {"full_name": "ProbabilityTheory.condCount", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [54, 5], "def_end_pos": [54, 14]}, {"full_name": "ProbabilityTheory.condCount", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [54, 5], "def_end_pos": [54, 14]}, {"full_name": "ProbabilityTheory.condCount", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [54, 5], "def_end_pos": [54, 14]}, {"full_name": "ProbabilityTheory.condCount_disjoint_union", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [170, 9], "def_end_pos": [170, 33]}, {"full_name": "inf_le_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [399, 9], "def_end_pos": [399, 21]}, {"full_name": "inf_le_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [399, 9], "def_end_pos": [399, 21]}, {"full_name": "Set.inter_union_compl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1900, 9], "def_end_pos": [1900, 26]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t\u271d u\u271d u t : Set \u03a9\nhs : Set.Finite s\n\u22a2 \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u + \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount s) u\u1d9c = \u2191\u2191(condCount s) t", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t\u271d u\u271d u t : Set \u03a9\nhs : Set.Finite s\nthis :\n  \u2191\u2191(condCount s) t =\n    \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n      \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)\n\u22a2 \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u + \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount s) u\u1d9c = \u2191\u2191(condCount s) t"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t\u271d u\u271d u t : Set \u03a9\nhs : Set.Finite s\nthis :\n  \u2191\u2191(condCount s) t =\n    \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n      \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)\n\u22a2 \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u + \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount s) u\u1d9c = \u2191\u2191(condCount s) t", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t\u271d u\u271d u t : Set \u03a9\nhs : Set.Finite s\nthis :\n  \u2191\u2191(condCount s) t =\n    \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n      \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)\n\u22a2 \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u + \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount s) u\u1d9c =\n    \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n      \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)"}, {"tactic": "simp [condCount_inter_self hs]", "annotated_tactic": ["simp [<a>condCount_inter_self</a> hs]", [{"full_name": "ProbabilityTheory.condCount_inter_self", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [100, 9], "def_end_pos": [100, 29]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t\u271d u\u271d u t : Set \u03a9\nhs : Set.Finite s\nthis :\n  \u2191\u2191(condCount s) t =\n    \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n      \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)\n\u22a2 \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount s) u + \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount s) u\u1d9c =\n    \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n      \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)", "state_after": "no goals"}, {"tactic": "rw [condCount_disjoint_union (hs.inter_of_left _) (hs.inter_of_left _)\n  (disjoint_compl_right.mono inf_le_right inf_le_right), Set.inter_union_compl]", "annotated_tactic": ["rw [<a>condCount_disjoint_union</a> (hs.inter_of_left _) (hs.inter_of_left _)\n      (disjoint_compl_right.mono <a>inf_le_right</a> <a>inf_le_right</a>), <a>Set.inter_union_compl</a>]", [{"full_name": "ProbabilityTheory.condCount_disjoint_union", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [170, 9], "def_end_pos": [170, 33]}, {"full_name": "inf_le_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [399, 9], "def_end_pos": [399, 21]}, {"full_name": "inf_le_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [399, 9], "def_end_pos": [399, 21]}, {"full_name": "Set.inter_union_compl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1900, 9], "def_end_pos": [1900, 26]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t\u271d u\u271d u t : Set \u03a9\nhs : Set.Finite s\n\u22a2 \u2191\u2191(condCount s) t =\n    \u2191\u2191(condCount (s \u2229 u)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n      \u2191\u2191(condCount (s \u2229 u\u1d9c)) t * \u2191\u2191(condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/NAry.lean", "full_name": "Set.image2_subset_iff", "start": [83, 1], "end": [84, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean", "full_name": "MvPolynomial.le_weightedTotalDegree", "start": [121, 1], "end": [123, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Iterate.lean", "full_name": "Function.Commute.iterate_pos_eq_iff_map_eq", "start": [220, 1], "end": [223, 48], "traced_tactics": [{"tactic": "simp only [le_antisymm_iff, h.iterate_pos_le_iff_map_le hf hg hn,\n  h.symm.iterate_pos_le_iff_map_le' hg hf hn]", "annotated_tactic": ["simp only [<a>le_antisymm_iff</a>, h.iterate_pos_le_iff_map_le hf hg hn,\n    h.symm.iterate_pos_le_iff_map_le' hg hf hn]", [{"full_name": "le_antisymm_iff", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [192, 9], "def_end_pos": [192, 24]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nf g : \u03b1 \u2192 \u03b1\nh : Commute f g\nhf : Monotone f\nhg : StrictMono g\nx : \u03b1\nn : \u2115\nhn : 0 < n\n\u22a2 f^[n] x = g^[n] x \u2194 f x = g x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.castIso_to_equiv", "start": [841, 1], "end": [843, 7], "traced_tactics": [{"tactic": "subst h", "annotated_tactic": ["subst h", []], "state_before": "n m : \u2115\nh : n = m\n\u22a2 (castIso h).toEquiv = Equiv.cast (_ : Fin n = Fin m)", "state_after": "n : \u2115\n\u22a2 (castIso (_ : n = n)).toEquiv = Equiv.cast (_ : Fin n = Fin n)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "n : \u2115\n\u22a2 (castIso (_ : n = n)).toEquiv = Equiv.cast (_ : Fin n = Fin n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Real/EReal.lean", "full_name": "EReal.coe_sub", "start": [763, 20], "end": [763, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf'_insert", "start": [950, 1], "end": [952, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "full_name": "MeasureTheory.LocallyIntegrableOn.add", "start": [154, 11], "end": [156, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Lie/IdealOperations.lean", "full_name": "LieSubmodule.bot_lie", "start": [139, 1], "end": [142, 69], "traced_tactics": [{"tactic": "suffices \u2045(\u22a5 : LieIdeal R L), N\u2046 \u2264 \u22a5 by exact le_bot_iff.mp this", "annotated_tactic": ["suffices \u2045(\u22a5 : <a>LieIdeal</a> R L), N\u2046 \u2264 \u22a5 by exact le_bot_iff.mp this", [{"full_name": "LieIdeal", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [228, 8], "def_end_pos": [228, 16]}]], "state_before": "R : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\n\u22a2 \u2045\u22a5, N\u2046 = \u22a5", "state_after": "R : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\n\u22a2 \u2045\u22a5, N\u2046 \u2264 \u22a5"}, {"tactic": "rw [lieIdeal_oper_eq_span, lieSpan_le]", "annotated_tactic": ["rw [<a>lieIdeal_oper_eq_span</a>, <a>lieSpan_le</a>]", [{"full_name": "LieSubmodule.lieIdeal_oper_eq_span", "def_path": "Mathlib/Algebra/Lie/IdealOperations.lean", "def_pos": [59, 9], "def_end_pos": [59, 30]}, {"full_name": "LieSubmodule.lieSpan_le", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [688, 9], "def_end_pos": [688, 19]}]], "state_before": "R : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\n\u22a2 \u2045\u22a5, N\u2046 \u2264 \u22a5", "state_after": "R : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\n\u22a2 {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} \u2286 \u2191\u22a5"}, {"tactic": "rintro m \u27e8\u27e8x, hx\u27e9, n, hn\u27e9", "annotated_tactic": ["rintro m \u27e8\u27e8x, hx\u27e9, n, hn\u27e9", []], "state_before": "R : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\n\u22a2 {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} \u2286 \u2191\u22a5", "state_after": "case intro.mk.intro\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\nm : M\nx : L\nhx : x \u2208 \u22a5\nn : { x // x \u2208 N }\nhn : \u2045\u2191{ val := x, property := hx }, \u2191n\u2046 = m\n\u22a2 m \u2208 \u2191\u22a5"}, {"tactic": "rw [\u2190 hn]", "annotated_tactic": ["rw [\u2190 hn]", []], "state_before": "case intro.mk.intro\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\nm : M\nx : L\nhx : x \u2208 \u22a5\nn : { x // x \u2208 N }\nhn : \u2045\u2191{ val := x, property := hx }, \u2191n\u2046 = m\n\u22a2 m \u2208 \u2191\u22a5", "state_after": "case intro.mk.intro\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\nm : M\nx : L\nhx : x \u2208 \u22a5\nn : { x // x \u2208 N }\nhn : \u2045\u2191{ val := x, property := hx }, \u2191n\u2046 = m\n\u22a2 \u2045\u2191{ val := x, property := hx }, \u2191n\u2046 \u2208 \u2191\u22a5"}, {"tactic": "change x \u2208 (\u22a5 : LieIdeal R L) at hx", "annotated_tactic": ["change x \u2208 (\u22a5 : <a>LieIdeal</a> R L) at hx", [{"full_name": "LieIdeal", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [228, 8], "def_end_pos": [228, 16]}]], "state_before": "case intro.mk.intro\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\nm : M\nx : L\nhx : x \u2208 \u22a5\nn : { x // x \u2208 N }\nhn : \u2045\u2191{ val := x, property := hx }, \u2191n\u2046 = m\n\u22a2 \u2045\u2191{ val := x, property := hx }, \u2191n\u2046 \u2208 \u2191\u22a5", "state_after": "case intro.mk.intro\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\nm : M\nx : L\nn : { x // x \u2208 N }\nhx : x \u2208 \u22a5\nhn : \u2045\u2191{ val := x, property := hx }, \u2191n\u2046 = m\n\u22a2 \u2045\u2191{ val := x, property := hx }, \u2191n\u2046 \u2208 \u2191\u22a5"}, {"tactic": "rw [mem_bot] at hx", "annotated_tactic": ["rw [<a>mem_bot</a>] at hx", [{"full_name": "LieSubmodule.mem_bot", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [376, 9], "def_end_pos": [376, 16]}]], "state_before": "case intro.mk.intro\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\nm : M\nx : L\nn : { x // x \u2208 N }\nhx : x \u2208 \u22a5\nhn : \u2045\u2191{ val := x, property := hx }, \u2191n\u2046 = m\n\u22a2 \u2045\u2191{ val := x, property := hx }, \u2191n\u2046 \u2208 \u2191\u22a5", "state_after": "case intro.mk.intro\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\nm : M\nx : L\nn : { x // x \u2208 N }\nhx\u271d : x \u2208 \u22a5\nhx : x = 0\nhn : \u2045\u2191{ val := x, property := hx\u271d }, \u2191n\u2046 = m\n\u22a2 \u2045\u2191{ val := x, property := hx\u271d }, \u2191n\u2046 \u2208 \u2191\u22a5"}, {"tactic": "simp [hx]", "annotated_tactic": ["simp [hx]", []], "state_before": "case intro.mk.intro\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\nm : M\nx : L\nn : { x // x \u2208 N }\nhx\u271d : x \u2208 \u22a5\nhx : x = 0\nhn : \u2045\u2191{ val := x, property := hx\u271d }, \u2191n\u2046 = m\n\u22a2 \u2045\u2191{ val := x, property := hx\u271d }, \u2191n\u2046 \u2208 \u2191\u22a5", "state_after": "no goals"}, {"tactic": "exact le_bot_iff.mp this", "annotated_tactic": ["exact le_bot_iff.mp this", []], "state_before": "R : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\nthis : \u2045\u22a5, N\u2046 \u2264 \u22a5\n\u22a2 \u2045\u22a5, N\u2046 = \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/QuadraticForm/Basic.lean", "full_name": "QuadraticForm.associated_left_inverse", "start": [849, 1], "end": [852, 15], "traced_tactics": [{"tactic": "rw [associated_toQuadraticForm, h.eq x y, \u2190 two_mul, \u2190 mul_assoc, invOf_mul_self,\n  one_mul]", "annotated_tactic": ["rw [<a>associated_toQuadraticForm</a>, h.eq x y, \u2190 <a>two_mul</a>, \u2190 <a>mul_assoc</a>, <a>invOf_mul_self</a>,\n      <a>one_mul</a>]", [{"full_name": "QuadraticForm.associated_toQuadraticForm", "def_path": "Mathlib/LinearAlgebra/QuadraticForm/Basic.lean", "def_pos": [844, 9], "def_end_pos": [844, 35]}, {"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "invOf_mul_self", "def_path": "Mathlib/Algebra/Invertible/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 23]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "S : Type u_1\nT : Type u_2\nR : Type u_3\nM : Type u_4\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : CommSemiring S\ninst\u271d\u00b9 : Algebra S R\ninst\u271d : Invertible 2\nB\u2081 : BilinForm R M\nQ : QuadraticForm R M\nh : BilinForm.IsSymm B\u2081\nx y : M\n\u22a2 bilin (\u2191(associatedHom S) (toQuadraticForm B\u2081)) x y = bilin B\u2081 x y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/FieldTheory/SplittingField/Construction.lean", "full_name": "Polynomial.SplittingField.splits", "start": [326, 11], "end": [327, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/MeanValue.lean", "full_name": "monotone_of_deriv_nonneg", "start": [938, 1], "end": [942, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/GradedMonoid.lean", "full_name": "List.dProdIndex_nil", "start": [392, 1], "end": [393, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Multiset/FinsetOps.lean", "full_name": "Multiset.ndinter_cons_of_not_mem", "start": [235, 1], "end": [236, 62], "traced_tactics": [{"tactic": "simp [ndinter, h]", "annotated_tactic": ["simp [<a>ndinter</a>, h]", [{"full_name": "Multiset.ndinter", "def_path": "Mathlib/Data/Multiset/FinsetOps.lean", "def_pos": [215, 5], "def_end_pos": [215, 12]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\ns\u271d : Multiset \u03b1\na : \u03b1\ns t : Multiset \u03b1\nh : \u00aca \u2208 t\n\u22a2 ndinter (a ::\u2098 s) t = ndinter s t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "full_name": "NormOneClass.nontrivial", "start": [146, 1], "end": [148, 57], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : One \u03b1\ninst\u271d : NormOneClass \u03b1\n\u22a2 \u20160\u2016 \u2260 \u20161\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/BigOperators/Basic.lean", "full_name": "Finset.prod_range_succ", "start": [1220, 1], "end": [1222, 45], "traced_tactics": [{"tactic": "simp only [mul_comm, prod_range_succ_comm]", "annotated_tactic": ["simp only [<a>mul_comm</a>, <a>prod_range_succ_comm</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "Finset.prod_range_succ_comm", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1213, 9], "def_end_pos": [1213, 29]}]], "state_before": "\u03b9 : Type u_1\n\u03b2 : Type u\n\u03b1 : Type v\n\u03b3 : Type w\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nn : \u2115\n\u22a2 \u220f x in range (n + 1), f x = (\u220f x in range n, f x) * f n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Field/Basic.lean", "full_name": "one_le_div_iff", "start": [840, 1], "end": [844, 37], "traced_tactics": [{"tactic": "rcases lt_trichotomy b 0 with (hb | rfl | hb)", "annotated_tactic": ["rcases <a>lt_trichotomy</a> b 0 with (hb | rfl | hb)", [{"full_name": "lt_trichotomy", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [311, 9], "def_end_pos": [311, 22]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\n\u22a2 1 \u2264 a / b \u2194 0 < b \u2227 b \u2264 a \u2228 b < 0 \u2227 a \u2264 b", "state_after": "case inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhb : b < 0\n\u22a2 1 \u2264 a / b \u2194 0 < b \u2227 b \u2264 a \u2228 b < 0 \u2227 a \u2264 b\n\ncase inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na c d : \u03b1\nn : \u2124\n\u22a2 1 \u2264 a / 0 \u2194 0 < 0 \u2227 0 \u2264 a \u2228 0 < 0 \u2227 a \u2264 0\n\ncase inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhb : 0 < b\n\u22a2 1 \u2264 a / b \u2194 0 < b \u2227 b \u2264 a \u2228 b < 0 \u2227 a \u2264 b"}, {"tactic": "simp [hb, hb.not_lt, one_le_div_of_neg]", "annotated_tactic": ["simp [hb, hb.not_lt, <a>one_le_div_of_neg</a>]", [{"full_name": "one_le_div_of_neg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [805, 9], "def_end_pos": [805, 26]}]], "state_before": "case inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhb : b < 0\n\u22a2 1 \u2264 a / b \u2194 0 < b \u2227 b \u2264 a \u2228 b < 0 \u2227 a \u2264 b", "state_after": "no goals"}, {"tactic": "simp [lt_irrefl, zero_lt_one.not_le, zero_lt_one]", "annotated_tactic": ["simp [<a>lt_irrefl</a>, zero_lt_one.not_le, <a>zero_lt_one</a>]", [{"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "case inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na c d : \u03b1\nn : \u2124\n\u22a2 1 \u2264 a / 0 \u2194 0 < 0 \u2227 0 \u2264 a \u2228 0 < 0 \u2227 a \u2264 0", "state_after": "no goals"}, {"tactic": "simp [hb, hb.not_lt, one_le_div]", "annotated_tactic": ["simp [hb, hb.not_lt, <a>one_le_div</a>]", [{"full_name": "one_le_div", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [422, 9], "def_end_pos": [422, 19]}]], "state_before": "case inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\na b c d : \u03b1\nn : \u2124\nhb : 0 < b\n\u22a2 1 \u2264 a / b \u2194 0 < b \u2227 b \u2264 a \u2228 b < 0 \u2227 a \u2264 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Closed/Monoidal.lean", "full_name": "CategoryTheory.MonoidalClosed.homEquiv_symm_apply_eq", "start": [176, 1], "end": [178, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Normed/Group/Hom.lean", "full_name": "NormedAddGroupHom.smul_apply", "start": [521, 1], "end": [522, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Matrix/Notation.lean", "full_name": "Matrix.vecMulVec_cons", "start": [365, 1], "end": [368, 51], "traced_tactics": [{"tactic": "ext i j", "annotated_tactic": ["ext i j", []], "state_before": "\u03b1 : Type u\no n m : \u2115\nm' : Type u\u2098\nn' : Type u\u2099\no' : Type u\u2092\na b : \u2115\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nv : m' \u2192 \u03b1\nx : \u03b1\nw : Fin n \u2192 \u03b1\n\u22a2 vecMulVec v (vecCons x w) = fun i => v i \u2022 vecCons x w", "state_after": "case a.h\n\u03b1 : Type u\no n m : \u2115\nm' : Type u\u2098\nn' : Type u\u2099\no' : Type u\u2092\na b : \u2115\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nv : m' \u2192 \u03b1\nx : \u03b1\nw : Fin n \u2192 \u03b1\ni : m'\nj : Fin (Nat.succ n)\n\u22a2 vecMulVec v (vecCons x w) i j = (v i \u2022 vecCons x w) j"}, {"tactic": "rw [vecMulVec_apply, Pi.smul_apply, smul_eq_mul]", "annotated_tactic": ["rw [<a>vecMulVec_apply</a>, <a>Pi.smul_apply</a>, <a>smul_eq_mul</a>]", [{"full_name": "Matrix.vecMulVec_apply", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [1654, 9], "def_end_pos": [1654, 24]}, {"full_name": "Pi.smul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [116, 60], "def_end_pos": [116, 70]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}]], "state_before": "case a.h\n\u03b1 : Type u\no n m : \u2115\nm' : Type u\u2098\nn' : Type u\u2099\no' : Type u\u2092\na b : \u2115\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nv : m' \u2192 \u03b1\nx : \u03b1\nw : Fin n \u2192 \u03b1\ni : m'\nj : Fin (Nat.succ n)\n\u22a2 vecMulVec v (vecCons x w) i j = (v i \u2022 vecCons x w) j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Rat/Floor.lean", "full_name": "Rat.num_lt_succ_floor_mul_den", "start": [109, 1], "end": [128, 47], "traced_tactics": [{"tactic": "suffices (q.num : \u211a) < (\u230aq\u230b + 1) * q.den by exact_mod_cast this", "annotated_tactic": ["suffices (q.num : \u211a) < (\u230aq\u230b + 1) * q.den by exact_mod_cast this", []], "state_before": "q : \u211a\n\u22a2 q.num < (\u230aq\u230b + 1) * \u2191q.den", "state_after": "q : \u211a\n\u22a2 \u2191q.num < (\u2191\u230aq\u230b + 1) * \u2191q.den"}, {"tactic": "suffices (q.num : \u211a) < (q - fract q + 1) * q.den by\n  have : (\u230aq\u230b : \u211a) = q - fract q := eq_sub_of_add_eq <| floor_add_fract q\n  rwa [this]", "annotated_tactic": ["suffices (q.num : \u211a) < (q - <a>fract</a> q + 1) * q.den by\n    have : (\u230aq\u230b : \u211a) = q - <a>fract</a> q := <a>eq_sub_of_add_eq</a> <| <a>floor_add_fract</a> q\n    rwa [this]", [{"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}, {"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}, {"full_name": "eq_sub_of_add_eq", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [756, 15], "def_end_pos": [756, 31]}, {"full_name": "Int.floor_add_fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [852, 9], "def_end_pos": [852, 24]}]], "state_before": "q : \u211a\n\u22a2 \u2191q.num < (\u2191\u230aq\u230b + 1) * \u2191q.den", "state_after": "q : \u211a\n\u22a2 \u2191q.num < (q - fract q + 1) * \u2191q.den"}, {"tactic": "suffices (q.num : \u211a) < q.num + (1 - fract q) * q.den by\n  have : (q - fract q + 1) * q.den = q.num + (1 - fract q) * q.den := by\n    calc\n      (q - fract q + 1) * q.den = (q + (1 - fract q)) * q.den := by ring\n      _ = q * q.den + (1 - fract q) * q.den := by rw [add_mul]\n      _ = q.num + (1 - fract q) * q.den := by simp\n  rwa [this]", "annotated_tactic": ["suffices (q.num : \u211a) < q.num + (1 - <a>fract</a> q) * q.den by\n    have : (q - <a>fract</a> q + 1) * q.den = q.num + (1 - <a>fract</a> q) * q.den := by\n      calc\n        (q - <a>fract</a> q + 1) * q.den = (q + (1 - <a>fract</a> q)) * q.den := by ring\n        _ = q * q.den + (1 - <a>fract</a> q) * q.den := by rw [<a>add_mul</a>]\n        _ = q.num + (1 - <a>fract</a> q) * q.den := by simp\n    rwa [this]", [{"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}, {"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}, {"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}, {"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}, {"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}, {"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}, {"full_name": "add_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [91, 7], "def_end_pos": [91, 14]}, {"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}]], "state_before": "q : \u211a\n\u22a2 \u2191q.num < (q - fract q + 1) * \u2191q.den", "state_after": "q : \u211a\n\u22a2 \u2191q.num < \u2191q.num + (1 - fract q) * \u2191q.den"}, {"tactic": "suffices 0 < (1 - fract q) * q.den by\n  rw [\u2190 sub_lt_iff_lt_add']\n  simpa", "annotated_tactic": ["suffices 0 < (1 - <a>fract</a> q) * q.den by\n    rw [\u2190 <a>sub_lt_iff_lt_add'</a>]\n    simpa", [{"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}, {"full_name": "sub_lt_iff_lt_add'", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [979, 3], "def_end_pos": [979, 14]}]], "state_before": "q : \u211a\n\u22a2 \u2191q.num < \u2191q.num + (1 - fract q) * \u2191q.den", "state_after": "q : \u211a\n\u22a2 0 < (1 - fract q) * \u2191q.den"}, {"tactic": "have : 0 < 1 - fract q := by\n  have : fract q < 1 := fract_lt_one q\n  have : 0 + fract q < 1 := by simp [this]\n  rwa [lt_sub_iff_add_lt]", "annotated_tactic": ["have : 0 < 1 - <a>fract</a> q := by\n    have : <a>fract</a> q < 1 := <a>fract_lt_one</a> q\n    have : 0 + <a>fract</a> q < 1 := by simp [this]\n    rwa [<a>lt_sub_iff_add_lt</a>]", [{"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}, {"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}, {"full_name": "Int.fract_lt_one", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [946, 9], "def_end_pos": [946, 21]}, {"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}, {"full_name": "lt_sub_iff_add_lt", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [906, 3], "def_end_pos": [906, 14]}]], "state_before": "q : \u211a\n\u22a2 0 < (1 - fract q) * \u2191q.den", "state_after": "q : \u211a\nthis : 0 < 1 - fract q\n\u22a2 0 < (1 - fract q) * \u2191q.den"}, {"tactic": "exact mul_pos this (by exact_mod_cast q.pos)", "annotated_tactic": ["exact <a>mul_pos</a> this (by exact_mod_cast q.pos)", [{"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [345, 7], "def_end_pos": [345, 14]}]], "state_before": "q : \u211a\nthis : 0 < 1 - fract q\n\u22a2 0 < (1 - fract q) * \u2191q.den", "state_after": "no goals"}, {"tactic": "exact_mod_cast this", "annotated_tactic": ["exact_mod_cast this", []], "state_before": "q : \u211a\nthis : \u2191q.num < (\u2191\u230aq\u230b + 1) * \u2191q.den\n\u22a2 q.num < (\u230aq\u230b + 1) * \u2191q.den", "state_after": "no goals"}, {"tactic": "have : (\u230aq\u230b : \u211a) = q - fract q := eq_sub_of_add_eq <| floor_add_fract q", "annotated_tactic": ["have : (\u230aq\u230b : \u211a) = q - <a>fract</a> q := <a>eq_sub_of_add_eq</a> <| <a>floor_add_fract</a> q", [{"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}, {"full_name": "eq_sub_of_add_eq", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [756, 15], "def_end_pos": [756, 31]}, {"full_name": "Int.floor_add_fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [852, 9], "def_end_pos": [852, 24]}]], "state_before": "q : \u211a\nthis : \u2191q.num < (q - fract q + 1) * \u2191q.den\n\u22a2 \u2191q.num < (\u2191\u230aq\u230b + 1) * \u2191q.den", "state_after": "q : \u211a\nthis\u271d : \u2191q.num < (q - fract q + 1) * \u2191q.den\nthis : \u2191\u230aq\u230b = q - fract q\n\u22a2 \u2191q.num < (\u2191\u230aq\u230b + 1) * \u2191q.den"}, {"tactic": "rwa [this]", "annotated_tactic": ["rwa [this]", []], "state_before": "q : \u211a\nthis\u271d : \u2191q.num < (q - fract q + 1) * \u2191q.den\nthis : \u2191\u230aq\u230b = q - fract q\n\u22a2 \u2191q.num < (\u2191\u230aq\u230b + 1) * \u2191q.den", "state_after": "no goals"}, {"tactic": "have : (q - fract q + 1) * q.den = q.num + (1 - fract q) * q.den := by\n  calc\n    (q - fract q + 1) * q.den = (q + (1 - fract q)) * q.den := by ring\n    _ = q * q.den + (1 - fract q) * q.den := by rw [add_mul]\n    _ = q.num + (1 - fract q) * q.den := by simp", "annotated_tactic": ["have : (q - <a>fract</a> q + 1) * q.den = q.num + (1 - <a>fract</a> q) * q.den := by\n      calc\n        (q - <a>fract</a> q + 1) * q.den = (q + (1 - <a>fract</a> q)) * q.den := by ring\n        _ = q * q.den + (1 - <a>fract</a> q) * q.den := by rw [<a>add_mul</a>]\n        _ = q.num + (1 - <a>fract</a> q) * q.den := by simp", [{"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}, {"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}, {"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}, {"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}, {"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}, {"full_name": "add_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [91, 7], "def_end_pos": [91, 14]}, {"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}]], "state_before": "q : \u211a\nthis : \u2191q.num < \u2191q.num + (1 - fract q) * \u2191q.den\n\u22a2 \u2191q.num < (q - fract q + 1) * \u2191q.den", "state_after": "q : \u211a\nthis\u271d : \u2191q.num < \u2191q.num + (1 - fract q) * \u2191q.den\nthis : (q - fract q + 1) * \u2191q.den = \u2191q.num + (1 - fract q) * \u2191q.den\n\u22a2 \u2191q.num < (q - fract q + 1) * \u2191q.den"}, {"tactic": "rwa [this]", "annotated_tactic": ["rwa [this]", []], "state_before": "q : \u211a\nthis\u271d : \u2191q.num < \u2191q.num + (1 - fract q) * \u2191q.den\nthis : (q - fract q + 1) * \u2191q.den = \u2191q.num + (1 - fract q) * \u2191q.den\n\u22a2 \u2191q.num < (q - fract q + 1) * \u2191q.den", "state_after": "no goals"}, {"tactic": "calc\n  (q - fract q + 1) * q.den = (q + (1 - fract q)) * q.den := by ring\n  _ = q * q.den + (1 - fract q) * q.den := by rw [add_mul]\n  _ = q.num + (1 - fract q) * q.den := by simp", "annotated_tactic": ["calc\n        (q - <a>fract</a> q + 1) * q.den = (q + (1 - <a>fract</a> q)) * q.den := by ring\n        _ = q * q.den + (1 - <a>fract</a> q) * q.den := by rw [<a>add_mul</a>]\n        _ = q.num + (1 - <a>fract</a> q) * q.den := by simp", [{"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}, {"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}, {"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}, {"full_name": "add_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [91, 7], "def_end_pos": [91, 14]}, {"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}]], "state_before": "q : \u211a\nthis : \u2191q.num < \u2191q.num + (1 - fract q) * \u2191q.den\n\u22a2 (q - fract q + 1) * \u2191q.den = \u2191q.num + (1 - fract q) * \u2191q.den", "state_after": "no goals"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "q : \u211a\nthis : \u2191q.num < \u2191q.num + (1 - fract q) * \u2191q.den\n\u22a2 (q - fract q + 1) * \u2191q.den = (q + (1 - fract q)) * \u2191q.den", "state_after": "no goals"}, {"tactic": "rw [add_mul]", "annotated_tactic": ["rw [<a>add_mul</a>]", [{"full_name": "add_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [91, 7], "def_end_pos": [91, 14]}]], "state_before": "q : \u211a\nthis : \u2191q.num < \u2191q.num + (1 - fract q) * \u2191q.den\n\u22a2 (q + (1 - fract q)) * \u2191q.den = q * \u2191q.den + (1 - fract q) * \u2191q.den", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "q : \u211a\nthis : \u2191q.num < \u2191q.num + (1 - fract q) * \u2191q.den\n\u22a2 q * \u2191q.den + (1 - fract q) * \u2191q.den = \u2191q.num + (1 - fract q) * \u2191q.den", "state_after": "no goals"}, {"tactic": "rw [\u2190 sub_lt_iff_lt_add']", "annotated_tactic": ["rw [\u2190 <a>sub_lt_iff_lt_add'</a>]", [{"full_name": "sub_lt_iff_lt_add'", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [979, 3], "def_end_pos": [979, 14]}]], "state_before": "q : \u211a\nthis : 0 < (1 - fract q) * \u2191q.den\n\u22a2 \u2191q.num < \u2191q.num + (1 - fract q) * \u2191q.den", "state_after": "q : \u211a\nthis : 0 < (1 - fract q) * \u2191q.den\n\u22a2 \u2191q.num - \u2191q.num < (1 - fract q) * \u2191q.den"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "q : \u211a\nthis : 0 < (1 - fract q) * \u2191q.den\n\u22a2 \u2191q.num - \u2191q.num < (1 - fract q) * \u2191q.den", "state_after": "no goals"}, {"tactic": "have : fract q < 1 := fract_lt_one q", "annotated_tactic": ["have : <a>fract</a> q < 1 := <a>fract_lt_one</a> q", [{"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}, {"full_name": "Int.fract_lt_one", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [946, 9], "def_end_pos": [946, 21]}]], "state_before": "q : \u211a\n\u22a2 0 < 1 - fract q", "state_after": "q : \u211a\nthis : fract q < 1\n\u22a2 0 < 1 - fract q"}, {"tactic": "have : 0 + fract q < 1 := by simp [this]", "annotated_tactic": ["have : 0 + <a>fract</a> q < 1 := by simp [this]", [{"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [629, 5], "def_end_pos": [629, 10]}]], "state_before": "q : \u211a\nthis : fract q < 1\n\u22a2 0 < 1 - fract q", "state_after": "q : \u211a\nthis\u271d : fract q < 1\nthis : 0 + fract q < 1\n\u22a2 0 < 1 - fract q"}, {"tactic": "rwa [lt_sub_iff_add_lt]", "annotated_tactic": ["rwa [<a>lt_sub_iff_add_lt</a>]", [{"full_name": "lt_sub_iff_add_lt", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [906, 3], "def_end_pos": [906, 14]}]], "state_before": "q : \u211a\nthis\u271d : fract q < 1\nthis : 0 + fract q < 1\n\u22a2 0 < 1 - fract q", "state_after": "no goals"}, {"tactic": "simp [this]", "annotated_tactic": ["simp [this]", []], "state_before": "q : \u211a\nthis : fract q < 1\n\u22a2 0 + fract q < 1", "state_after": "no goals"}, {"tactic": "exact_mod_cast q.pos", "annotated_tactic": ["exact_mod_cast q.pos", []], "state_before": "q : \u211a\nthis : 0 < 1 - fract q\n\u22a2 0 < \u2191q.den", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/NAry.lean", "full_name": "Filter.map\u2082_pure_left", "start": [178, 1], "end": [182, 97], "traced_tactics": [{"tactic": "rw [image2_singleton_left, image_subset_iff]", "annotated_tactic": ["rw [<a>image2_singleton_left</a>, <a>image_subset_iff</a>]", [{"full_name": "Set.image2_singleton_left", "def_path": "Mathlib/Data/Set/NAry.lean", "def_pos": [195, 9], "def_end_pos": [195, 30]}, {"full_name": "Set.image_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [497, 9], "def_end_pos": [497, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\n\u03b3' : Type u_6\n\u03b4 : Type u_7\n\u03b4' : Type u_8\n\u03b5 : Type u_9\n\u03b5' : Type u_10\nm : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nf f\u2081 f\u2082 : Filter \u03b1\ng g\u2081 g\u2082 : Filter \u03b2\nh\u271d h\u2081 h\u2082 : Filter \u03b3\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nu\u271d : Set \u03b3\nv : Set \u03b4\na : \u03b1\nb : \u03b2\nc : \u03b3\nu : Set \u03b3\nh : u \u2208 map (fun b => m a b) g\n\u22a2 image2 m {a} ((fun b => m a b) \u207b\u00b9' u) \u2286 u", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.prod_eq_zero", "start": [1069, 1], "end": [1071, 50], "traced_tactics": [{"tactic": "lift f to \u03b9 \u2192 Type u using fun _ => trivial", "annotated_tactic": ["lift f to \u03b9 \u2192 Type u using fun _ => <a>trivial</a>", [{"full_name": "trivial", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [514, 31], "def_end_pos": [514, 38]}]], "state_before": "\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Cardinal.{u}\n\u22a2 prod f = 0 \u2194 \u2203 i, f i = 0", "state_after": "case intro\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Type u\n\u22a2 (prod fun i => #(f i)) = 0 \u2194 \u2203 i, (fun i => #(f i)) i = 0"}, {"tactic": "simp only [mk_eq_zero_iff, \u2190 mk_pi, isEmpty_pi]", "annotated_tactic": ["simp only [<a>mk_eq_zero_iff</a>, \u2190 <a>mk_pi</a>, <a>isEmpty_pi</a>]", [{"full_name": "Cardinal.mk_eq_zero_iff", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [394, 9], "def_end_pos": [394, 23]}, {"full_name": "Cardinal.mk_pi", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1048, 9], "def_end_pos": [1048, 14]}, {"full_name": "isEmpty_pi", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [152, 9], "def_end_pos": [152, 19]}]], "state_before": "case intro\n\u03b1 \u03b2 : Type u\n\u03b9 : Type u_1\nf : \u03b9 \u2192 Type u\n\u22a2 (prod fun i => #(f i)) = 0 \u2194 \u2203 i, (fun i => #(f i)) i = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM_mono", "start": [986, 1], "end": [989, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "full_name": "parallelepiped_comp_equiv", "start": [62, 1], "end": [80, 84], "traced_tactics": [{"tactic": "simp only [parallelepiped]", "annotated_tactic": ["simp only [<a>parallelepiped</a>]", [{"full_name": "parallelepiped", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [44, 5], "def_end_pos": [44, 19]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\n\u22a2 parallelepiped (v \u2218 \u2191e) = parallelepiped v", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\n\u22a2 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 = (fun t => \u2211 i : \u03b9, t i \u2022 v i) '' Icc 0 1"}, {"tactic": "let K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a : \u03b9' => \u211d) e", "annotated_tactic": ["let K : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := <a>Equiv.piCongrLeft'</a> (fun _a : \u03b9' => \u211d) e", [{"full_name": "Equiv.piCongrLeft'", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1823, 5], "def_end_pos": [1823, 17]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\n\u22a2 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 = (fun t => \u2211 i : \u03b9, t i \u2022 v i) '' Icc 0 1", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\n\u22a2 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 = (fun t => \u2211 i : \u03b9, t i \u2022 v i) '' Icc 0 1"}, {"tactic": "rw [this, \u2190 image_comp]", "annotated_tactic": ["rw [this, \u2190 <a>image_comp</a>]", [{"full_name": "Set.image_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [293, 9], "def_end_pos": [293, 19]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nthis : Icc 0 1 = \u2191K '' Icc 0 1\n\u22a2 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 = (fun t => \u2211 i : \u03b9, t i \u2022 v i) '' Icc 0 1", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nthis : Icc 0 1 = \u2191K '' Icc 0 1\n\u22a2 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 = (fun t => \u2211 i : \u03b9, t i \u2022 v i) \u2218 \u2191K '' Icc 0 1"}, {"tactic": "congr 1 with x", "annotated_tactic": ["congr 1 with x", []], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nthis : Icc 0 1 = \u2191K '' Icc 0 1\n\u22a2 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 = (fun t => \u2211 i : \u03b9, t i \u2022 v i) \u2218 \u2191K '' Icc 0 1", "state_after": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nthis : Icc 0 1 = \u2191K '' Icc 0 1\nx : E\n\u22a2 x \u2208 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 \u2194 x \u2208 (fun t => \u2211 i : \u03b9, t i \u2022 v i) \u2218 \u2191K '' Icc 0 1"}, {"tactic": "have := fun z : \u03b9' \u2192 \u211d => e.symm.sum_comp fun i => z i \u2022 v (e i)", "annotated_tactic": ["have := fun z : \u03b9' \u2192 \u211d => e.symm.sum_comp fun i => z i \u2022 v (e i)", []], "state_before": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nthis : Icc 0 1 = \u2191K '' Icc 0 1\nx : E\n\u22a2 x \u2208 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 \u2194 x \u2208 (fun t => \u2211 i : \u03b9, t i \u2022 v i) \u2218 \u2191K '' Icc 0 1", "state_after": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nthis\u271d : Icc 0 1 = \u2191K '' Icc 0 1\nx : E\nthis : \u2200 (z : \u03b9' \u2192 \u211d), \u2211 i : \u03b9, z (\u2191e.symm i) \u2022 v (\u2191e (\u2191e.symm i)) = \u2211 i : \u03b9', z i \u2022 v (\u2191e i)\n\u22a2 x \u2208 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 \u2194 x \u2208 (fun t => \u2211 i : \u03b9, t i \u2022 v i) \u2218 \u2191K '' Icc 0 1"}, {"tactic": "simp_rw [Equiv.apply_symm_apply] at this", "annotated_tactic": ["simp_rw [<a>Equiv.apply_symm_apply</a>] at this", [{"full_name": "Equiv.apply_symm_apply", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [280, 17], "def_end_pos": [280, 33]}]], "state_before": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nthis\u271d : Icc 0 1 = \u2191K '' Icc 0 1\nx : E\nthis : \u2200 (z : \u03b9' \u2192 \u211d), \u2211 i : \u03b9, z (\u2191e.symm i) \u2022 v (\u2191e (\u2191e.symm i)) = \u2211 i : \u03b9', z i \u2022 v (\u2191e i)\n\u22a2 x \u2208 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 \u2194 x \u2208 (fun t => \u2211 i : \u03b9, t i \u2022 v i) \u2218 \u2191K '' Icc 0 1", "state_after": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nthis\u271d : Icc 0 1 = \u2191K '' Icc 0 1\nx : E\nthis : \u2200 (z : \u03b9' \u2192 \u211d), \u2211 x : \u03b9, z (\u2191e.symm x) \u2022 v x = \u2211 x : \u03b9', z x \u2022 v (\u2191e x)\n\u22a2 x \u2208 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 \u2194 x \u2208 (fun t => \u2211 i : \u03b9, t i \u2022 v i) \u2218 \u2191K '' Icc 0 1"}, {"tactic": "simp_rw [Function.comp_apply, mem_image, mem_Icc, Equiv.piCongrLeft'_apply, this]", "annotated_tactic": ["simp_rw [<a>Function.comp_apply</a>, <a>mem_image</a>, <a>mem_Icc</a>, <a>Equiv.piCongrLeft'_apply</a>, this]", [{"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}, {"full_name": "Set.mem_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}, {"full_name": "Equiv.piCongrLeft'_apply", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1822, 3], "def_end_pos": [1822, 8]}]], "state_before": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nthis\u271d : Icc 0 1 = \u2191K '' Icc 0 1\nx : E\nthis : \u2200 (z : \u03b9' \u2192 \u211d), \u2211 x : \u03b9, z (\u2191e.symm x) \u2022 v x = \u2211 x : \u03b9', z x \u2022 v (\u2191e x)\n\u22a2 x \u2208 (fun a => \u2211 x : \u03b9', a x \u2022 (v \u2218 \u2191e) x) '' Icc 0 1 \u2194 x \u2208 (fun t => \u2211 i : \u03b9, t i \u2022 v i) \u2218 \u2191K '' Icc 0 1", "state_after": "no goals"}, {"tactic": "rw [\u2190 Equiv.preimage_eq_iff_eq_image]", "annotated_tactic": ["rw [\u2190 <a>Equiv.preimage_eq_iff_eq_image</a>]", [{"full_name": "Equiv.preimage_eq_iff_eq_image", "def_path": "Mathlib/Logic/Equiv/Set.lean", "def_pos": [131, 9], "def_end_pos": [131, 33]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\n\u22a2 Icc 0 1 = \u2191K '' Icc 0 1", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\n\u22a2 \u2191K \u207b\u00b9' Icc 0 1 = Icc 0 1"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\n\u22a2 \u2191K \u207b\u00b9' Icc 0 1 = Icc 0 1", "state_after": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nx : \u03b9' \u2192 \u211d\n\u22a2 x \u2208 \u2191K \u207b\u00b9' Icc 0 1 \u2194 x \u2208 Icc 0 1"}, {"tactic": "simp only [mem_preimage, mem_Icc, Pi.le_def, Pi.zero_apply, Equiv.piCongrLeft'_apply,\n  Pi.one_apply]", "annotated_tactic": ["simp only [<a>mem_preimage</a>, <a>mem_Icc</a>, <a>Pi.le_def</a>, <a>Pi.zero_apply</a>, <a>Equiv.piCongrLeft'_apply</a>,\n      <a>Pi.one_apply</a>]", [{"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "Set.mem_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}, {"full_name": "Pi.le_def", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [814, 9], "def_end_pos": [814, 18]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [46, 3], "def_end_pos": [46, 14]}, {"full_name": "Equiv.piCongrLeft'_apply", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1822, 3], "def_end_pos": [1822, 8]}, {"full_name": "Pi.one_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [47, 9], "def_end_pos": [47, 18]}]], "state_before": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nx : \u03b9' \u2192 \u211d\n\u22a2 x \u2208 \u2191K \u207b\u00b9' Icc 0 1 \u2194 x \u2208 Icc 0 1", "state_after": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nx : \u03b9' \u2192 \u211d\n\u22a2 ((\u2200 (i : \u03b9), 0 \u2264 x (\u2191e.symm i)) \u2227 \u2200 (i : \u03b9), x (\u2191e.symm i) \u2264 1) \u2194 (\u2200 (i : \u03b9'), 0 \u2264 x i) \u2227 \u2200 (i : \u03b9'), x i \u2264 1"}, {"tactic": "refine'\n  \u27e8fun h => \u27e8fun i => _, fun i => _\u27e9, fun h =>\n    \u27e8fun i => h.1 (e.symm i), fun i => h.2 (e.symm i)\u27e9\u27e9", "annotated_tactic": ["refine'\n      \u27e8fun h => \u27e8fun i => _, fun i => _\u27e9, fun h =>\n        \u27e8fun i => h.1 (e.symm i), fun i => h.2 (e.symm i)\u27e9\u27e9", []], "state_before": "case h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nx : \u03b9' \u2192 \u211d\n\u22a2 ((\u2200 (i : \u03b9), 0 \u2264 x (\u2191e.symm i)) \u2227 \u2200 (i : \u03b9), x (\u2191e.symm i) \u2264 1) \u2194 (\u2200 (i : \u03b9'), 0 \u2264 x i) \u2227 \u2200 (i : \u03b9'), x i \u2264 1", "state_after": "case h.refine'_1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nx : \u03b9' \u2192 \u211d\nh : (\u2200 (i : \u03b9), 0 \u2264 x (\u2191e.symm i)) \u2227 \u2200 (i : \u03b9), x (\u2191e.symm i) \u2264 1\ni : \u03b9'\n\u22a2 0 \u2264 x i\n\ncase h.refine'_2\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nx : \u03b9' \u2192 \u211d\nh : (\u2200 (i : \u03b9), 0 \u2264 x (\u2191e.symm i)) \u2227 \u2200 (i : \u03b9), x (\u2191e.symm i) \u2264 1\ni : \u03b9'\n\u22a2 x i \u2264 1"}, {"tactic": "simpa only [Equiv.symm_apply_apply] using h.1 (e i)", "annotated_tactic": ["simpa only [<a>Equiv.symm_apply_apply</a>] using h.1 (e i)", [{"full_name": "Equiv.symm_apply_apply", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [283, 17], "def_end_pos": [283, 33]}]], "state_before": "case h.refine'_1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nx : \u03b9' \u2192 \u211d\nh : (\u2200 (i : \u03b9), 0 \u2264 x (\u2191e.symm i)) \u2227 \u2200 (i : \u03b9), x (\u2191e.symm i) \u2264 1\ni : \u03b9'\n\u22a2 0 \u2264 x i", "state_after": "no goals"}, {"tactic": "simpa only [Equiv.symm_apply_apply] using h.2 (e i)", "annotated_tactic": ["simpa only [<a>Equiv.symm_apply_apply</a>] using h.2 (e i)", [{"full_name": "Equiv.symm_apply_apply", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [283, 17], "def_end_pos": [283, 33]}]], "state_before": "case h.refine'_2\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2075 : Fintype \u03b9\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \u211d F\nv : \u03b9 \u2192 E\ne : \u03b9' \u2243 \u03b9\nK : (\u03b9' \u2192 \u211d) \u2243 (\u03b9 \u2192 \u211d) := Equiv.piCongrLeft' (fun _a => \u211d) e\nx : \u03b9' \u2192 \u211d\nh : (\u2200 (i : \u03b9), 0 \u2264 x (\u2191e.symm i)) \u2227 \u2200 (i : \u03b9), x (\u2191e.symm i) \u2264 1\ni : \u03b9'\n\u22a2 x i \u2264 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/ContinuousOn.lean", "full_name": "nhdsWithin_pi_eq", "start": [321, 1], "end": [329, 26], "traced_tactics": [{"tactic": "simp only [nhdsWithin, nhds_pi, Filter.pi, pi_def, \u2190 iInf_principal_finite hI, comap_inf,\n  comap_principal, eval]", "annotated_tactic": ["simp only [<a>nhdsWithin</a>, <a>nhds_pi</a>, <a>Filter.pi</a>, <a>pi_def</a>, \u2190 <a>iInf_principal_finite</a> hI, <a>comap_inf</a>,\n    <a>comap_principal</a>, <a>eval</a>]", [{"full_name": "nhdsWithin", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [838, 5], "def_end_pos": [838, 15]}, {"full_name": "nhds_pi", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1227, 9], "def_end_pos": [1227, 16]}, {"full_name": "Filter.pi", "def_path": "Mathlib/Order/Filter/Pi.lean", "def_pos": [36, 5], "def_end_pos": [36, 7]}, {"full_name": "Set.pi_def", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [2163, 9], "def_end_pos": [2163, 15]}, {"full_name": "Filter.iInf_principal_finite", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1056, 9], "def_end_pos": [1056, 30]}, {"full_name": "Filter.comap_inf", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2268, 17], "def_end_pos": [2268, 26]}, {"full_name": "Filter.comap_principal", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2211, 9], "def_end_pos": [2211, 24]}, {"full_name": "Function.eval", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [29, 24], "def_end_pos": [29, 28]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_5\n\u03b1 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03b1 i)\nI : Set \u03b9\nhI : Set.Finite I\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nx : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 \ud835\udcdd[Set.pi I s] x = (\u2a05 i \u2208 I, comap (fun x => x i) (\ud835\udcdd[s i] x i)) \u2293 \u2a05 i, \u2a05 (_ : \u00aci \u2208 I), comap (fun x => x i) (\ud835\udcdd (x i))", "state_after": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_5\n\u03b1 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03b1 i)\nI : Set \u03b9\nhI : Set.Finite I\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nx : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (\u2a05 i, comap (fun f => f i) (\ud835\udcdd (x i))) \u2293 \u2a05 i \u2208 I, \ud835\udcdf ((fun f => f i) \u207b\u00b9' s i) =\n    (\u2a05 i \u2208 I, comap (fun x => x i) (\ud835\udcdd (x i)) \u2293 \ud835\udcdf ((fun x => x i) \u207b\u00b9' s i)) \u2293\n      \u2a05 i, \u2a05 (_ : \u00aci \u2208 I), comap (fun x => x i) (\ud835\udcdd (x i))"}, {"tactic": "rw [iInf_split _ fun i => i \u2208 I, inf_right_comm]", "annotated_tactic": ["rw [<a>iInf_split</a> _ fun i => i \u2208 I, <a>inf_right_comm</a>]", [{"full_name": "iInf_split", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1488, 9], "def_end_pos": [1488, 19]}, {"full_name": "inf_right_comm", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [532, 9], "def_end_pos": [532, 23]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_5\n\u03b1 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03b1 i)\nI : Set \u03b9\nhI : Set.Finite I\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nx : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (\u2a05 i, comap (fun f => f i) (\ud835\udcdd (x i))) \u2293 \u2a05 i \u2208 I, \ud835\udcdf ((fun f => f i) \u207b\u00b9' s i) =\n    (\u2a05 i \u2208 I, comap (fun x => x i) (\ud835\udcdd (x i)) \u2293 \ud835\udcdf ((fun x => x i) \u207b\u00b9' s i)) \u2293\n      \u2a05 i, \u2a05 (_ : \u00aci \u2208 I), comap (fun x => x i) (\ud835\udcdd (x i))", "state_after": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_5\n\u03b1 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03b1 i)\nI : Set \u03b9\nhI : Set.Finite I\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nx : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 ((\u2a05 i \u2208 I, comap (fun f => f i) (\ud835\udcdd (x i))) \u2293 \u2a05 i \u2208 I, \ud835\udcdf ((fun f => f i) \u207b\u00b9' s i)) \u2293\n      \u2a05 i, \u2a05 (_ : \u00aci \u2208 I), comap (fun f => f i) (\ud835\udcdd (x i)) =\n    (\u2a05 i \u2208 I, comap (fun x => x i) (\ud835\udcdd (x i)) \u2293 \ud835\udcdf ((fun x => x i) \u207b\u00b9' s i)) \u2293\n      \u2a05 i, \u2a05 (_ : \u00aci \u2208 I), comap (fun x => x i) (\ud835\udcdd (x i))"}, {"tactic": "simp only [iInf_inf_eq]", "annotated_tactic": ["simp only [<a>iInf_inf_eq</a>]", [{"full_name": "iInf_inf_eq", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1298, 9], "def_end_pos": [1298, 20]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_5\n\u03b1 : \u03b9 \u2192 Type u_6\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03b1 i)\nI : Set \u03b9\nhI : Set.Finite I\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\nx : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 ((\u2a05 i \u2208 I, comap (fun f => f i) (\ud835\udcdd (x i))) \u2293 \u2a05 i \u2208 I, \ud835\udcdf ((fun f => f i) \u207b\u00b9' s i)) \u2293\n      \u2a05 i, \u2a05 (_ : \u00aci \u2208 I), comap (fun f => f i) (\ud835\udcdd (x i)) =\n    (\u2a05 i \u2208 I, comap (fun x => x i) (\ud835\udcdd (x i)) \u2293 \ud835\udcdf ((fun x => x i) \u207b\u00b9' s i)) \u2293\n      \u2a05 i, \u2a05 (_ : \u00aci \u2208 I), comap (fun x => x i) (\ud835\udcdd (x i))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/Measure.lean", "full_name": "MeasureTheory.map_mul_right_eq_self", "start": [91, 1], "end": [92, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Real/ENNReal.lean", "full_name": "ENNReal.mul_left_mono", "start": [1012, 1], "end": [1012, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/UpperLower/Basic.lean", "full_name": "UpperSet.prod_sup", "start": [1832, 1], "end": [1833, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Star/SelfAdjoint.lean", "full_name": "selfAdjoint.val_one", "start": [343, 1], "end": [344, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/NAry.lean", "full_name": "Filter.map\u2082_map_left_anticomm", "start": [395, 1], "end": [398, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Instances/ENNReal.lean", "full_name": "ENNReal.tendsto_sub", "start": [303, 1], "end": [321, 40], "traced_tactics": [{"tactic": "simp only at h", "annotated_tactic": ["simp only at h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nh : \u22a4 \u2260 \u22a4 \u2228 \u22a4 \u2260 \u22a4\n\u22a2 Tendsto (fun p => p.1 - p.2) (\ud835\udcdd (\u22a4, \u22a4)) (\ud835\udcdd (\u22a4 - \u22a4))", "state_after": "no goals"}, {"tactic": "rw [top_sub_coe, tendsto_nhds_top_iff_nnreal]", "annotated_tactic": ["rw [<a>top_sub_coe</a>, <a>tendsto_nhds_top_iff_nnreal</a>]", [{"full_name": "ENNReal.top_sub_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1141, 17], "def_end_pos": [1141, 28]}, {"full_name": "ENNReal.tendsto_nhds_top_iff_nnreal", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [160, 9], "def_end_pos": [160, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nb : \u211d\u22650\nx\u271d : \u22a4 \u2260 \u22a4 \u2228 \u2191b \u2260 \u22a4\n\u22a2 Tendsto (fun p => p.1 - p.2) (\ud835\udcdd (\u22a4, \u2191b)) (\ud835\udcdd (\u22a4 - \u2191b))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nb : \u211d\u22650\nx\u271d : \u22a4 \u2260 \u22a4 \u2228 \u2191b \u2260 \u22a4\n\u22a2 \u2200 (x : \u211d\u22650), \u2200\u1da0 (a : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e) in \ud835\udcdd (\u22a4, \u2191b), \u2191x < a.1 - a.2"}, {"tactic": "refine fun x => ((lt_mem_nhds <| @coe_lt_top (b + 1 + x)).prod_nhds\n  (ge_mem_nhds <| coe_lt_coe.2 <| lt_add_one b)).mono fun y hy => ?_", "annotated_tactic": ["refine fun x => ((<a>lt_mem_nhds</a> <| @<a>coe_lt_top</a> (b + 1 + x)).<a>prod_nhds</a>\n      (<a>ge_mem_nhds</a> <| <a>coe_lt_coe</a>.2 <| <a>lt_add_one</a> b)).<a>mono</a> fun y hy => ?_", [{"full_name": "lt_mem_nhds", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [897, 9], "def_end_pos": [897, 20]}, {"full_name": "ENNReal.coe_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [308, 17], "def_end_pos": [308, 27]}, {"full_name": "Filter.Eventually.prod_nhds", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [573, 9], "def_end_pos": [573, 36]}, {"full_name": "ge_mem_nhds", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 20]}, {"full_name": "ENNReal.coe_lt_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [352, 28], "def_end_pos": [352, 38]}, {"full_name": "lt_add_one", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [20, 7], "def_end_pos": [20, 17]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nb : \u211d\u22650\nx\u271d : \u22a4 \u2260 \u22a4 \u2228 \u2191b \u2260 \u22a4\n\u22a2 \u2200 (x : \u211d\u22650), \u2200\u1da0 (a : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e) in \ud835\udcdd (\u22a4, \u2191b), \u2191x < a.1 - a.2", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx\u271d\u00b9 y\u271d z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nb : \u211d\u22650\nx\u271d : \u22a4 \u2260 \u22a4 \u2228 \u2191b \u2260 \u22a4\nx : \u211d\u22650\ny : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e\nhy : \u2191(b + 1 + x) < y.1 \u2227 y.2 \u2264 \u2191(b + 1)\n\u22a2 \u2191x < y.1 - y.2"}, {"tactic": "rw [lt_tsub_iff_left]", "annotated_tactic": ["rw [<a>lt_tsub_iff_left</a>]", [{"full_name": "lt_tsub_iff_left", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [422, 9], "def_end_pos": [422, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx\u271d\u00b9 y\u271d z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nb : \u211d\u22650\nx\u271d : \u22a4 \u2260 \u22a4 \u2228 \u2191b \u2260 \u22a4\nx : \u211d\u22650\ny : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e\nhy : \u2191(b + 1 + x) < y.1 \u2227 y.2 \u2264 \u2191(b + 1)\n\u22a2 \u2191x < y.1 - y.2", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx\u271d\u00b9 y\u271d z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nb : \u211d\u22650\nx\u271d : \u22a4 \u2260 \u22a4 \u2228 \u2191b \u2260 \u22a4\nx : \u211d\u22650\ny : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e\nhy : \u2191(b + 1 + x) < y.1 \u2227 y.2 \u2264 \u2191(b + 1)\n\u22a2 y.2 + \u2191x < y.1"}, {"tactic": "calc y.2 + x \u2264 \u2191(b + 1) + x := add_le_add_right hy.2 _\n_ < y.1 := hy.1", "annotated_tactic": ["calc y.2 + x \u2264 \u2191(b + 1) + x := <a>add_le_add_right</a> hy.2 _\n    _ < y.1 := hy.1", [{"full_name": "add_le_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [66, 15], "def_end_pos": [66, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx\u271d\u00b9 y\u271d z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nb : \u211d\u22650\nx\u271d : \u22a4 \u2260 \u22a4 \u2228 \u2191b \u2260 \u22a4\nx : \u211d\u22650\ny : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e\nhy : \u2191(b + 1 + x) < y.1 \u2227 y.2 \u2264 \u2191(b + 1)\n\u22a2 y.2 + \u2191x < y.1", "state_after": "no goals"}, {"tactic": "rw [sub_top]", "annotated_tactic": ["rw [<a>sub_top</a>]", [{"full_name": "ENNReal.sub_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [1145, 9], "def_end_pos": [1145, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\na : \u211d\u22650\nx\u271d : \u2191a \u2260 \u22a4 \u2228 \u22a4 \u2260 \u22a4\n\u22a2 Tendsto (fun p => p.1 - p.2) (\ud835\udcdd (\u2191a, \u22a4)) (\ud835\udcdd (\u2191a - \u22a4))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\na : \u211d\u22650\nx\u271d : \u2191a \u2260 \u22a4 \u2228 \u22a4 \u2260 \u22a4\n\u22a2 Tendsto (fun p => p.1 - p.2) (\ud835\udcdd (\u2191a, \u22a4)) (\ud835\udcdd 0)"}, {"tactic": "refine (tendsto_pure.2 ?_).mono_right (pure_le_nhds _)", "annotated_tactic": ["refine (<a>tendsto_pure</a>.2 ?_).<a>mono_right</a> (<a>pure_le_nhds</a> _)", [{"full_name": "Filter.tendsto_pure", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3167, 17], "def_end_pos": [3167, 29]}, {"full_name": "Filter.Tendsto.mono_right", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3041, 9], "def_end_pos": [3041, 27]}, {"full_name": "pure_le_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1063, 9], "def_end_pos": [1063, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\na : \u211d\u22650\nx\u271d : \u2191a \u2260 \u22a4 \u2228 \u22a4 \u2260 \u22a4\n\u22a2 Tendsto (fun p => p.1 - p.2) (\ud835\udcdd (\u2191a, \u22a4)) (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\na : \u211d\u22650\nx\u271d : \u2191a \u2260 \u22a4 \u2228 \u22a4 \u2260 \u22a4\n\u22a2 \u2200\u1da0 (x : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e) in \ud835\udcdd (\u2191a, \u22a4), x.1 - x.2 = 0"}, {"tactic": "exact ((gt_mem_nhds <| coe_lt_coe.2 <| lt_add_one a).prod_nhds\n  (lt_mem_nhds <| @coe_lt_top (a + 1))).mono fun x hx =>\n    tsub_eq_zero_iff_le.2 (hx.1.trans hx.2).le", "annotated_tactic": ["exact ((<a>gt_mem_nhds</a> <| <a>coe_lt_coe</a>.2 <| <a>lt_add_one</a> a).<a>prod_nhds</a>\n      (<a>lt_mem_nhds</a> <| @<a>coe_lt_top</a> (a + 1))).<a>mono</a> fun x hx =>\n        <a>tsub_eq_zero_iff_le</a>.2 (hx.1.<a>trans</a> hx.2).<a>le</a>", [{"full_name": "gt_mem_nhds", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [905, 9], "def_end_pos": [905, 20]}, {"full_name": "ENNReal.coe_lt_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [352, 28], "def_end_pos": [352, 38]}, {"full_name": "lt_add_one", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [20, 7], "def_end_pos": [20, 17]}, {"full_name": "Filter.Eventually.prod_nhds", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [573, 9], "def_end_pos": [573, 36]}, {"full_name": "lt_mem_nhds", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [897, 9], "def_end_pos": [897, 20]}, {"full_name": "ENNReal.coe_lt_top", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [308, 17], "def_end_pos": [308, 27]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}, {"full_name": "tsub_eq_zero_iff_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [324, 9], "def_end_pos": [324, 28]}, {"full_name": "LT.lt.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [144, 7], "def_end_pos": [144, 18]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\na : \u211d\u22650\nx\u271d : \u2191a \u2260 \u22a4 \u2228 \u22a4 \u2260 \u22a4\n\u22a2 \u2200\u1da0 (x : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e) in \ud835\udcdd (\u2191a, \u22a4), x.1 - x.2 = 0", "state_after": "no goals"}, {"tactic": "exact continuous_sub.tendsto (a, b)", "annotated_tactic": ["exact continuous_sub.tendsto (a, b)", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na\u271d b\u271d c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\na b : \u211d\u22650\nx\u271d : \u2191a \u2260 \u22a4 \u2228 \u2191b \u2260 \u22a4\n\u22a2 Tendsto (fun a => a.1 - a.2) (\ud835\udcdd (a, b)) (\ud835\udcdd (a - b))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.monotone_filter_left", "start": [1960, 1], "end": [1960, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Option/Lemmas.lean", "full_name": "Option.orElse_none", "start": [173, 9], "end": [173, 84], "traced_tactics": [{"tactic": "cases x <;> rfl", "annotated_tactic": ["cases x <;> rfl", []], "state_before": "\u03b1 : Type u_1\nx : Option \u03b1\n\u22a2 (HOrElse.hOrElse x fun x => none) = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.sum_measure_preimage_singleton", "start": [229, 1], "end": [232, 43], "traced_tactics": [{"tactic": "simp only [\u2190 measure_biUnion_finset (pairwiseDisjoint_fiber f s) hf,\n  Finset.set_biUnion_preimage_singleton]", "annotated_tactic": ["simp only [\u2190 <a>measure_biUnion_finset</a> (<a>pairwiseDisjoint_fiber</a> f s) hf,\n    <a>Finset.set_biUnion_preimage_singleton</a>]", [{"full_name": "MeasureTheory.measure_biUnion_finset", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [194, 9], "def_end_pos": [194, 31]}, {"full_name": "Set.pairwiseDisjoint_fiber", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [353, 9], "def_end_pos": [353, 31]}, {"full_name": "Finset.set_biUnion_preimage_singleton", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [2104, 9], "def_end_pos": [2104, 39]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns\u271d s\u2081 s\u2082 t : Set \u03b1\ns : Finset \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (y : \u03b2), y \u2208 s \u2192 MeasurableSet (f \u207b\u00b9' {y})\n\u22a2 \u2211 b in s, \u2191\u2191\u03bc (f \u207b\u00b9' {b}) = \u2191\u2191\u03bc (f \u207b\u00b9' \u2191s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "full_name": "ContinuousLinearMap.map_smul_of_tower", "start": [525, 1], "end": [528, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Disjoint.lean", "full_name": "Disjoint.dual", "start": [402, 1], "end": [404, 5], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Init/Data/Int/CompLemmas.lean", "full_name": "Int.natAbs_bit1_nonneg", "start": [153, 11], "end": [155, 95], "traced_tactics": [{"tactic": "rw [Int.natAbs_add_nonneg (Int.bit0_nonneg h) (le_of_lt Int.zero_lt_one), Int.natAbs_bit0]", "annotated_tactic": ["rw [<a>Int.natAbs_add_nonneg</a> (<a>Int.bit0_nonneg</a> h) (<a>le_of_lt</a> <a>Int.zero_lt_one</a>), <a>Int.natAbs_bit0</a>]", [{"full_name": "Int.natAbs_add_nonneg", "def_path": "Mathlib/Init/Data/Int/CompLemmas.lean", "def_pos": [122, 19], "def_end_pos": [122, 36]}, {"full_name": "Int.bit0_nonneg", "def_path": "Mathlib/Init/Data/Int/CompLemmas.lean", "def_pos": [72, 19], "def_end_pos": [72, 30]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "Int.zero_lt_one", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [655, 19], "def_end_pos": [655, 30]}, {"full_name": "Int.natAbs_bit0", "def_path": "Mathlib/Init/Data/Int/CompLemmas.lean", "def_pos": [140, 19], "def_end_pos": [140, 30]}]], "state_before": "a : \u2124\nh : 0 \u2264 a\n\u22a2 natAbs (bit0 a + 1) = bit0 (natAbs a) + natAbs 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Subsemigroup/Operations.lean", "full_name": "Subsemigroup.le_comap_of_map_le", "start": [287, 1], "end": [288, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/UniformSpace/AbstractCompletion.lean", "full_name": "AbstractCompletion.extend_comp_coe", "start": [165, 1], "end": [168, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/PellMatiyasevic.lean", "full_name": "Pell.pellZd_succ", "start": [220, 1], "end": [220, 95], "traced_tactics": [{"tactic": "simp [Zsqrtd.ext]", "annotated_tactic": ["simp [<a>Zsqrtd.ext</a>]", [{"full_name": "Zsqrtd.ext", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [43, 9], "def_end_pos": [43, 12]}]], "state_before": "a : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 pellZd a1 (n + 1) = pellZd a1 n * { re := \u2191a, im := 1 }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/WithBot.lean", "full_name": "WithTop.ofDual_apply_top", "start": [695, 1], "end": [696, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Logic/Unique.lean", "full_name": "Pi.default_def", "start": [189, 1], "end": [191, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Init/Data/Bool/Lemmas.lean", "full_name": "Bool.ite_eq_true_distrib", "start": [176, 1], "end": [177, 99], "traced_tactics": [{"tactic": "by_cases c <;> simp [*]", "annotated_tactic": ["by_cases c <;> simp [*]", []], "state_before": "c : Prop\ninst\u271d : Decidable c\na b : Bool\n\u22a2 ((if c then a else b) = true) = if c then a = true else b = true", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "iSup_emptyset", "start": [1463, 1], "end": [1463, 74], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b2 \u2192 \u03b1\n\u22a2 \u2a06 x \u2208 \u2205, f x = \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.singleton_eq", "start": [246, 1], "end": [247, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/Forall2.lean", "full_name": "List.forall\u2082_eq_eq_eq", "start": [66, 1], "end": [74, 25], "traced_tactics": [{"tactic": "funext a b", "annotated_tactic": ["funext a b", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 (Forall\u2082 fun x x_1 => x = x_1) = Eq", "state_after": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : List \u03b1\n\u22a2 Forall\u2082 (fun x x_1 => x = x_1) a b = (a = b)"}, {"tactic": "apply propext", "annotated_tactic": ["apply <a>propext</a>", [{"full_name": "propext", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1142, 7], "def_end_pos": [1142, 14]}]], "state_before": "case h.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : List \u03b1\n\u22a2 Forall\u2082 (fun x x_1 => x = x_1) a b = (a = b)", "state_after": "case h.h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : List \u03b1\n\u22a2 Forall\u2082 (fun x x_1 => x = x_1) a b \u2194 a = b"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h.h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : List \u03b1\n\u22a2 Forall\u2082 (fun x x_1 => x = x_1) a b \u2194 a = b", "state_after": "case h.h.a.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : List \u03b1\n\u22a2 Forall\u2082 (fun x x_1 => x = x_1) a b \u2192 a = b\n\ncase h.h.a.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : List \u03b1\n\u22a2 a = b \u2192 Forall\u2082 (fun x x_1 => x = x_1) a b"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case h.h.a.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : List \u03b1\n\u22a2 Forall\u2082 (fun x x_1 => x = x_1) a b \u2192 a = b", "state_after": "case h.h.a.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : List \u03b1\nh : Forall\u2082 (fun x x_1 => x = x_1) a b\n\u22a2 a = b"}, {"tactic": "induction h", "annotated_tactic": ["induction h", []], "state_before": "case h.h.a.mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : List \u03b1\nh : Forall\u2082 (fun x x_1 => x = x_1) a b\n\u22a2 a = b", "state_after": "case h.h.a.mp.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : List \u03b1\n\u22a2 [] = []\n\ncase h.h.a.mp.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : List \u03b1\na\u271d\u00b2 b\u271d : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d\u00b9 : a\u271d\u00b2 = b\u271d\na\u271d : Forall\u2082 (fun x x_1 => x = x_1) l\u2081\u271d l\u2082\u271d\na_ih\u271d : l\u2081\u271d = l\u2082\u271d\n\u22a2 a\u271d\u00b2 :: l\u2081\u271d = b\u271d :: l\u2082\u271d"}, {"tactic": "simp only [*]", "annotated_tactic": ["simp only [*]", []], "state_before": "case h.h.a.mp.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : List \u03b1\na\u271d\u00b2 b\u271d : \u03b1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na\u271d\u00b9 : a\u271d\u00b2 = b\u271d\na\u271d : Forall\u2082 (fun x x_1 => x = x_1) l\u2081\u271d l\u2082\u271d\na_ih\u271d : l\u2081\u271d = l\u2082\u271d\n\u22a2 a\u271d\u00b2 :: l\u2081\u271d = b\u271d :: l\u2082\u271d", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.h.a.mp.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : List \u03b1\n\u22a2 [] = []", "state_after": "no goals"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "case h.h.a.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : List \u03b1\n\u22a2 a = b \u2192 Forall\u2082 (fun x x_1 => x = x_1) a b", "state_after": "case h.h.a.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na : List \u03b1\n\u22a2 Forall\u2082 (fun x x_1 => x = x_1) a a"}, {"tactic": "exact forall\u2082_refl _", "annotated_tactic": ["exact <a>forall\u2082_refl</a> _", [{"full_name": "List.forall\u2082_refl", "def_path": "Mathlib/Data/List/Forall2.lean", "def_pos": [61, 9], "def_end_pos": [61, 21]}]], "state_before": "case h.h.a.mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na : List \u03b1\n\u22a2 Forall\u2082 (fun x x_1 => x = x_1) a a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.setToFun_add_left'", "start": [1320, 1], "end": [1326, 44], "traced_tactics": [{"tactic": "by_cases hf : Integrable f \u03bc", "annotated_tactic": ["by_cases hf : <a>Integrable</a> f \u03bc", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u2077 : 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[]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/AdjoinRoot.lean", "full_name": "AdjoinRoot.Minpoly.toAdjoin.apply_X", "start": [631, 1], "end": [633, 18], "traced_tactics": [{"tactic": "simp [toAdjoin]", "annotated_tactic": ["simp [<a>toAdjoin</a>]", [{"full_name": "AdjoinRoot.Minpoly.toAdjoin", "def_path": "Mathlib/RingTheory/AdjoinRoot.lean", "def_pos": [617, 5], "def_end_pos": [617, 21]}]], "state_before": "R : Type u\nS : Type v\nK : Type w\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra R S\nx : S\n\u22a2 \u2191(toAdjoin R x) (\u2191(mk (minpoly R x)) X) = { val := x, property := (_ : x \u2208 adjoin R {x}) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "full_name": "Orthonormal.toSubtypeRange", "start": [895, 1], "end": [897, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/InnerProductSpace/Calculus.lean", "full_name": "LocalHomeomorph.contDiff_unitBallBall_symm", "start": [429, 1], "end": [430, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/CircleTransform.lean", "full_name": "Complex.continuous_circleTransform", "start": [77, 1], "end": [85, 52], "traced_tactics": [{"tactic": "apply_rules [Continuous.smul, continuous_const]", "annotated_tactic": ["apply_rules [<a>Continuous.smul</a>, <a>continuous_const</a>]", [{"full_name": "Continuous.smul", "def_path": "Mathlib/Topology/Algebra/MulAction.lean", "def_pos": [120, 9], "def_end_pos": [120, 24]}, {"full_name": "continuous_const", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1723, 9], "def_end_pos": [1723, 25]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous (circleTransform R z w f)", "state_after": "case hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => deriv (circleMap z R) x\n\ncase hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => (circleMap z R x - w)\u207b\u00b9\n\ncase hg.hg.hg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => f (circleMap z R x)"}, {"tactic": "simp_rw [deriv_circleMap]", "annotated_tactic": ["simp_rw [<a>deriv_circleMap</a>]", [{"full_name": "deriv_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [195, 9], "def_end_pos": [195, 24]}]], "state_before": "case hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => deriv (circleMap z R) x\n\ncase hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => (circleMap z R x - w)\u207b\u00b9\n\ncase hg.hg.hg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => f (circleMap z R x)", "state_after": "case hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => circleMap 0 R x * I\n\ncase hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => (circleMap z R x - w)\u207b\u00b9\n\ncase hg.hg.hg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => f (circleMap z R x)"}, {"tactic": "apply_rules [Continuous.mul, continuous_circleMap 0 R, continuous_const]", "annotated_tactic": ["apply_rules [<a>Continuous.mul</a>, <a>continuous_circleMap</a> 0 R, <a>continuous_const</a>]", [{"full_name": "Continuous.mul", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [87, 9], "def_end_pos": [87, 23]}, {"full_name": "continuous_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [185, 9], "def_end_pos": [185, 29]}, {"full_name": "continuous_const", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1723, 9], "def_end_pos": [1723, 25]}]], "state_before": "case hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => circleMap 0 R x * I\n\ncase hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => (circleMap z R x - w)\u207b\u00b9\n\ncase hg.hg.hg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => f (circleMap z R x)", "state_after": "case hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => (circleMap z R x - w)\u207b\u00b9\n\ncase hg.hg.hg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => f (circleMap z R x)"}, {"tactic": "apply continuous_circleMap_inv hw", "annotated_tactic": ["apply <a>continuous_circleMap_inv</a> hw", [{"full_name": "continuous_circleMap_inv", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [213, 9], "def_end_pos": [213, 33]}]], "state_before": "case hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => (circleMap z R x - w)\u207b\u00b9", "state_after": "no goals"}, {"tactic": "apply ContinuousOn.comp_continuous hf (continuous_circleMap z R)", "annotated_tactic": ["apply <a>ContinuousOn.comp_continuous</a> hf (<a>continuous_circleMap</a> z R)", [{"full_name": "ContinuousOn.comp_continuous", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [962, 9], "def_end_pos": [962, 37]}, {"full_name": "continuous_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [185, 9], "def_end_pos": [185, 29]}]], "state_before": "case hg.hg.hg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => f (circleMap z R x)", "state_after": "case hg.hg.hg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 \u2200 (x : \u211d), circleMap z R x \u2208 sphere z R"}, {"tactic": "exact fun _ => (circleMap_mem_sphere _ hR.le) _", "annotated_tactic": ["exact fun _ => (<a>circleMap_mem_sphere</a> _ hR.le) _", [{"full_name": "circleMap_mem_sphere", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [122, 9], "def_end_pos": [122, 29]}]], "state_before": "case hg.hg.hg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 \u2200 (x : \u211d), circleMap z R x \u2208 sphere z R", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "Algebra.adjoin_eq_range", "start": [1602, 1], "end": [1604, 64], "traced_tactics": [{"tactic": "rw [\u2190 Algebra.adjoin_range_eq_range_aeval, Subtype.range_coe]", "annotated_tactic": ["rw [\u2190 <a>Algebra.adjoin_range_eq_range_aeval</a>, <a>Subtype.range_coe</a>]", [{"full_name": "Algebra.adjoin_range_eq_range_aeval", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [1596, 9], "def_end_pos": [1596, 51]}, {"full_name": "Subtype.range_coe", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 18]}]], "state_before": "R : Type u\nS\u2081 : Type v\nS\u2082 : Type w\nS\u2083 : Type x\n\u03c3 : Type u_1\na a' a\u2081 a\u2082 : R\ne : \u2115\nn m : \u03c3\ns\u271d : \u03c3 \u2192\u2080 \u2115\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommSemiring S\u2081\np q : MvPolynomial \u03c3 R\ninst\u271d\u00b9 : Algebra R S\u2081\ninst\u271d : CommSemiring S\u2082\nf : \u03c3 \u2192 S\u2081\ns : Set S\u2081\n\u22a2 Algebra.adjoin R s = AlgHom.range (aeval Subtype.val)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Category/Ring/Constructions.lean", "full_name": "CommRingCat.pushoutCocone_pt", "start": [75, 1], "end": [80, 6], "traced_tactics": [{"tactic": "letI := f.toAlgebra", "annotated_tactic": ["letI := f.toAlgebra", []], "state_before": "R A B : CommRingCat\nf : R \u27f6 A\ng : R \u27f6 B\n\u22a2 CommRingCat", "state_after": "R A B : CommRingCat\nf : R \u27f6 A\ng : R \u27f6 B\nthis : Algebra \u2191R \u2191A := RingHom.toAlgebra f\n\u22a2 CommRingCat"}, {"tactic": "letI := g.toAlgebra", "annotated_tactic": ["letI := g.toAlgebra", []], "state_before": "R A B : CommRingCat\nf : R \u27f6 A\ng : R \u27f6 B\nthis : Algebra \u2191R \u2191A := RingHom.toAlgebra f\n\u22a2 CommRingCat", "state_after": "R A B : CommRingCat\nf : R \u27f6 A\ng : R \u27f6 B\nthis\u271d : Algebra \u2191R \u2191A := RingHom.toAlgebra f\nthis : Algebra \u2191R \u2191B := RingHom.toAlgebra g\n\u22a2 CommRingCat"}, {"tactic": "exact CommRingCat.of (A \u2297[R] B)", "annotated_tactic": ["exact <a>CommRingCat.of</a> (A \u2297[R] B)", [{"full_name": "CommRingCat.of", "def_path": "Mathlib/Algebra/Category/Ring/Basic.lean", "def_pos": [434, 5], "def_end_pos": [434, 7]}]], "state_before": "R A B : CommRingCat\nf : R \u27f6 A\ng : R \u27f6 B\nthis\u271d : Algebra \u2191R \u2191A := RingHom.toAlgebra f\nthis : Algebra \u2191R \u2191B := RingHom.toAlgebra g\n\u22a2 CommRingCat", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Hom/Lattice.lean", "full_name": "SupHom.dual_id", "start": [1366, 1], "end": [1367, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/TypeVec.lean", "full_name": "TypeVec.lastFun_subtypeVal", "start": [687, 1], "end": [689, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Extr.lean", "full_name": "isMinFilter_dual_iff", "start": [206, 1], "end": [207, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Sites/Subsheaf.lean", "full_name": "CategoryTheory.GrothendieckTopology.Subpresheaf.sheafify_sheafify", "start": [270, 1], "end": [272, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.scanl_cons", "start": [2536, 1], "end": [2537, 70], "traced_tactics": [{"tactic": "simp only [scanl, eq_self_iff_true, singleton_append, and_self_iff]", "annotated_tactic": ["simp only [<a>scanl</a>, <a>eq_self_iff_true</a>, <a>singleton_append</a>, <a>and_self_iff</a>]", [{"full_name": "List.scanl", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [643, 13], "def_end_pos": [643, 18]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "List.singleton_append", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [55, 22], "def_end_pos": [55, 38]}, {"full_name": "and_self_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [155, 9], "def_end_pos": [155, 21]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\na : \u03b1\nl : List \u03b1\n\u22a2 scanl f b (a :: l) = [b] ++ scanl f (f b a) l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Preadditive/Basic.lean", "full_name": "CategoryTheory.Preadditive.forkOfKernelFork_\u03b9", "start": [325, 1], "end": [326, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "full_name": "USize.modn_toNat", "start": [750, 9], "end": [751, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Jordan/Basic.lean", "full_name": "aux1", "start": [197, 9], "end": [208, 26], "traced_tactics": [{"tactic": "rw [add_lie, add_lie]", "annotated_tactic": ["rw [<a>add_lie</a>, <a>add_lie</a>]", [{"full_name": "add_lie", "def_path": "Mathlib/Algebra/Lie/Basic.lean", "def_pos": [113, 9], "def_end_pos": [113, 16]}, {"full_name": "add_lie", "def_path": "Mathlib/Algebra/Lie/Basic.lean", "def_pos": [113, 9], "def_end_pos": [113, 16]}]], "state_before": "A : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a + \u2191L b + \u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 =\n    \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n          \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n        (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n            \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n          \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n      (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n        \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046)", "state_after": "A : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 +\n        \u2045\u2191L b, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 +\n      \u2045\u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 =\n    \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n          \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n        (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n            \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n          \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n      (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n        \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046)"}, {"tactic": "iterate 15 rw [lie_add]", "annotated_tactic": ["iterate 15 rw [<a>lie_add</a>]", [{"full_name": "lie_add", "def_path": "Mathlib/Algebra/Lie/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 16]}]], "state_before": "A : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 +\n        \u2045\u2191L b, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 +\n      \u2045\u2191L c, \u2191L (a * a) + \u2191L (b * b) + \u2191L (c * c) + 2 \u2022 \u2191L (a * b) + 2 \u2022 \u2191L (c * a) + 2 \u2022 \u2191L (b * c)\u2046 =\n    \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n          \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n        (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n            \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n          \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n      (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n        \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046)", "state_after": "no goals"}, {"tactic": "rw [lie_add]", "annotated_tactic": ["rw [<a>lie_add</a>]", [{"full_name": "lie_add", "def_path": "Mathlib/Algebra/Lie/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 16]}]], "state_before": "A : Type u_1\ninst\u271d\u00b9 : NonUnitalNonAssocRing A\ninst\u271d : IsCommJordan A\na b c : A\n\u22a2 \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n          \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n        (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n            \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n          \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n      (\u2045\u2191L c, \u2191L (a * a) + \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n        \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046) =\n    \u2045\u2191L a, \u2191L (a * a)\u2046 + \u2045\u2191L a, \u2191L (b * b)\u2046 + \u2045\u2191L a, \u2191L (c * c)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L a, 2 \u2022 \u2191L (c * a)\u2046 +\n          \u2045\u2191L a, 2 \u2022 \u2191L (b * c)\u2046 +\n        (\u2045\u2191L b, \u2191L (a * a)\u2046 + \u2045\u2191L b, \u2191L (b * b)\u2046 + \u2045\u2191L b, \u2191L (c * c)\u2046 + \u2045\u2191L b, 2 \u2022 \u2191L (a * b)\u2046 +\n            \u2045\u2191L b, 2 \u2022 \u2191L (c * a)\u2046 +\n          \u2045\u2191L b, 2 \u2022 \u2191L (b * c)\u2046) +\n      (\u2045\u2191L c, \u2191L (a * a)\u2046 + \u2045\u2191L c, \u2191L (b * b)\u2046 + \u2045\u2191L c, \u2191L (c * c)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (a * b)\u2046 + \u2045\u2191L c, 2 \u2022 \u2191L (c * a)\u2046 +\n        \u2045\u2191L c, 2 \u2022 \u2191L (b * c)\u2046)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "full_name": "Std.RBNode.mem_insert_of_mem", "start": [567, 1], "end": [578, 37], "traced_tactics": [{"tactic": "match e : zoom (cmp v) t with\n| (nil, p) =>\n  let \u27e8_, _, h\u2081, h\u2082\u27e9 := exists_insert_toList_zoom_nil ht e\n  simp [\u2190 mem_toList, h\u2081] at h\n  simp [\u2190 mem_toList, h\u2082]; cases h <;> simp [*]\n| (node .., p) =>\n  let \u27e8_, _, h\u2081, h\u2082\u27e9 := exists_insert_toList_zoom_node ht e\n  simp [\u2190 mem_toList, h\u2081] at h\n  simp [\u2190 mem_toList, h\u2082]; rcases h with h|h|h <;> simp [*]\n  exact .inr (Path.zoom_zoomed\u2081 e)", "annotated_tactic": ["match e : <a>zoom</a> (cmp v) t with\n  | (<a>nil</a>, p) =>\n    let \u27e8_, _, h\u2081, h\u2082\u27e9 := <a>exists_insert_toList_zoom_nil</a> ht e\n    simp [\u2190 <a>mem_toList</a>, h\u2081] at h\n    simp [\u2190 <a>mem_toList</a>, h\u2082]; cases h <;> simp [*]\n  | (<a>node</a> .., p) =>\n    let \u27e8_, _, h\u2081, h\u2082\u27e9 := <a>exists_insert_toList_zoom_node</a> ht e\n    simp [\u2190 <a>mem_toList</a>, h\u2081] at h\n    simp [\u2190 <a>mem_toList</a>, h\u2082]; rcases h with h|h|h <;> simp [*]\n    exact .inr (<a>Path.zoom_zoomed\u2081</a> e)", [{"full_name": "Std.RBNode.zoom", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [451, 19], "def_end_pos": [451, 23]}, {"full_name": "Std.RBNode.nil", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [46, 5], "def_end_pos": [46, 8]}, {"full_name": "Std.RBNode.exists_insert_toList_zoom_nil", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [546, 9], "def_end_pos": [546, 38]}, {"full_name": "Std.RBNode.mem_toList", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [370, 17], "def_end_pos": [370, 27]}, {"full_name": "Std.RBNode.mem_toList", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [370, 17], "def_end_pos": [370, 27]}, {"full_name": "Std.RBNode.node", "def_path": "lake-packages/std/Std/Data/RBMap/Basic.lean", "def_pos": [50, 5], "def_end_pos": [50, 9]}, {"full_name": "Std.RBNode.exists_insert_toList_zoom_node", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [555, 9], "def_end_pos": [555, 39]}, {"full_name": "Std.RBNode.mem_toList", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [370, 17], "def_end_pos": [370, 27]}, {"full_name": "Std.RBNode.mem_toList", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [370, 17], "def_end_pos": [370, 27]}, {"full_name": "Std.RBNode.Path.zoom_zoomed\u2081", "def_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "def_pos": [218, 9], "def_end_pos": [218, 21]}]], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nh : v' \u2208 t\n\u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq", "state_after": "no goals"}, {"tactic": "let \u27e8_, _, h\u2081, h\u2082\u27e9 := exists_insert_toList_zoom_nil ht e", "annotated_tactic": ["let \u27e8_, _, h\u2081, h\u2082\u27e9 := <a>exists_insert_toList_zoom_nil</a> ht e", [{"full_name": "Std.RBNode.exists_insert_toList_zoom_nil", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [546, 9], "def_end_pos": [546, 38]}]], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nh : v' \u2208 t\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (nil, p)\n\u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq", "state_after": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nh : v' \u2208 t\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (nil, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\n\u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq"}, {"tactic": "simp [\u2190 mem_toList, h\u2081] at h", "annotated_tactic": ["simp [\u2190 <a>mem_toList</a>, h\u2081] at h", [{"full_name": "Std.RBNode.mem_toList", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [370, 17], "def_end_pos": [370, 27]}]], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nh : v' \u2208 t\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (nil, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\n\u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq", "state_after": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (nil, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\nh : v' \u2208 w\u271d\u00b9 \u2228 v' \u2208 w\u271d\n\u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq"}, {"tactic": "simp [\u2190 mem_toList, h\u2082]", "annotated_tactic": ["simp [\u2190 <a>mem_toList</a>, h\u2082]", [{"full_name": "Std.RBNode.mem_toList", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [370, 17], "def_end_pos": [370, 27]}]], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (nil, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\nh : v' \u2208 w\u271d\u00b9 \u2228 v' \u2208 w\u271d\n\u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq", "state_after": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (nil, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\nh : v' \u2208 w\u271d\u00b9 \u2228 v' \u2208 w\u271d\n\u22a2 (v' \u2208 w\u271d\u00b9 \u2228 v' = v \u2228 v' \u2208 w\u271d) \u2228 cmp v v' = Ordering.eq"}, {"tactic": "cases h <;> simp [*]", "annotated_tactic": ["cases h <;> simp [*]", []], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (nil, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\nh : v' \u2208 w\u271d\u00b9 \u2228 v' \u2208 w\u271d\n\u22a2 (v' \u2208 w\u271d\u00b9 \u2228 v' = v \u2228 v' \u2208 w\u271d) \u2228 cmp v v' = Ordering.eq", "state_after": "no goals"}, {"tactic": "let \u27e8_, _, h\u2081, h\u2082\u27e9 := exists_insert_toList_zoom_node ht e", "annotated_tactic": ["let \u27e8_, _, h\u2081, h\u2082\u27e9 := <a>exists_insert_toList_zoom_node</a> ht e", [{"full_name": "Std.RBNode.exists_insert_toList_zoom_node", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [555, 9], "def_end_pos": [555, 39]}]], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nh : v' \u2208 t\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p)\n\u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq", "state_after": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nh : v' \u2208 t\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\n\u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq"}, {"tactic": "simp [\u2190 mem_toList, h\u2081] at h", "annotated_tactic": ["simp [\u2190 <a>mem_toList</a>, h\u2081] at h", [{"full_name": "Std.RBNode.mem_toList", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [370, 17], "def_end_pos": [370, 27]}]], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nh : v' \u2208 t\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\n\u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq", "state_after": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\nh : v' \u2208 w\u271d\u00b9 \u2228 v' = v\u271d \u2228 v' \u2208 w\u271d\n\u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq"}, {"tactic": "simp [\u2190 mem_toList, h\u2082]", "annotated_tactic": ["simp [\u2190 <a>mem_toList</a>, h\u2082]", [{"full_name": "Std.RBNode.mem_toList", "def_path": "lake-packages/std/Std/Data/RBMap/Lemmas.lean", "def_pos": [370, 17], "def_end_pos": [370, 27]}]], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\nh : v' \u2208 w\u271d\u00b9 \u2228 v' = v\u271d \u2228 v' \u2208 w\u271d\n\u22a2 v' \u2208 insert cmp t v \u2228 cmp v v' = Ordering.eq", "state_after": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\nh : v' \u2208 w\u271d\u00b9 \u2228 v' = v\u271d \u2228 v' \u2208 w\u271d\n\u22a2 (v' \u2208 w\u271d\u00b9 \u2228 v' = v \u2228 v' \u2208 w\u271d) \u2228 cmp v v' = Ordering.eq"}, {"tactic": "rcases h with h|h|h <;> simp [*]", "annotated_tactic": ["rcases h with h|h|h <;> simp [*]", []], "state_before": "\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\nh : v' \u2208 w\u271d\u00b9 \u2228 v' = v\u271d \u2228 v' \u2208 w\u271d\n\u22a2 (v' \u2208 w\u271d\u00b9 \u2228 v' = v \u2228 v' \u2208 w\u271d) \u2228 cmp v v' = Ordering.eq", "state_after": "case inr.inl\n\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\nh : v' = v\u271d\n\u22a2 (v\u271d \u2208 w\u271d\u00b9 \u2228 v\u271d = v \u2228 v\u271d \u2208 w\u271d) \u2228 cmp v v\u271d = Ordering.eq"}, {"tactic": "exact .inr (Path.zoom_zoomed\u2081 e)", "annotated_tactic": ["exact .inr (<a>Path.zoom_zoomed\u2081</a> e)", [{"full_name": "Std.RBNode.Path.zoom_zoomed\u2081", "def_path": "lake-packages/std/Std/Data/RBMap/Alter.lean", "def_pos": [218, 9], "def_end_pos": [218, 21]}]], "state_before": "case inr.inl\n\u03b1 : Type u_1\nc : RBColor\nn : Nat\nv' : \u03b1\ncmp : \u03b1 \u2192 \u03b1 \u2192 Ordering\nv : \u03b1\nt : RBNode \u03b1\nht : Balanced t c n\nc\u271d : RBColor\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\np : Path \u03b1\ne : zoom (cmp v) t Path.root = (node c\u271d l\u271d v\u271d r\u271d, p)\nw\u271d\u00b9 w\u271d : List \u03b1\nh\u2081 : toList t = w\u271d\u00b9 ++ v\u271d :: w\u271d\nh\u2082 : toList (insert cmp t v) = w\u271d\u00b9 ++ v :: w\u271d\nh : v' = v\u271d\n\u22a2 (v\u271d \u2208 w\u271d\u00b9 \u2228 v\u271d = v \u2228 v\u271d \u2208 w\u271d) \u2228 cmp v v\u271d = Ordering.eq", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/SuccPred/Basic.lean", "full_name": "Order.pred_le_iff_eq_or_le", "start": [799, 1], "end": [802, 65], "traced_tactics": [{"tactic": "by_cases ha : IsMin a", "annotated_tactic": ["by_cases ha : <a>IsMin</a> a", [{"full_name": "IsMin", "def_path": "Mathlib/Order/Max.lean", "def_pos": [202, 5], "def_end_pos": [202, 10]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\n\u22a2 pred a \u2264 b \u2194 b = pred a \u2228 a \u2264 b", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nha : IsMin a\n\u22a2 pred a \u2264 b \u2194 b = pred a \u2228 a \u2264 b\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nha : \u00acIsMin a\n\u22a2 pred a \u2264 b \u2194 b = pred a \u2228 a \u2264 b"}, {"tactic": "rw [ha.pred_eq, or_iff_right_of_imp ge_of_eq]", "annotated_tactic": ["rw [ha.pred_eq, <a>or_iff_right_of_imp</a> <a>ge_of_eq</a>]", [{"full_name": "or_iff_right_of_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [289, 9], "def_end_pos": [289, 28]}, {"full_name": "ge_of_eq", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [352, 9], "def_end_pos": [352, 17]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nha : IsMin a\n\u22a2 pred a \u2264 b \u2194 b = pred a \u2228 a \u2264 b", "state_after": "no goals"}, {"tactic": "rw [\u2190 pred_lt_iff_of_not_isMin ha, le_iff_eq_or_lt, eq_comm]", "annotated_tactic": ["rw [\u2190 <a>pred_lt_iff_of_not_isMin</a> ha, <a>le_iff_eq_or_lt</a>, <a>eq_comm</a>]", [{"full_name": "Order.pred_lt_iff_of_not_isMin", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [638, 9], "def_end_pos": [638, 33]}, {"full_name": "le_iff_eq_or_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [390, 9], "def_end_pos": [390, 24]}, {"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nha : \u00acIsMin a\n\u22a2 pred a \u2264 b \u2194 b = pred a \u2228 a \u2264 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Surreal/Basic.lean", "full_name": "SetTheory.PGame.numeric_zero", "start": [205, 1], "end": [206, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Dynamics/PeriodicPts.lean", "full_name": "Function.minimalPeriod_id", "start": [367, 1], "end": [369, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean", "full_name": "CliffordAlgebraComplex.Q_apply", "start": [135, 1], "end": [136, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Torsion.lean", "full_name": "IsTorsion.quotient_iff", "start": [123, 1], "end": [125, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Polynomial/Basic.lean", "full_name": "Polynomial.linearIndependent_powers_iff_aeval", "start": [987, 1], "end": [992, 16], "traced_tactics": [{"tactic": "rw [linearIndependent_iff]", "annotated_tactic": ["rw [<a>linearIndependent_iff</a>]", [{"full_name": "linearIndependent_iff", "def_path": "Mathlib/LinearAlgebra/LinearIndependent.lean", "def_pos": [105, 9], "def_end_pos": [105, 30]}]], "state_before": "R : Type u\nS : Type u_1\n\u03c3 : Type v\nM : Type w\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : M \u2192\u2097[R] M\nv : M\n\u22a2 (LinearIndependent R fun n => \u2191(f ^ n) v) \u2194 \u2200 (p : R[X]), \u2191(\u2191(aeval f) p) v = 0 \u2192 p = 0", "state_after": "R : Type u\nS : Type u_1\n\u03c3 : Type v\nM : Type w\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : M \u2192\u2097[R] M\nv : M\n\u22a2 (\u2200 (l : \u2115 \u2192\u2080 R), \u2191(Finsupp.total \u2115 ((fun x => M) v) R fun n => \u2191(f ^ n) v) l = 0 \u2192 l = 0) \u2194\n    \u2200 (p : R[X]), \u2191(\u2191(aeval f) p) v = 0 \u2192 p = 0"}, {"tactic": "simp only [Finsupp.total_apply, aeval_endomorphism, forall_iff_forall_finsupp, Sum, support,\n  coeff, ofFinsupp_eq_zero]", "annotated_tactic": ["simp only [<a>Finsupp.total_apply</a>, <a>aeval_endomorphism</a>, <a>forall_iff_forall_finsupp</a>, <a>Sum</a>, <a>support</a>,\n    <a>coeff</a>, <a>ofFinsupp_eq_zero</a>]", [{"full_name": "Finsupp.total_apply", "def_path": "Mathlib/LinearAlgebra/Finsupp.lean", "def_pos": [549, 9], "def_end_pos": [549, 20]}, {"full_name": "Polynomial.aeval_endomorphism", "def_path": "Mathlib/Data/Polynomial/AlgebraMap.lean", "def_pos": [511, 9], "def_end_pos": [511, 27]}, {"full_name": "Polynomial.forall_iff_forall_finsupp", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [87, 9], "def_end_pos": [87, 34]}, {"full_name": "Sum", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [102, 11], "def_end_pos": [102, 14]}, {"full_name": "Polynomial.support", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [404, 5], "def_end_pos": [404, 12]}, {"full_name": "Polynomial.coeff", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [665, 5], "def_end_pos": [665, 10]}, {"full_name": "Polynomial.ofFinsupp_eq_zero", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [277, 9], "def_end_pos": [277, 26]}]], "state_before": "R : Type u\nS : Type u_1\n\u03c3 : Type v\nM : Type w\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : M \u2192\u2097[R] M\nv : M\n\u22a2 (\u2200 (l : \u2115 \u2192\u2080 R), \u2191(Finsupp.total \u2115 ((fun x => M) v) R fun n => \u2191(f ^ n) v) l = 0 \u2192 l = 0) \u2194\n    \u2200 (p : R[X]), \u2191(\u2191(aeval f) p) v = 0 \u2192 p = 0", "state_after": "R : Type u\nS : Type u_1\n\u03c3 : Type v\nM : Type w\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : M \u2192\u2097[R] M\nv : M\n\u22a2 (\u2200 (l : \u2115 \u2192\u2080 R), (Finsupp.sum l fun i a => a \u2022 \u2191(f ^ i) v) = 0 \u2192 l = 0) \u2194\n    \u2200 (q : AddMonoidAlgebra R \u2115), (sum { toFinsupp := q } fun n b => b \u2022 \u2191(f ^ n) v) = 0 \u2192 q = 0"}, {"tactic": "exact Iff.rfl", "annotated_tactic": ["exact <a>Iff.rfl</a>", 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Category.{u_2, u_1} C\n\u03c6 : V \u2964q C\n\u22a2 \u2200 (X : V), (of \u22d9q (lift \u03c6).toPrefunctor).obj X = \u03c6.obj X\n\ncase h_map\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03c6 : V \u2964q C\n\u22a2 \u2200 (X Y : V) (f : X \u27f6 Y),\n    (of \u22d9q (lift \u03c6).toPrefunctor).map f =\n      Eq.recOn (_ : \u03c6.obj Y = (of \u22d9q (lift \u03c6).toPrefunctor).obj Y)\n        (Eq.recOn (_ : \u03c6.obj X = (of \u22d9q (lift \u03c6).toPrefunctor).obj X) (\u03c6.map f))"}, {"tactic": "rintro X", "annotated_tactic": ["rintro X", []], "state_before": "case h_obj\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03c6 : V \u2964q C\n\u22a2 \u2200 (X : V), (of \u22d9q (lift \u03c6).toPrefunctor).obj X = \u03c6.obj X", "state_after": "case h_obj\nV : Type u\u2081\ninst\u271d\u00b9 : Quiver V\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\n\u03c6 : V \u2964q C\nX : 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= x\n\u22a2 IsPath ({ fst := a, snd := i } :: xs) x \u2192 IsPath xs (f i)"}, {"tactic": "rintro (_ | \u27e8_, _, _, _, rfl, hp\u27e9)", "annotated_tactic": ["rintro (_ | \u27e8_, _, _, _, rfl, hp\u27e9)", []], "state_before": "F : PFunctor.{u}\nX : Type u_1\nf\u271d : X \u2192 \u2191F X\nxs : Path F\na : F.A\nf : B F a \u2192 M F\ni : B F a\nx : M F\nh : M.mk { fst := a, snd := f } = x\n\u22a2 IsPath ({ fst := a, snd := i } :: xs) x \u2192 IsPath xs (f i)", "state_after": "case cons\nF : PFunctor.{u}\nX : Type u_1\nf\u271d\u00b9 : X \u2192 \u2191F X\nxs : Path F\na : F.A\nf : B F a \u2192 M F\ni : B F a\nf\u271d : B F a \u2192 M F\nhp : IsPath xs (f\u271d i)\nh : M.mk { fst := a, snd := f } = M.mk { fst := a, snd := f\u271d }\n\u22a2 IsPath xs (f i)"}, {"tactic": "cases mk_inj h", "annotated_tactic": ["cases <a>mk_inj</a> h", [{"full_name": "PFunctor.M.mk_inj", "def_path": "Mathlib/Data/PFunctor/Univariate/M.lean", "def_pos": [365, 9], "def_end_pos": [365, 15]}]], "state_before": 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"3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Logic/Equiv/Defs.lean", "full_name": "Equiv.eq_symm_comp", "start": [594, 1], "end": [595, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Regular/SMul.lean", "full_name": "Units.isSMulRegular", "start": [248, 1], "end": [249, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finite/Card.lean", "full_name": "Finite.one_lt_card", "start": [88, 1], "end": [89, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean", "full_name": "Ideal.hasBasis_nhds_adic", "start": [107, 1], "end": [112, 41], "traced_tactics": [{"tactic": "letI := I.adicTopology", "annotated_tactic": ["letI := I.adicTopology", []], "state_before": "R : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nx : R\n\u22a2 HasBasis (\ud835\udcdd x) (fun _n => True) fun n => (fun y => x + y) '' \u2191(I ^ n)", "state_after": "R : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nx : R\nthis : TopologicalSpace R := adicTopology I\n\u22a2 HasBasis (\ud835\udcdd x) (fun _n => True) fun n => (fun y => x + y) '' \u2191(I ^ n)"}, {"tactic": "have := I.hasBasis_nhds_zero_adic.map fun y => x + y", "annotated_tactic": ["have := I.hasBasis_nhds_zero_adic.map fun y => x + y", []], "state_before": "R : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nx : R\nthis : TopologicalSpace R := adicTopology I\n\u22a2 HasBasis (\ud835\udcdd x) (fun _n => True) fun n => (fun y => x + y) '' \u2191(I ^ n)", "state_after": "R : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nx : R\nthis\u271d : TopologicalSpace R := adicTopology I\nthis : HasBasis (Filter.map (fun y => x + y) (\ud835\udcdd 0)) (fun _n => True) fun i => (fun y => x + y) '' \u2191(I ^ i)\n\u22a2 HasBasis (\ud835\udcdd x) (fun _n => True) fun n => (fun y => x + y) '' \u2191(I ^ n)"}, {"tactic": "rwa [map_add_left_nhds_zero x] at this", "annotated_tactic": ["rwa [<a>map_add_left_nhds_zero</a> x] at this", [{"full_name": "map_add_left_nhds_zero", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [822, 3], "def_end_pos": [822, 14]}]], "state_before": "R : Type u_1\ninst\u271d : CommRing R\nI : Ideal R\nx : R\nthis\u271d : TopologicalSpace R := adicTopology I\nthis : HasBasis (Filter.map (fun y => x + y) (\ud835\udcdd 0)) (fun _n => True) fun i => (fun y => x + y) '' \u2191(I ^ i)\n\u22a2 HasBasis (\ud835\udcdd x) (fun _n => True) fun n => (fun y => x + y) '' \u2191(I ^ n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Game/PGame.lean", "full_name": "SetTheory.PGame.lf_def", "start": [638, 1], "end": [645, 33], "traced_tactics": [{"tactic": "rw [lf_iff_exists_le]", "annotated_tactic": ["rw [<a>lf_iff_exists_le</a>]", [{"full_name": "SetTheory.PGame.lf_iff_exists_le", "def_path": "Mathlib/SetTheory/Game/PGame.lean", "def_pos": [455, 9], "def_end_pos": [455, 25]}]], "state_before": "x y : PGame\n\u22a2 x \u29cf y \u2194\n    (\u2203 i,\n        (\u2200 (i' : LeftMoves x), moveLeft x i' \u29cf moveLeft y i) \u2227\n          \u2200 (j : RightMoves (moveLeft y i)), x \u29cf moveRight (moveLeft y i) j) \u2228\n      \u2203 j,\n        (\u2200 (i : LeftMoves (moveRight x j)), moveLeft (moveRight x j) i \u29cf y) \u2227\n          \u2200 (j' : RightMoves y), moveRight x j \u29cf moveRight y j'", "state_after": "x y : PGame\n\u22a2 ((\u2203 i, x \u2264 moveLeft y i) \u2228 \u2203 j, moveRight x j \u2264 y) \u2194\n    (\u2203 i,\n        (\u2200 (i' : LeftMoves x), moveLeft x i' \u29cf moveLeft y i) \u2227\n          \u2200 (j : RightMoves (moveLeft y i)), x \u29cf moveRight (moveLeft y i) j) \u2228\n      \u2203 j,\n        (\u2200 (i : LeftMoves (moveRight x j)), moveLeft (moveRight x j) i \u29cf y) \u2227\n          \u2200 (j' : RightMoves y), moveRight x j \u29cf moveRight y j'"}, {"tactic": "conv =>\n  lhs\n  simp only [le_iff_forall_lf]", "annotated_tactic": ["conv =>\n    lhs\n    simp only [<a>le_iff_forall_lf</a>]", [{"full_name": "SetTheory.PGame.le_iff_forall_lf", "def_path": "Mathlib/SetTheory/Game/PGame.lean", "def_pos": [431, 9], "def_end_pos": [431, 25]}]], "state_before": "x y : PGame\n\u22a2 ((\u2203 i, x \u2264 moveLeft y i) \u2228 \u2203 j, moveRight x j \u2264 y) \u2194\n    (\u2203 i,\n        (\u2200 (i' : LeftMoves x), moveLeft x i' \u29cf moveLeft y i) \u2227\n          \u2200 (j : RightMoves (moveLeft y i)), x \u29cf moveRight (moveLeft y i) j) \u2228\n      \u2203 j,\n        (\u2200 (i : LeftMoves (moveRight x j)), moveLeft (moveRight x j) i \u29cf y) \u2227\n          \u2200 (j' : RightMoves y), moveRight x j \u29cf moveRight y j'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "full_name": "ENNReal.lintegral_mul_le_Lp_mul_Lq", "start": [159, 1], "end": [175, 101], "traced_tactics": [{"tactic": "by_cases hf_zero : \u222b\u207b a, f a ^ p \u2202\u03bc = 0", "annotated_tactic": ["by_cases hf_zero : \u222b\u207b a, f a ^ p \u2202\u03bc = 0", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)"}, {"tactic": "by_cases hg_zero : \u222b\u207b a, g a ^ q \u2202\u03bc = 0", "annotated_tactic": ["by_cases hg_zero : \u222b\u207b a, g a ^ q \u2202\u03bc = 0", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)"}, {"tactic": "by_cases hf_top : \u222b\u207b a, f a ^ p \u2202\u03bc = \u22a4", "annotated_tactic": ["by_cases hf_top : \u222b\u207b a, f a ^ p \u2202\u03bc = \u22a4", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)"}, {"tactic": "by_cases hg_top : \u222b\u207b a, g a ^ q \u2202\u03bc = \u22a4", "annotated_tactic": ["by_cases hg_top : \u222b\u207b a, g a ^ q \u2202\u03bc = \u22a4", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)"}, {"tactic": "exact ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_ne_top hpq hf hf_top hg_top hf_zero hg_zero", "annotated_tactic": ["exact <a>ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_ne_top</a> hpq hf hf_top hg_top hf_zero hg_zero", [{"full_name": "ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_ne_top", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [105, 9], "def_end_pos": [105, 56]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "state_after": "no goals"}, {"tactic": "refine' Eq.trans_le _ (zero_le _)", "annotated_tactic": ["refine' <a>Eq.trans_le</a> _ (<a>zero_le</a> _)", [{"full_name": "Eq.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [219, 7], "def_end_pos": [219, 18]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = 0"}, {"tactic": "exact lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero hpq.nonneg hf hf_zero", "annotated_tactic": ["exact <a>lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero</a> hpq.nonneg hf hf_zero", [{"full_name": "ENNReal.lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [137, 9], "def_end_pos": [137, 56]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = 0", "state_after": "no goals"}, {"tactic": "refine' Eq.trans_le _ (zero_le _)", "annotated_tactic": ["refine' <a>Eq.trans_le</a> _ (<a>zero_le</a> _)", [{"full_name": "Eq.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [219, 7], "def_end_pos": [219, 18]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [217, 30], "def_end_pos": [217, 37]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = 0"}, {"tactic": "rw [mul_comm]", "annotated_tactic": ["rw [<a>mul_comm</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc = 0", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (g * f) a \u2202\u03bc = 0"}, {"tactic": "exact lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero hpq.symm.nonneg hg hg_zero", "annotated_tactic": ["exact <a>lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero</a> hpq.symm.nonneg hg hg_zero", [{"full_name": "ENNReal.lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [137, 9], "def_end_pos": [137, 56]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\n\u22a2 \u222b\u207b (a : \u03b1), (g * f) a \u2202\u03bc = 0", "state_after": "no goals"}, {"tactic": "exact lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top hpq.pos hpq.symm.nonneg hf_top hg_zero", "annotated_tactic": ["exact <a>lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top</a> hpq.pos hpq.symm.nonneg hf_top hg_zero", [{"full_name": "ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [147, 9], "def_end_pos": [147, 56]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "state_after": "no goals"}, {"tactic": "rw [mul_comm, mul_comm ((\u222b\u207b a : \u03b1, f a ^ p \u2202\u03bc) ^ (1 / p))]", "annotated_tactic": ["rw [<a>mul_comm</a>, <a>mul_comm</a> ((\u222b\u207b a : \u03b1, f a ^ p \u2202\u03bc) ^ (1 / p))]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (f * g) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p) * (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q)", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (g * f) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p)"}, {"tactic": "exact lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top hpq.symm.pos hpq.nonneg hg_top hf_zero", "annotated_tactic": ["exact <a>lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top</a> hpq.symm.pos hpq.nonneg hg_top hf_zero", [{"full_name": "ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [147, 9], "def_end_pos": [147, 56]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc : Measure \u03b1\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhg : AEMeasurable g\nhf_zero : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = 0\nhg_zero : \u00ac\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = 0\nhf_top : \u00ac\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc = \u22a4\nhg_top : \u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (a : \u03b1), (g * f) a \u2202\u03bc \u2264 (\u222b\u207b (a : \u03b1), g a ^ q \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc) ^ (1 / p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Semicontinuous.lean", "full_name": "IsClosed.upperSemicontinuousOn_indicator", "start": [760, 1], "end": [762, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Lie/Normalizer.lean", "full_name": "LieSubalgebra.mem_normalizer_iff", "start": [141, 1], "end": [144, 40], "traced_tactics": [{"tactic": "rw [mem_normalizer_iff']", "annotated_tactic": ["rw [<a>mem_normalizer_iff'</a>]", [{"full_name": "LieSubalgebra.mem_normalizer_iff'", "def_path": "Mathlib/Algebra/Lie/Normalizer.lean", "def_pos": [137, 9], "def_end_pos": [137, 28]}]], "state_before": "R : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nx : L\n\u22a2 x \u2208 normalizer H \u2194 \u2200 (y : L), y \u2208 H \u2192 \u2045x, y\u2046 \u2208 H", "state_after": "R : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nx : L\n\u22a2 (\u2200 (y : L), y \u2208 H \u2192 \u2045y, x\u2046 \u2208 H) \u2194 \u2200 (y : L), y \u2208 H \u2192 \u2045x, y\u2046 \u2208 H"}, {"tactic": "refine' forall\u2082_congr fun y hy => _", "annotated_tactic": ["refine' <a>forall\u2082_congr</a> fun y hy => _", [{"full_name": "forall\u2082_congr", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [383, 9], "def_end_pos": [383, 22]}]], "state_before": "R : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nx : L\n\u22a2 (\u2200 (y : L), y \u2208 H \u2192 \u2045y, x\u2046 \u2208 H) \u2194 \u2200 (y : L), y \u2208 H \u2192 \u2045x, y\u2046 \u2208 H", "state_after": "R : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nx y : L\nhy : y \u2208 H\n\u22a2 \u2045y, x\u2046 \u2208 H \u2194 \u2045x, y\u2046 \u2208 H"}, {"tactic": "rw [\u2190 lie_skew, neg_mem_iff (G := L)]", "annotated_tactic": ["rw [\u2190 <a>lie_skew</a>, <a>neg_mem_iff</a> (G := L)]", [{"full_name": "lie_skew", "def_path": "Mathlib/Algebra/Lie/Basic.lean", "def_pos": [156, 9], "def_end_pos": [156, 17]}, {"full_name": "neg_mem_iff", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [129, 3], "def_end_pos": [129, 14]}]], "state_before": "R : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nH : LieSubalgebra R L\nx y : L\nhy : y \u2208 H\n\u22a2 \u2045y, x\u2046 \u2208 H \u2194 \u2045x, y\u2046 \u2208 H", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "full_name": "Real.Angle.sign_coe_nonneg_of_nonneg_of_le_pi", "start": [976, 1], "end": [979, 45], "traced_tactics": [{"tactic": "rw [sign, sign_nonneg_iff]", "annotated_tactic": ["rw [<a>sign</a>, <a>sign_nonneg_iff</a>]", [{"full_name": "Real.Angle.sign", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [849, 5], "def_end_pos": [849, 9]}, {"full_name": "sign_nonneg_iff", "def_path": "Mathlib/Data/Sign.lean", "def_pos": [360, 9], "def_end_pos": [360, 24]}]], "state_before": "\u03b8 : \u211d\nh0 : 0 \u2264 \u03b8\nhpi : \u03b8 \u2264 \u03c0\n\u22a2 0 \u2264 sign \u2191\u03b8", "state_after": "\u03b8 : \u211d\nh0 : 0 \u2264 \u03b8\nhpi : \u03b8 \u2264 \u03c0\n\u22a2 0 \u2264 sin \u2191\u03b8"}, {"tactic": "exact sin_nonneg_of_nonneg_of_le_pi h0 hpi", "annotated_tactic": ["exact <a>sin_nonneg_of_nonneg_of_le_pi</a> h0 hpi", [{"full_name": "Real.sin_nonneg_of_nonneg_of_le_pi", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "def_pos": [433, 9], "def_end_pos": [433, 38]}]], "state_before": "\u03b8 : \u211d\nh0 : 0 \u2264 \u03b8\nhpi : \u03b8 \u2264 \u03c0\n\u22a2 0 \u2264 sin \u2191\u03b8", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/Colex.lean", "full_name": "Colex.lt_singleton_iff_mem_lt", "start": [271, 1], "end": [283, 30], "traced_tactics": [{"tactic": "simp only [lt_def, mem_singleton, \u2190 and_assoc, exists_eq_right]", "annotated_tactic": ["simp only [<a>lt_def</a>, <a>mem_singleton</a>, \u2190 <a>and_assoc</a>, <a>exists_eq_right</a>]", [{"full_name": "Colex.lt_def", "def_path": "Mathlib/Combinatorics/Colex.lean", "def_pos": [91, 9], "def_end_pos": [91, 21]}, {"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 22]}, {"full_name": "and_assoc", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [177, 9], "def_end_pos": [177, 18]}, {"full_name": "exists_eq_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [462, 17], "def_end_pos": [462, 32]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\n\u22a2 toColex s < toColex {r} \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x < r", "state_after": "\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\n\u22a2 (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x < r"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\n\u22a2 (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x < r", "state_after": "case mp\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\n\u22a2 (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s \u2192 \u2200 (x : \u03b1), x \u2208 s \u2192 x < r\n\ncase mpr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\n\u22a2 (\u2200 (x : \u03b1), x \u2208 s \u2192 x < r) \u2192 (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s"}, {"tactic": "intro t x hx", "annotated_tactic": ["intro t x hx", []], "state_before": "case mp\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\n\u22a2 (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s \u2192 \u2200 (x : \u03b1), x \u2208 s \u2192 x < r", "state_after": "case mp\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\nt : (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x < r"}, {"tactic": "rw [\u2190 not_le]", "annotated_tactic": ["rw [\u2190 <a>not_le</a>]", [{"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "case mp\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\nt : (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x < r", "state_after": "case mp\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\nt : (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u00acr \u2264 x"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mp\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\nt : (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u00acr \u2264 x", "state_after": "case mp\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\nt : (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s\nx : \u03b1\nhx : x \u2208 s\nh : r \u2264 x\n\u22a2 False"}, {"tactic": "rcases lt_or_eq_of_le h with (h\u2081 | rfl)", "annotated_tactic": ["rcases <a>lt_or_eq_of_le</a> h with (h\u2081 | rfl)", [{"full_name": "lt_or_eq_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [228, 9], "def_end_pos": [228, 23]}]], "state_before": "case mp\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\nt : (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s\nx : \u03b1\nhx : x \u2208 s\nh : r \u2264 x\n\u22a2 False", "state_after": "case mp.inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\nt : (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s\nx : \u03b1\nhx : x \u2208 s\nh : r \u2264 x\nh\u2081 : r < x\n\u22a2 False\n\ncase mp.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\nt : (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s\nhx : r \u2208 s\nh : r \u2264 r\n\u22a2 False"}, {"tactic": "exact ne_of_irrefl h\u2081 ((t.1 h\u2081).1 hx).symm", "annotated_tactic": ["exact <a>ne_of_irrefl</a> h\u2081 ((t.1 h\u2081).1 hx).<a>symm</a>", [{"full_name": "ne_of_irrefl", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [133, 9], "def_end_pos": [133, 21]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case mp.inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\nt : (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s\nx : \u03b1\nhx : x \u2208 s\nh : r \u2264 x\nh\u2081 : r < x\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exact t.2 hx", "annotated_tactic": ["exact t.2 hx", []], "state_before": "case mp.inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\nt : (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s\nhx : r \u2208 s\nh : r \u2264 r\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exact fun h =>\n  \u27e8fun {z} hz => \u27e8fun i => (asymm hz (h _ i)).elim, fun i => (hz.ne' i).elim\u27e9,\n      by simpa using h r\u27e9", "annotated_tactic": ["exact fun h =>\n      \u27e8fun {z} hz => \u27e8fun i => (<a>asymm</a> hz (h _ i)).<a>elim</a>, fun i => (hz.ne' i).<a>elim</a>\u27e9,\n          by simpa using h r\u27e9", [{"full_name": "asymm", "def_path": "Mathlib/Init/Algebra/Classes.lean", "def_pos": [320, 9], "def_end_pos": [320, 14]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case mpr\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\n\u22a2 (\u2200 (x : \u03b1), x \u2208 s \u2192 x < r) \u2192 (\u2200 {x : \u03b1}, r < x \u2192 (x \u2208 s \u2194 x = r)) \u2227 \u00acr \u2208 s", "state_after": "no goals"}, {"tactic": "simpa using h r", "annotated_tactic": ["simpa using h r", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nr : \u03b1\ns : Finset \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 x < r\n\u22a2 \u00acr \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Monotone/Basic.lean", "full_name": "Subsingleton.strictMono", "start": [533, 11], "end": [534, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "full_name": "SimpleGraph.Walk.fst_mem_support_of_mem_edges", "start": [795, 1], "end": [801, 49], "traced_tactics": [{"tactic": "obtain \u27e8d, hd, he\u27e9 := List.mem_map.mp he", "annotated_tactic": ["obtain \u27e8d, hd, he\u27e9 := List.mem_map.mp he", []], "state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nt u v w : V\np : Walk G v w\nhe : Quotient.mk (Sym2.Rel.setoid V) (t, u) \u2208 edges p\n\u22a2 t \u2208 support p", "state_after": "case intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nt u v w : V\np : Walk G v w\nhe\u271d : Quotient.mk (Sym2.Rel.setoid V) (t, u) \u2208 edges p\nd : Dart G\nhd : d \u2208 darts p\nhe : Dart.edge d = Quotient.mk (Sym2.Rel.setoid V) (t, u)\n\u22a2 t \u2208 support p"}, {"tactic": "rw [dart_edge_eq_mk'_iff'] at he", "annotated_tactic": ["rw [<a>dart_edge_eq_mk'_iff'</a>] at he", [{"full_name": "SimpleGraph.dart_edge_eq_mk'_iff'", "def_path": "Mathlib/Combinatorics/SimpleGraph/Basic.lean", "def_pos": [803, 9], "def_end_pos": [803, 30]}]], "state_before": "case intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nt u v w : V\np : Walk G v w\nhe\u271d : Quotient.mk (Sym2.Rel.setoid V) (t, u) \u2208 edges p\nd : Dart G\nhd : d \u2208 darts p\nhe : Dart.edge d = Quotient.mk (Sym2.Rel.setoid V) (t, u)\n\u22a2 t \u2208 support p", "state_after": "case intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nt u v w : V\np : Walk G v w\nhe\u271d : Quotient.mk (Sym2.Rel.setoid V) (t, u) \u2208 edges p\nd : Dart G\nhd : d \u2208 darts p\nhe : d.toProd.1 = t \u2227 d.toProd.2 = u \u2228 d.toProd.1 = u \u2227 d.toProd.2 = t\n\u22a2 t \u2208 support p"}, {"tactic": "rcases he with (\u27e8rfl, rfl\u27e9 | \u27e8rfl, rfl\u27e9)", "annotated_tactic": ["rcases he with (\u27e8rfl, rfl\u27e9 | \u27e8rfl, rfl\u27e9)", []], "state_before": "case intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nt u v w : V\np : Walk G v w\nhe\u271d : Quotient.mk (Sym2.Rel.setoid V) (t, u) \u2208 edges p\nd : Dart G\nhd : d \u2208 darts p\nhe : d.toProd.1 = t \u2227 d.toProd.2 = u \u2228 d.toProd.1 = u \u2227 d.toProd.2 = t\n\u22a2 t \u2208 support p", "state_after": "case intro.intro.inl.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\np : Walk G v w\nd : Dart G\nhd : d \u2208 darts p\nhe : Quotient.mk (Sym2.Rel.setoid V) (d.toProd.1, d.toProd.2) \u2208 edges p\n\u22a2 d.toProd.1 \u2208 support p\n\ncase intro.intro.inr.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\np : Walk G v w\nd : Dart G\nhd : d \u2208 darts p\nhe : Quotient.mk (Sym2.Rel.setoid V) (d.toProd.2, d.toProd.1) \u2208 edges p\n\u22a2 d.toProd.2 \u2208 support p"}, {"tactic": "exact dart_fst_mem_support_of_mem_darts _ hd", "annotated_tactic": ["exact <a>dart_fst_mem_support_of_mem_darts</a> _ hd", [{"full_name": "SimpleGraph.Walk.dart_fst_mem_support_of_mem_darts", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [781, 9], "def_end_pos": [781, 42]}]], "state_before": "case intro.intro.inl.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\np : Walk G v w\nd : Dart G\nhd : d \u2208 darts p\nhe : Quotient.mk (Sym2.Rel.setoid V) (d.toProd.1, d.toProd.2) \u2208 edges p\n\u22a2 d.toProd.1 \u2208 support p", "state_after": "no goals"}, {"tactic": "exact dart_snd_mem_support_of_mem_darts _ hd", "annotated_tactic": ["exact <a>dart_snd_mem_support_of_mem_darts</a> _ hd", [{"full_name": "SimpleGraph.Walk.dart_snd_mem_support_of_mem_darts", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [790, 9], "def_end_pos": [790, 42]}]], "state_before": "case intro.intro.inr.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\np : Walk G v w\nd : Dart G\nhd : d \u2208 darts p\nhe : Quotient.mk (Sym2.Rel.setoid V) (d.toProd.2, d.toProd.1) \u2208 edges p\n\u22a2 d.toProd.2 \u2208 support p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/DirectSum/Basic.lean", "full_name": "DirectSum.induction_on", "start": [166, 11], "end": [171, 16], "traced_tactics": [{"tactic": "apply DFinsupp.induction x H_zero", "annotated_tactic": ["apply <a>DFinsupp.induction</a> x H_zero", [{"full_name": "DFinsupp.induction", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [964, 19], "def_end_pos": [964, 28]}]], "state_before": "\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\nC : (\u2a01 (i : \u03b9), \u03b2 i) \u2192 Prop\nx : \u2a01 (i : \u03b9), \u03b2 i\nH_zero : C 0\nH_basic : \u2200 (i : \u03b9) (x : \u03b2 i), C (\u2191(of \u03b2 i) x)\nH_plus : \u2200 (x y : \u2a01 (i : \u03b9), \u03b2 i), C x \u2192 C y \u2192 C (x + y)\n\u22a2 C x", "state_after": "\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\nC : (\u2a01 (i : \u03b9), \u03b2 i) \u2192 Prop\nx : \u2a01 (i : \u03b9), \u03b2 i\nH_zero : C 0\nH_basic : \u2200 (i : \u03b9) (x : \u03b2 i), C (\u2191(of \u03b2 i) x)\nH_plus : \u2200 (x y : \u2a01 (i : \u03b9), \u03b2 i), C x \u2192 C y \u2192 C (x + y)\n\u22a2 \u2200 (i : \u03b9) (b : (fun i => \u03b2 i) i) (f : \u03a0\u2080 (i : \u03b9), (fun i => \u03b2 i) i),\n    \u2191f i = 0 \u2192 b \u2260 0 \u2192 C f \u2192 C (DFinsupp.single i b + f)"}, {"tactic": "intro i b f h1 h2 ih", "annotated_tactic": ["intro i b f h1 h2 ih", []], "state_before": "\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\nC : (\u2a01 (i : \u03b9), \u03b2 i) \u2192 Prop\nx : \u2a01 (i : \u03b9), \u03b2 i\nH_zero : C 0\nH_basic : \u2200 (i : \u03b9) (x : \u03b2 i), C (\u2191(of \u03b2 i) x)\nH_plus : \u2200 (x y : \u2a01 (i : \u03b9), \u03b2 i), C x \u2192 C y \u2192 C (x + y)\n\u22a2 \u2200 (i : \u03b9) (b : (fun i => \u03b2 i) i) (f : \u03a0\u2080 (i : \u03b9), (fun i => \u03b2 i) i),\n    \u2191f i = 0 \u2192 b \u2260 0 \u2192 C f \u2192 C (DFinsupp.single i b + f)", "state_after": "\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\nC : (\u2a01 (i : \u03b9), \u03b2 i) \u2192 Prop\nx : \u2a01 (i : \u03b9), \u03b2 i\nH_zero : C 0\nH_basic : \u2200 (i : \u03b9) (x : \u03b2 i), C (\u2191(of \u03b2 i) x)\nH_plus : \u2200 (x y : \u2a01 (i : \u03b9), \u03b2 i), C x \u2192 C y \u2192 C (x + y)\ni : \u03b9\nb : \u03b2 i\nf : \u03a0\u2080 (i : \u03b9), (fun i => \u03b2 i) i\nh1 : \u2191f i = 0\nh2 : b \u2260 0\nih : C f\n\u22a2 C (DFinsupp.single i b + f)"}, {"tactic": "solve_by_elim", "annotated_tactic": ["solve_by_elim", []], "state_before": "\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type w\ninst\u271d : (i : \u03b9) \u2192 AddCommMonoid (\u03b2 i)\nC : (\u2a01 (i : \u03b9), \u03b2 i) \u2192 Prop\nx : \u2a01 (i : \u03b9), \u03b2 i\nH_zero : C 0\nH_basic : \u2200 (i : \u03b9) (x : \u03b2 i), C (\u2191(of \u03b2 i) x)\nH_plus : \u2200 (x y : \u2a01 (i : \u03b9), \u03b2 i), C x \u2192 C y \u2192 C (x + y)\ni : \u03b9\nb : \u03b2 i\nf : \u03a0\u2080 (i : \u03b9), (fun i => \u03b2 i) i\nh1 : \u2191f i = 0\nh2 : b \u2260 0\nih : C f\n\u22a2 C (DFinsupp.single i b + f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Variables.lean", "full_name": "MvPolynomial.vars_one", "start": [366, 1], "end": [367, 9], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Finrank.lean", "full_name": "Submodule.lt_of_le_of_finrank_lt_finrank", "start": [290, 1], "end": [292, 53], "traced_tactics": [{"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "K : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\ns t : Submodule K V\nle : s \u2264 t\nlt : finrank K { x // x \u2208 s } < finrank K { x // x \u2208 t }\nh : s = t\n\u22a2 finrank K { x // x \u2208 s } = finrank K { x // x \u2208 t }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean", "full_name": "CategoryTheory.Limits.Multicoequalizer.multicofork_\u03b9_app_right", "start": [853, 1], "end": [855, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Logic/Unique.lean", "full_name": "unique_subtype_iff_exists_unique", "start": [71, 1], "end": [76, 23], "traced_tactics": [{"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Sort u_1\np : \u03b1 \u2192 Prop\nx\u271d\u00b9 : \u2203! a, p a\na : \u03b1\nha : (fun a => p a) a\nhe : \u2200 (y : \u03b1), (fun a => p a) y \u2192 y = a\nx\u271d : Subtype p\nb : \u03b1\nhb : p b\n\u22a2 { val := b, property := hb } = default", "state_after": "case e_val\n\u03b1 : Sort u_1\np : \u03b1 \u2192 Prop\nx\u271d\u00b9 : \u2203! a, p a\na : \u03b1\nha : (fun a => p a) a\nhe : \u2200 (y : \u03b1), (fun a => p a) y \u2192 y = a\nx\u271d : Subtype p\nb : \u03b1\nhb : p b\n\u22a2 b = a"}, {"tactic": "exact he b hb", "annotated_tactic": ["exact he b hb", []], "state_before": "case e_val\n\u03b1 : Sort u_1\np : \u03b1 \u2192 Prop\nx\u271d\u00b9 : \u2203! a, p a\na : \u03b1\nha : (fun a => p a) a\nhe : \u2200 (y : \u03b1), (fun a => p a) y \u2192 y = a\nx\u271d : Subtype p\nb : \u03b1\nhb : p b\n\u22a2 b = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/Forall2.lean", "full_name": "List.rel_reverse", "start": [253, 1], "end": [257, 68], "traced_tactics": [{"tactic": "simp only [reverse_cons]", "annotated_tactic": ["simp only [<a>reverse_cons</a>]", [{"full_name": "List.reverse_cons", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [174, 17], "def_end_pos": [174, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nb\u271d : \u03b2\nl\u2081\u271d : List \u03b1\nl\u2082\u271d : List \u03b2\nh\u2081 : R a\u271d b\u271d\nh\u2082 : Forall\u2082 R l\u2081\u271d l\u2082\u271d\n\u22a2 Forall\u2082 R (reverse (a\u271d :: l\u2081\u271d)) (reverse (b\u271d :: l\u2082\u271d))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nb\u271d : \u03b2\nl\u2081\u271d : List \u03b1\nl\u2082\u271d : List \u03b2\nh\u2081 : R a\u271d b\u271d\nh\u2082 : Forall\u2082 R l\u2081\u271d l\u2082\u271d\n\u22a2 Forall\u2082 R (reverse l\u2081\u271d ++ [a\u271d]) (reverse l\u2082\u271d ++ [b\u271d])"}, {"tactic": "exact rel_append (rel_reverse h\u2082) (Forall\u2082.cons h\u2081 Forall\u2082.nil)", "annotated_tactic": ["exact <a>rel_append</a> (rel_reverse h\u2082) (<a>Forall\u2082.cons</a> h\u2081 <a>Forall\u2082.nil</a>)", [{"full_name": "List.rel_append", "def_path": "Mathlib/Data/List/Forall2.lean", "def_pos": [248, 9], "def_end_pos": [248, 19]}, {"full_name": "List.Forall\u2082.cons", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [910, 5], "def_end_pos": [910, 9]}, {"full_name": "List.Forall\u2082.nil", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [907, 5], "def_end_pos": [907, 8]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nb\u271d : \u03b2\nl\u2081\u271d : List \u03b1\nl\u2082\u271d : List \u03b2\nh\u2081 : R a\u271d b\u271d\nh\u2082 : Forall\u2082 R l\u2081\u271d l\u2082\u271d\n\u22a2 Forall\u2082 R (reverse l\u2081\u271d ++ [a\u271d]) (reverse l\u2082\u271d ++ [b\u271d])", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "full_name": "Real.rpow_zero_pos", "start": [100, 1], "end": [100, 59], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "x : \u211d\n\u22a2 0 < x ^ 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "full_name": "mul_eq_zero_comm", "start": [258, 1], "end": [259, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Variance.lean", "full_name": "ProbabilityTheory.evariance_mul", "start": [162, 1], "end": [174, 72], "traced_tactics": [{"tactic": "rw [evariance, evariance, \u2190 lintegral_const_mul' _ _ ENNReal.ofReal_lt_top.ne]", "annotated_tactic": ["rw [<a>evariance</a>, <a>evariance</a>, \u2190 <a>lintegral_const_mul'</a> _ _ ENNReal.ofReal_lt_top.ne]", [{"full_name": "ProbabilityTheory.evariance", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [55, 5], "def_end_pos": [55, 14]}, {"full_name": "ProbabilityTheory.evariance", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [55, 5], "def_end_pos": [55, 14]}, {"full_name": "MeasureTheory.lintegral_const_mul'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [711, 9], "def_end_pos": [711, 29]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nc : \u211d\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u22a2 evariance (fun \u03c9 => c * X \u03c9) \u03bc = ENNReal.ofReal (c ^ 2) * evariance X \u03bc", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nc : \u211d\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016c * X \u03c9 - \u222b (x : \u03a9), c * X x \u2202\u03bc\u2016\u208a ^ 2 \u2202\u03bc =\n    \u222b\u207b (a : \u03a9), ENNReal.ofReal (c ^ 2) * \u2191\u2016X a - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2 \u2202\u03bc"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nc : \u211d\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191\u2016c * X \u03c9 - \u222b (x : \u03a9), c * X x \u2202\u03bc\u2016\u208a ^ 2 \u2202\u03bc =\n    \u222b\u207b (a : \u03a9), ENNReal.ofReal (c ^ 2) * \u2191\u2016X a - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2 \u2202\u03bc", "state_after": "case e_f\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nc : \u211d\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u22a2 (fun \u03c9 => \u2191\u2016c * X \u03c9 - \u222b (x : \u03a9), c * X x \u2202\u03bc\u2016\u208a ^ 2) = fun a => ENNReal.ofReal (c ^ 2) * \u2191\u2016X a - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2"}, {"tactic": "ext1 \u03c9", "annotated_tactic": ["ext1 \u03c9", []], "state_before": "case e_f\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nc : \u211d\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u22a2 (fun \u03c9 => \u2191\u2016c * X \u03c9 - \u222b (x : \u03a9), c * X x \u2202\u03bc\u2016\u208a ^ 2) = fun a => ENNReal.ofReal (c ^ 2) * \u2191\u2016X a - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2", "state_after": "case e_f.h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nc : \u211d\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u03c9 : \u03a9\n\u22a2 \u2191\u2016c * X \u03c9 - \u222b (x : \u03a9), c * X x \u2202\u03bc\u2016\u208a ^ 2 = ENNReal.ofReal (c ^ 2) * \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2"}, {"tactic": "rw [ENNReal.ofReal, \u2190 ENNReal.coe_pow, \u2190 ENNReal.coe_pow, \u2190 ENNReal.coe_mul]", "annotated_tactic": ["rw [<a>ENNReal.ofReal</a>, \u2190 <a>ENNReal.coe_pow</a>, \u2190 <a>ENNReal.coe_pow</a>, \u2190 <a>ENNReal.coe_mul</a>]", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.coe_pow", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [554, 9], "def_end_pos": [554, 16]}, {"full_name": "ENNReal.coe_pow", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [554, 9], "def_end_pos": [554, 16]}, {"full_name": "ENNReal.coe_mul", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [390, 9], "def_end_pos": [390, 16]}]], "state_before": "case e_f.h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nc : \u211d\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u03c9 : \u03a9\n\u22a2 \u2191\u2016c * X \u03c9 - \u222b (x : \u03a9), c * X x \u2202\u03bc\u2016\u208a ^ 2 = ENNReal.ofReal (c ^ 2) * \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2", "state_after": "case e_f.h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nc : \u211d\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u03c9 : \u03a9\n\u22a2 \u2191(\u2016c * X \u03c9 - \u222b (x : \u03a9), c * X x \u2202\u03bc\u2016\u208a ^ 2) = \u2191(Real.toNNReal (c ^ 2) * \u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case e_f.h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nc : \u211d\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u03c9 : \u03a9\n\u22a2 \u2191(\u2016c * X \u03c9 - \u222b (x : \u03a9), c * X x \u2202\u03bc\u2016\u208a ^ 2) = \u2191(Real.toNNReal (c ^ 2) * \u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2)", "state_after": "case e_f.h.e_a\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nc : \u211d\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u03c9 : \u03a9\n\u22a2 \u2016c * X \u03c9 - \u222b (x : \u03a9), c * X x \u2202\u03bc\u2016\u208a ^ 2 = Real.toNNReal (c ^ 2) * \u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2"}, {"tactic": "rw [\u2190 sq_abs, \u2190 Real.rpow_two, Real.toNNReal_rpow_of_nonneg (abs_nonneg _), NNReal.rpow_two,\n  \u2190 mul_pow, Real.toNNReal_mul_nnnorm _ (abs_nonneg _)]", "annotated_tactic": ["rw [\u2190 <a>sq_abs</a>, \u2190 <a>Real.rpow_two</a>, <a>Real.toNNReal_rpow_of_nonneg</a> (<a>abs_nonneg</a> _), <a>NNReal.rpow_two</a>,\n    \u2190 <a>mul_pow</a>, <a>Real.toNNReal_mul_nnnorm</a> _ (<a>abs_nonneg</a> _)]", [{"full_name": "sq_abs", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [680, 9], "def_end_pos": [680, 15]}, {"full_name": "Real.rpow_two", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [362, 9], "def_end_pos": [362, 17]}, {"full_name": "Real.toNNReal_rpow_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [333, 9], "def_end_pos": [333, 44]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}, {"full_name": "NNReal.rpow_two", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [132, 9], "def_end_pos": [132, 17]}, {"full_name": "mul_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 16]}, {"full_name": "Real.toNNReal_mul_nnnorm", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [809, 9], "def_end_pos": [809, 28]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}]], "state_before": "case e_f.h.e_a\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nc : \u211d\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u03c9 : \u03a9\n\u22a2 \u2016c * X \u03c9 - \u222b (x : \u03a9), c * X x \u2202\u03bc\u2016\u208a ^ 2 = Real.toNNReal (c ^ 2) * \u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2", "state_after": "case e_f.h.e_a\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nc : \u211d\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u03c9 : \u03a9\n\u22a2 \u2016c * X \u03c9 - \u222b (x : \u03a9), c * X x \u2202\u03bc\u2016\u208a ^ 2 = \u2016|c| * (X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc)\u2016\u208a ^ 2"}, {"tactic": "conv_rhs => rw [\u2190 nnnorm_norm, norm_mul, norm_abs_eq_norm, \u2190 norm_mul, nnnorm_norm, mul_sub]", "annotated_tactic": ["conv_rhs => rw [\u2190 <a>nnnorm_norm</a>, <a>norm_mul</a>, <a>norm_abs_eq_norm</a>, \u2190 <a>norm_mul</a>, <a>nnnorm_norm</a>, <a>mul_sub</a>]", [{"full_name": "nnnorm_norm", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [842, 9], "def_end_pos": [842, 20]}, {"full_name": "norm_mul", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}, {"full_name": "norm_abs_eq_norm", "def_path": "Mathlib/Analysis/Normed/Order/Lattice.lean", "def_pos": [114, 9], "def_end_pos": [114, 25]}, {"full_name": "norm_mul", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}, {"full_name": "nnnorm_norm", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [842, 9], "def_end_pos": [842, 20]}, {"full_name": "mul_sub", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [365, 7], "def_end_pos": [365, 14]}]], "state_before": "case e_f.h.e_a\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nc : \u211d\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u03c9 : \u03a9\n\u22a2 \u2016c * X \u03c9 - \u222b (x : \u03a9), c * X x \u2202\u03bc\u2016\u208a ^ 2 = \u2016|c| * (X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc)\u2016\u208a ^ 2", "state_after": "case e_f.h.e_a\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nc : \u211d\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u03c9 : \u03a9\n\u22a2 \u2016c * X \u03c9 - \u222b (x : \u03a9), c * X x \u2202\u03bc\u2016\u208a ^ 2 = \u2016c * X \u03c9 - c * \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case e_f.h.e_a\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nc : \u211d\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u03c9 : \u03a9\n\u22a2 \u2016c * X \u03c9 - \u222b (x : \u03a9), c * X x \u2202\u03bc\u2016\u208a ^ 2 = \u2016c * X \u03c9 - c * \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2", "state_after": "case e_f.h.e_a.e_a.e_a.e_a\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nc : \u211d\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u03c9 : \u03a9\n\u22a2 \u222b (x : \u03a9), c * X x \u2202\u03bc = c * \u222b (x : \u03a9), X x \u2202\u03bc"}, {"tactic": "rw [mul_comm]", "annotated_tactic": ["rw [<a>mul_comm</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "case e_f.h.e_a.e_a.e_a.e_a\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nc : \u211d\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u03c9 : \u03a9\n\u22a2 \u222b (x : \u03a9), c * X x \u2202\u03bc = c * \u222b (x : \u03a9), X x \u2202\u03bc", "state_after": "case e_f.h.e_a.e_a.e_a.e_a\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nc : \u211d\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u03c9 : \u03a9\n\u22a2 \u222b (x : \u03a9), c * X x \u2202\u03bc = (\u222b (x : \u03a9), X x \u2202\u03bc) * c"}, {"tactic": "simp_rw [\u2190 smul_eq_mul, \u2190 integral_smul_const, smul_eq_mul, mul_comm]", "annotated_tactic": ["simp_rw [\u2190 <a>smul_eq_mul</a>, \u2190 <a>integral_smul_const</a>, <a>smul_eq_mul</a>, <a>mul_comm</a>]", [{"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "integral_smul_const", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [1257, 9], "def_end_pos": [1257, 28]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "case e_f.h.e_a.e_a.e_a.e_a\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX\u271d : \u03a9 \u2192 \u211d\n\u03bc\u271d : Measure \u03a9\nc : \u211d\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\n\u03c9 : \u03a9\n\u22a2 \u222b (x : \u03a9), c * X x \u2202\u03bc = (\u222b (x : \u03a9), X x \u2202\u03bc) * c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/Prime.lean", "full_name": "Nat.Prime.mod_two_eq_one_iff_ne_two", "start": [514, 1], "end": [517, 12], "traced_tactics": [{"tactic": "refine' \u27e8fun h hf => _, (Nat.Prime.eq_two_or_odd <| @Fact.out p.Prime _).resolve_left\u27e9", "annotated_tactic": ["refine' \u27e8fun h hf => _, (<a>Nat.Prime.eq_two_or_odd</a> <| @<a>Fact.out</a> p.Prime _).<a>resolve_left</a>\u27e9", [{"full_name": "Nat.Prime.eq_two_or_odd", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [492, 9], "def_end_pos": [492, 28]}, {"full_name": "Fact.out", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [118, 3], "def_end_pos": [118, 6]}, {"full_name": "Or.resolve_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [268, 9], "def_end_pos": [268, 24]}]], "state_before": "p : \u2115\ninst\u271d : Fact (Prime p)\n\u22a2 p % 2 = 1 \u2194 p \u2260 2", "state_after": "p : \u2115\ninst\u271d : Fact (Prime p)\nh : p % 2 = 1\nhf : p = 2\n\u22a2 False"}, {"tactic": "rw [hf] at h", "annotated_tactic": ["rw [hf] at h", []], "state_before": "p : \u2115\ninst\u271d : Fact (Prime p)\nh : p % 2 = 1\nhf : p = 2\n\u22a2 False", "state_after": "p : \u2115\ninst\u271d : Fact (Prime p)\nh : 2 % 2 = 1\nhf : p = 2\n\u22a2 False"}, {"tactic": "simp at h", "annotated_tactic": ["simp at h", []], "state_before": "p : \u2115\ninst\u271d : Fact (Prime p)\nh : 2 % 2 = 1\nhf : p = 2\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Prod.lean", "full_name": "Set.offDiag_singleton", "start": [605, 1], "end": [605, 73], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\ns t : Set \u03b1\nx : \u03b1 \u00d7 \u03b1\na\u271d a : \u03b1\n\u22a2 offDiag {a} = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "full_name": "IsOpen.measure_eq_zero_iff", "start": [57, 1], "end": [59, 37], "traced_tactics": [{"tactic": "simpa only [not_lt, nonpos_iff_eq_zero, not_nonempty_iff_eq_empty] using\n  not_congr (hU.measure_pos_iff \u03bc)", "annotated_tactic": ["simpa only [<a>not_lt</a>, <a>nonpos_iff_eq_zero</a>, <a>not_nonempty_iff_eq_empty</a>] using\n    <a>not_congr</a> (hU.measure_pos_iff \u03bc)", [{"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}, {"full_name": "nonpos_iff_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [237, 3], "def_end_pos": [237, 14]}, {"full_name": "Set.not_nonempty_iff_eq_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [605, 9], "def_end_pos": [605, 34]}, {"full_name": "not_congr", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [22, 9], "def_end_pos": [22, 18]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\nm : MeasurableSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : T2Space Y\n\u03bc \u03bd : Measure X\ninst\u271d : IsOpenPosMeasure \u03bc\ns U F : Set X\nx : X\nhU : IsOpen U\n\u22a2 \u2191\u2191\u03bc U = 0 \u2194 U = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/MonoidAlgebra/Support.lean", "full_name": "MonoidAlgebra.support_mul_single_subset", "start": [39, 1], "end": [52, 53], "traced_tactics": [{"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "k : Type u\u2081\nG : Type u\u2082\ninst\u271d\u00b2 : Semiring k\ninst\u271d\u00b9 : DecidableEq G\ninst\u271d : Mul G\nf : MonoidAlgebra k G\nr : k\na : G\n\u22a2 (f * single a r).support \u2286 image (fun x => x * a) f.support", "state_after": "k : Type u\u2081\nG : Type u\u2082\ninst\u271d\u00b2 : Semiring k\ninst\u271d\u00b9 : DecidableEq G\ninst\u271d : Mul G\nf : MonoidAlgebra k G\nr : k\na x : G\nhx : x \u2208 (f * single a r).support\n\u22a2 x \u2208 image (fun x => x * a) f.support"}, {"tactic": "contrapose hx", "annotated_tactic": ["contrapose hx", []], "state_before": "k : Type u\u2081\nG : Type u\u2082\ninst\u271d\u00b2 : Semiring k\ninst\u271d\u00b9 : DecidableEq G\ninst\u271d : Mul G\nf : MonoidAlgebra k G\nr : k\na x : G\nhx : x \u2208 (f * single a r).support\n\u22a2 x \u2208 image (fun x => x * a) f.support", "state_after": "k : Type u\u2081\nG : Type u\u2082\ninst\u271d\u00b2 : Semiring k\ninst\u271d\u00b9 : DecidableEq G\ninst\u271d : Mul G\nf : MonoidAlgebra k G\nr : k\na x : G\nhx : \u00acx \u2208 image (fun x => x * a) f.support\n\u22a2 \u00acx \u2208 (f * single a r).support"}, {"tactic": "have : \u2200 y, y * a = x \u2192 f y = 0 := by\n  simpa only [not_and', mem_image, mem_support_iff, exists_prop, not_exists,\n    Classical.not_not] using hx", "annotated_tactic": ["have : \u2200 y, y * a = x \u2192 f y = 0 := by\n    simpa only [<a>not_and'</a>, <a>mem_image</a>, <a>mem_support_iff</a>, <a>exists_prop</a>, <a>not_exists</a>,\n      <a>Classical.not_not</a>] using hx", [{"full_name": "not_and'", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [318, 9], "def_end_pos": [318, 17]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "Finsupp.mem_support_iff", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [186, 9], "def_end_pos": [186, 24]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"full_name": "Classical.not_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [683, 24], "def_end_pos": [683, 31]}]], "state_before": "k : Type u\u2081\nG : Type u\u2082\ninst\u271d\u00b2 : Semiring k\ninst\u271d\u00b9 : DecidableEq G\ninst\u271d : Mul G\nf : MonoidAlgebra k G\nr : k\na x : G\nhx : \u00acx \u2208 image (fun x => x * a) f.support\n\u22a2 \u00acx \u2208 (f * single a r).support", "state_after": "k : Type u\u2081\nG : Type u\u2082\ninst\u271d\u00b2 : Semiring k\ninst\u271d\u00b9 : DecidableEq G\ninst\u271d : Mul G\nf : MonoidAlgebra k G\nr : k\na x : G\nhx : \u00acx \u2208 image (fun x => x * a) f.support\nthis : \u2200 (y : G), y * a = x \u2192 \u2191f y = 0\n\u22a2 \u00acx \u2208 (f * single a r).support"}, {"tactic": "simp only [mem_support_iff, mul_apply, sum_single_index, zero_mul, ite_self, sum_zero,\n  Classical.not_not]", "annotated_tactic": ["simp only [<a>mem_support_iff</a>, <a>mul_apply</a>, <a>sum_single_index</a>, <a>zero_mul</a>, <a>ite_self</a>, <a>sum_zero</a>,\n    <a>Classical.not_not</a>]", [{"full_name": "Finsupp.mem_support_iff", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [186, 9], "def_end_pos": [186, 24]}, {"full_name": "MonoidAlgebra.mul_apply", "def_path": "Mathlib/Algebra/MonoidAlgebra/Basic.lean", "def_pos": [417, 9], "def_end_pos": [417, 18]}, {"full_name": "MonoidAlgebra.sum_single_index", "def_path": "Mathlib/Algebra/MonoidAlgebra/Basic.lean", "def_pos": [121, 9], "def_end_pos": [121, 25]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "ite_self", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [81, 17], "def_end_pos": [81, 25]}, {"full_name": "Finsupp.sum_zero", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [356, 9], "def_end_pos": [356, 17]}, {"full_name": "Classical.not_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [683, 24], "def_end_pos": [683, 31]}]], "state_before": "k : Type u\u2081\nG : Type u\u2082\ninst\u271d\u00b2 : Semiring k\ninst\u271d\u00b9 : DecidableEq G\ninst\u271d : Mul G\nf : MonoidAlgebra k G\nr : k\na x : G\nhx : \u00acx \u2208 image (fun x => x * a) f.support\nthis : \u2200 (y : G), y * a = x \u2192 \u2191f y = 0\n\u22a2 \u00acx \u2208 (f * single a r).support", "state_after": "k : Type u\u2081\nG : Type u\u2082\ninst\u271d\u00b2 : Semiring k\ninst\u271d\u00b9 : DecidableEq G\ninst\u271d : Mul G\nf : MonoidAlgebra k G\nr : k\na x : G\nhx : \u00acx \u2208 image (fun x => x * a) f.support\nthis : \u2200 (y : G), y * a = x \u2192 \u2191f y = 0\n\u22a2 (sum f fun a\u2081 b\u2081 => sum (single a r) fun a\u2082 b\u2082 => if a\u2081 * a\u2082 = x then b\u2081 * b\u2082 else 0) = 0"}, {"tactic": "exact\n  Finset.sum_eq_zero\n    (by\n      simp (config := { contextual := true }) only [this, sum_single_index, ite_eq_right_iff,\n        eq_self_iff_true, imp_true_iff, zero_mul])", "annotated_tactic": ["exact\n    <a>Finset.sum_eq_zero</a>\n      (by\n        simp (config := { contextual := <a>true</a> }) only [this, <a>sum_single_index</a>, <a>ite_eq_right_iff</a>,\n          <a>eq_self_iff_true</a>, <a>imp_true_iff</a>, <a>zero_mul</a>])", [{"full_name": "Finset.sum_eq_zero", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [728, 3], "def_end_pos": [728, 14]}, {"full_name": "Bool.true", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [549, 5], "def_end_pos": [549, 9]}, {"full_name": "MonoidAlgebra.sum_single_index", "def_path": "Mathlib/Algebra/MonoidAlgebra/Basic.lean", "def_pos": [121, 9], "def_end_pos": [121, 25]}, {"full_name": "ite_eq_right_iff", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1162, 17], "def_end_pos": [1162, 33]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "imp_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [116, 9], "def_end_pos": [116, 21]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}]], "state_before": "k : Type u\u2081\nG : Type u\u2082\ninst\u271d\u00b2 : Semiring k\ninst\u271d\u00b9 : DecidableEq G\ninst\u271d : Mul G\nf : MonoidAlgebra k G\nr : k\na x : G\nhx : \u00acx \u2208 image (fun x => x * a) f.support\nthis : \u2200 (y : G), y * a = x \u2192 \u2191f y = 0\n\u22a2 (sum f fun a\u2081 b\u2081 => sum (single a r) fun a\u2082 b\u2082 => if a\u2081 * a\u2082 = x then b\u2081 * b\u2082 else 0) = 0", "state_after": "no goals"}, {"tactic": "simpa only [not_and', mem_image, mem_support_iff, exists_prop, not_exists,\n  Classical.not_not] using hx", "annotated_tactic": ["simpa only [<a>not_and'</a>, <a>mem_image</a>, <a>mem_support_iff</a>, <a>exists_prop</a>, <a>not_exists</a>,\n      <a>Classical.not_not</a>] using hx", [{"full_name": "not_and'", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [318, 9], "def_end_pos": [318, 17]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [330, 9], "def_end_pos": [330, 18]}, {"full_name": "Finsupp.mem_support_iff", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [186, 9], "def_end_pos": [186, 24]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "not_exists", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [422, 17], "def_end_pos": [422, 27]}, {"full_name": "Classical.not_not", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [683, 24], "def_end_pos": [683, 31]}]], "state_before": "k : Type u\u2081\nG : Type u\u2082\ninst\u271d\u00b2 : Semiring k\ninst\u271d\u00b9 : DecidableEq G\ninst\u271d : Mul G\nf : MonoidAlgebra k G\nr : k\na x : G\nhx : \u00acx \u2208 image (fun x => x * a) f.support\n\u22a2 \u2200 (y : G), y * a = x \u2192 \u2191f y = 0", "state_after": "no goals"}, {"tactic": "simp (config := { contextual := true }) only [this, sum_single_index, ite_eq_right_iff,\n  eq_self_iff_true, imp_true_iff, zero_mul]", "annotated_tactic": ["simp (config := { contextual := <a>true</a> }) only [this, <a>sum_single_index</a>, <a>ite_eq_right_iff</a>,\n          <a>eq_self_iff_true</a>, <a>imp_true_iff</a>, <a>zero_mul</a>]", [{"full_name": "Bool.true", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [549, 5], "def_end_pos": [549, 9]}, {"full_name": "MonoidAlgebra.sum_single_index", "def_path": "Mathlib/Algebra/MonoidAlgebra/Basic.lean", "def_pos": [121, 9], "def_end_pos": [121, 25]}, {"full_name": "ite_eq_right_iff", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1162, 17], "def_end_pos": [1162, 33]}, {"full_name": "eq_self_iff_true", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [86, 9], "def_end_pos": [86, 25]}, {"full_name": "imp_true_iff", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [116, 9], "def_end_pos": [116, 21]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}]], "state_before": "k : Type u\u2081\nG : Type u\u2082\ninst\u271d\u00b2 : Semiring k\ninst\u271d\u00b9 : DecidableEq G\ninst\u271d : Mul G\nf : MonoidAlgebra k G\nr : k\na x : G\nhx : \u00acx \u2208 image (fun x => x * a) f.support\nthis : \u2200 (y : G), y * a = x \u2192 \u2191f y = 0\n\u22a2 \u2200 (x_1 : G),\n    x_1 \u2208 f.support \u2192 (fun a\u2081 b\u2081 => sum (single a r) fun a\u2082 b\u2082 => if a\u2081 * a\u2082 = x then b\u2081 * b\u2082 else 0) x_1 (\u2191f x_1) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean", "full_name": "NonUnitalSubsemiring.mem_closure_iff", "start": [662, 1], "end": [664, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "Complex.exp_bound", "start": [1621, 1], "end": [1650, 40], "traced_tactics": [{"tactic": "rw [\u2190 lim_const (abv := Complex.abs) (\u2211 m in range n, _), exp, sub_eq_add_neg,\n  \u2190 lim_neg, lim_add, \u2190 lim_abs]", "annotated_tactic": ["rw [\u2190 <a>lim_const</a> (abv := <a>Complex.abs</a>) (\u2211 m in <a>range</a> n, _), <a>exp</a>, <a>sub_eq_add_neg</a>,\n    \u2190 <a>lim_neg</a>, <a>lim_add</a>, \u2190 <a>lim_abs</a>]", [{"full_name": "CauSeq.lim_const", "def_path": "Mathlib/Data/Real/CauSeqCompletion.lean", "def_pos": [360, 9], "def_end_pos": [360, 18]}, {"full_name": "Complex.abs", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [966, 19], "def_end_pos": [966, 37]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "Complex.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [379, 5], "def_end_pos": [379, 8]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}, {"full_name": "CauSeq.lim_neg", "def_path": "Mathlib/Data/Real/CauSeqCompletion.lean", "def_pos": [390, 9], "def_end_pos": [390, 16]}, {"full_name": "CauSeq.lim_add", "def_path": "Mathlib/Data/Real/CauSeqCompletion.lean", "def_pos": [364, 9], "def_end_pos": [364, 16]}, {"full_name": "Complex.lim_abs", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [1246, 9], "def_end_pos": [1246, 16]}]], "state_before": "x : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\n\u22a2 \u2191abs (cexp x - \u2211 m in range n, x ^ m / \u2191(Nat.factorial m)) \u2264\n    \u2191abs x ^ n * (\u2191(Nat.succ n) * (\u2191(Nat.factorial n) * \u2191n)\u207b\u00b9)", "state_after": "x : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\n\u22a2 CauSeq.lim (cauSeqAbs (exp' x + -const (\u2191abs) (\u2211 m in range n, x ^ m / \u2191(Nat.factorial m)))) \u2264\n    \u2191abs x ^ n * (\u2191(Nat.succ n) * (\u2191(Nat.factorial n) * \u2191n)\u207b\u00b9)"}, {"tactic": "refine' lim_le (CauSeq.le_of_exists \u27e8n, fun j hj => _\u27e9)", "annotated_tactic": ["refine' <a>lim_le</a> (<a>CauSeq.le_of_exists</a> \u27e8n, fun j hj => _\u27e9)", [{"full_name": "CauSeq.lim_le", "def_path": "Mathlib/Data/Real/CauSeqCompletion.lean", "def_pos": [446, 9], "def_end_pos": [446, 15]}, {"full_name": "CauSeq.le_of_exists", "def_path": "Mathlib/Data/Real/CauSeq.lean", "def_pos": [790, 9], "def_end_pos": [790, 21]}]], "state_before": "x : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\n\u22a2 CauSeq.lim (cauSeqAbs (exp' x + -const (\u2191abs) (\u2211 m in range n, x ^ m / \u2191(Nat.factorial m)))) \u2264\n    \u2191abs x ^ n * (\u2191(Nat.succ n) * (\u2191(Nat.factorial n) * \u2191n)\u207b\u00b9)", "state_after": "x : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\n\u22a2 \u2191(cauSeqAbs (exp' x + -const (\u2191abs) (\u2211 m in range n, x ^ m / \u2191(Nat.factorial m)))) j \u2264\n    \u2191(const abs' (\u2191abs x ^ n * (\u2191(Nat.succ n) * (\u2191(Nat.factorial n) * \u2191n)\u207b\u00b9))) j"}, {"tactic": "simp_rw [\u2190 sub_eq_add_neg]", "annotated_tactic": ["simp_rw [\u2190 <a>sub_eq_add_neg</a>]", [{"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}]], "state_before": "x : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\n\u22a2 \u2191(cauSeqAbs (exp' x + -const (\u2191abs) (\u2211 m in range n, x ^ m / \u2191(Nat.factorial m)))) j \u2264\n    \u2191(const abs' (\u2191abs x ^ n * (\u2191(Nat.succ n) * (\u2191(Nat.factorial n) * \u2191n)\u207b\u00b9))) j", "state_after": "x : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\n\u22a2 \u2191(cauSeqAbs (exp' x - const (\u2191abs) (\u2211 m in range n, x ^ m / \u2191(Nat.factorial m)))) j \u2264\n    \u2191(const abs' (\u2191abs x ^ n * (\u2191(Nat.succ n) * (\u2191(Nat.factorial n) * \u2191n)\u207b\u00b9))) j"}, {"tactic": "show\n  abs ((\u2211 m in range j, x ^ m / m.factorial) - \u2211 m in range n, x ^ m / m.factorial) \u2264\n    abs x ^ n * ((n.succ : \u211d) * (n.factorial * n : \u211d)\u207b\u00b9)", "annotated_tactic": ["show\n    <a>abs</a> ((\u2211 m in <a>range</a> j, x ^ m / m.factorial) - \u2211 m in <a>range</a> n, x ^ m / m.factorial) \u2264\n      <a>abs</a> x ^ n * ((n.succ : \u211d) * (n.factorial * n : \u211d)\u207b\u00b9)", [{"full_name": "Complex.abs", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [966, 19], "def_end_pos": [966, 37]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "Complex.abs", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [966, 19], "def_end_pos": [966, 37]}]], "state_before": "x : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\n\u22a2 \u2191(cauSeqAbs (exp' x - const (\u2191abs) (\u2211 m in range n, x ^ m / \u2191(Nat.factorial m)))) j \u2264\n    \u2191(const abs' (\u2191abs x ^ n * (\u2191(Nat.succ n) * (\u2191(Nat.factorial n) * \u2191n)\u207b\u00b9))) j", "state_after": "x : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\n\u22a2 \u2191abs (\u2211 m in range j, x ^ m / \u2191(Nat.factorial m) - \u2211 m in range n, x ^ m / \u2191(Nat.factorial m)) \u2264\n    \u2191abs x ^ n * (\u2191(Nat.succ n) * (\u2191(Nat.factorial n) * \u2191n)\u207b\u00b9)"}, {"tactic": "rw [sum_range_sub_sum_range hj]", "annotated_tactic": ["rw [<a>sum_range_sub_sum_range</a> hj]", [{"full_name": "Finset.sum_range_sub_sum_range", "def_path": "Mathlib/Algebra/BigOperators/Intervals.lean", "def_pos": [109, 3], "def_end_pos": [109, 14]}]], "state_before": "x : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\n\u22a2 \u2191abs (\u2211 m in range j, x ^ m / \u2191(Nat.factorial m) - \u2211 m in range n, x ^ m / \u2191(Nat.factorial m)) \u2264\n    \u2191abs x ^ n * (\u2191(Nat.succ n) * (\u2191(Nat.factorial n) * \u2191n)\u207b\u00b9)", "state_after": "x : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\n\u22a2 \u2191abs (\u2211 k in filter (fun k => n \u2264 k) (range j), x ^ k / \u2191(Nat.factorial k)) \u2264\n    \u2191abs x ^ n * (\u2191(Nat.succ n) * (\u2191(Nat.factorial n) * \u2191n)\u207b\u00b9)"}, {"tactic": "refine' congr_arg abs (sum_congr rfl fun m hm => _)", "annotated_tactic": ["refine' <a>congr_arg</a> <a>abs</a> (<a>sum_congr</a> <a>rfl</a> fun m hm => _)", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "Complex.abs", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [966, 19], "def_end_pos": [966, 37]}, {"full_name": "Finset.sum_congr", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [376, 3], "def_end_pos": [376, 14]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "x : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\n\u22a2 \u2191abs (\u2211 m in filter (fun k => n \u2264 k) (range j), x ^ m / \u2191(Nat.factorial m)) =\n    \u2191abs (\u2211 m in filter (fun k => n \u2264 k) (range j), x ^ n * (x ^ (m - n) / \u2191(Nat.factorial m)))", "state_after": "x : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\nm : \u2115\nhm : m \u2208 filter (fun k => n \u2264 k) (range j)\n\u22a2 x ^ m / \u2191(Nat.factorial m) = x ^ n * (x ^ (m - n) / \u2191(Nat.factorial m))"}, {"tactic": "rw [mem_filter, mem_range] at hm", "annotated_tactic": ["rw [<a>mem_filter</a>, <a>mem_range</a>] at hm", [{"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2708, 9], "def_end_pos": [2708, 19]}, {"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3037, 9], "def_end_pos": [3037, 18]}]], "state_before": "x : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\nm : \u2115\nhm : m \u2208 filter (fun k => n \u2264 k) (range j)\n\u22a2 x ^ m / \u2191(Nat.factorial m) = x ^ n * (x ^ (m - n) / \u2191(Nat.factorial m))", "state_after": "x : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\nm : \u2115\nhm : m < j \u2227 n \u2264 m\n\u22a2 x ^ m / \u2191(Nat.factorial m) = x ^ n * (x ^ (m - n) / \u2191(Nat.factorial m))"}, {"tactic": "rw [\u2190 mul_div_assoc, \u2190 pow_add, add_tsub_cancel_of_le hm.2]", "annotated_tactic": ["rw [\u2190 <a>mul_div_assoc</a>, \u2190 <a>pow_add</a>, <a>add_tsub_cancel_of_le</a> hm.2]", [{"full_name": "mul_div_assoc", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 22]}, {"full_name": "pow_add", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 16]}, {"full_name": "add_tsub_cancel_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [24, 9], "def_end_pos": [24, 30]}]], "state_before": "x : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\nm : \u2115\nhm : m < j \u2227 n \u2264 m\n\u22a2 x ^ m / \u2191(Nat.factorial m) = x ^ n * (x ^ (m - n) / \u2191(Nat.factorial m))", "state_after": "no goals"}, {"tactic": "simp_rw [map_mul, map_pow, map_div\u2080, abs_cast_nat]", "annotated_tactic": ["simp_rw [<a>map_mul</a>, <a>map_pow</a>, <a>map_div\u2080</a>, <a>abs_cast_nat</a>]", [{"full_name": "map_mul", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [299, 9], "def_end_pos": [299, 16]}, {"full_name": "map_pow", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [435, 9], "def_end_pos": [435, 16]}, {"full_name": "map_div\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [246, 9], "def_end_pos": [246, 17]}, {"full_name": "Complex.abs_cast_nat", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [1139, 9], "def_end_pos": [1139, 21]}]], "state_before": "x : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\n\u22a2 \u2211 m in filter (fun k => n \u2264 k) (range j), \u2191abs (x ^ n * (x ^ (m - n) / \u2191(Nat.factorial m))) \u2264\n    \u2211 m in filter (fun k => n \u2264 k) (range j), \u2191abs x ^ n * (1 / \u2191(Nat.factorial m))", "state_after": "x : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\n\u22a2 \u2211 x_1 in filter (fun k => n \u2264 k) (range j), \u2191abs x ^ n * (\u2191abs (x ^ (x_1 - n)) / \u2191(Nat.factorial x_1)) \u2264\n    \u2211 x_1 in filter (fun k => n \u2264 k) (range j), \u2191abs x ^ n * (1 / \u2191(Nat.factorial x_1))"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "x : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\n\u22a2 \u2211 x_1 in filter (fun k => n \u2264 k) (range j), \u2191abs x ^ n * (\u2191abs (x ^ (x_1 - n)) / \u2191(Nat.factorial x_1)) \u2264\n    \u2211 x_1 in filter (fun k => n \u2264 k) (range j), \u2191abs x ^ n * (1 / \u2191(Nat.factorial x_1))", "state_after": "case h.h.h\nx : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\ni\u271d : \u2115\na\u271d : i\u271d \u2208 filter (fun k => n \u2264 k) (range j)\n\u22a2 \u2191abs (x ^ (i\u271d - n)) \u2264 1"}, {"tactic": "rw [abv_pow abs]", "annotated_tactic": ["rw [<a>abv_pow</a> <a>abs</a>]", [{"full_name": "IsAbsoluteValue.abv_pow", "def_path": "Mathlib/Algebra/Order/AbsoluteValue.lean", "def_pos": [388, 9], "def_end_pos": [388, 16]}, {"full_name": "Complex.abs", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [966, 19], "def_end_pos": [966, 37]}]], "state_before": "case h.h.h\nx : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\ni\u271d : \u2115\na\u271d : i\u271d \u2208 filter (fun k => n \u2264 k) (range j)\n\u22a2 \u2191abs (x ^ (i\u271d - n)) \u2264 1", "state_after": "case h.h.h\nx : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\ni\u271d : \u2115\na\u271d : i\u271d \u2208 filter (fun k => n \u2264 k) (range j)\n\u22a2 \u2191abs x ^ (i\u271d - n) \u2264 1"}, {"tactic": "exact pow_le_one _ (abs.nonneg _) hx", "annotated_tactic": ["exact <a>pow_le_one</a> _ (abs.nonneg _) hx", [{"full_name": "pow_le_one", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [411, 9], "def_end_pos": [411, 19]}]], "state_before": "case h.h.h\nx : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\ni\u271d : \u2115\na\u271d : i\u271d \u2208 filter (fun k => n \u2264 k) (range j)\n\u22a2 \u2191abs x ^ (i\u271d - n) \u2264 1", "state_after": "no goals"}, {"tactic": "simp [abs_mul, abv_pow abs, abs_div, mul_sum.symm]", "annotated_tactic": ["simp [<a>abs_mul</a>, <a>abv_pow</a> <a>abs</a>, <a>abs_div</a>, mul_sum.symm]", [{"full_name": "abs_mul", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [33, 9], "def_end_pos": [33, 16]}, {"full_name": "IsAbsoluteValue.abv_pow", "def_path": "Mathlib/Algebra/Order/AbsoluteValue.lean", "def_pos": [388, 9], "def_end_pos": [388, 16]}, {"full_name": "Complex.abs", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [966, 19], "def_end_pos": [966, 37]}, {"full_name": "abs_div", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [1002, 9], "def_end_pos": [1002, 16]}]], "state_before": "x : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\n\u22a2 \u2211 m in filter (fun k => n \u2264 k) (range j), \u2191abs x ^ n * (1 / \u2191(Nat.factorial m)) =\n    \u2191abs x ^ n * \u2211 m in filter (fun k => n \u2264 k) (range j), 1 / \u2191(Nat.factorial m)", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "x : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\n\u22a2 \u2191abs x ^ n * \u2211 m in filter (fun k => n \u2264 k) (range j), 1 / \u2191(Nat.factorial m) \u2264\n    \u2191abs x ^ n * (\u2191(Nat.succ n) * (\u2191(Nat.factorial n) * \u2191n)\u207b\u00b9)", "state_after": "case h\nx : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\n\u22a2 \u2211 m in filter (fun k => n \u2264 k) (range j), 1 / \u2191(Nat.factorial m) \u2264 \u2191(Nat.succ n) * (\u2191(Nat.factorial n) * \u2191n)\u207b\u00b9"}, {"tactic": "exact sum_div_factorial_le _ _ hn", "annotated_tactic": ["exact <a>sum_div_factorial_le</a> _ _ hn", [{"full_name": "Complex.sum_div_factorial_le", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1583, 9], "def_end_pos": [1583, 29]}]], "state_before": "case h\nx : \u2102\nhx : \u2191abs x \u2264 1\nn : \u2115\nhn : 0 < n\nj : \u2115\nhj : j \u2265 n\n\u22a2 \u2211 m in filter (fun k => n \u2264 k) (range j), 1 / \u2191(Nat.factorial m) \u2264 \u2191(Nat.succ n) * (\u2191(Nat.factorial n) * \u2191n)\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/LocalHomeomorph.lean", "full_name": "LocalHomeomorph.EqOnSource.eqOn", "start": [966, 1], "end": [967, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.restrict_finset_biUnion_congr", "start": [1850, 1], "end": [1855, 34], "traced_tactics": [{"tactic": "induction' s using Finset.induction_on with i s _ hs", "annotated_tactic": ["induction' s using <a>Finset.induction_on</a> with i s _ hs", [{"full_name": "Finset.induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1251, 19], "def_end_pos": [1251, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ns : Finset \u03b9\nt : \u03b9 \u2192 Set \u03b1\n\u22a2 restrict \u03bc (\u22c3 i \u2208 s, t i) = restrict \u03bd (\u22c3 i \u2208 s, t i) \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 restrict \u03bc (t i) = restrict \u03bd (t i)", "state_after": "case empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : \u03b9 \u2192 Set \u03b1\n\u22a2 restrict \u03bc (\u22c3 i \u2208 \u2205, t i) = restrict \u03bd (\u22c3 i \u2208 \u2205, t i) \u2194 \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 restrict \u03bc (t i) = restrict \u03bd (t i)\n\ncase insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nt : \u03b9 \u2192 Set \u03b1\ni : \u03b9\ns : Finset \u03b9\na\u271d : \u00aci \u2208 s\nhs : restrict \u03bc (\u22c3 i \u2208 s, t i) = restrict \u03bd (\u22c3 i \u2208 s, t i) \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 restrict \u03bc (t i) = restrict \u03bd (t i)\n\u22a2 restrict \u03bc (\u22c3 i_1 \u2208 insert i s, t i_1) = restrict \u03bd (\u22c3 i_1 \u2208 insert i s, t i_1) \u2194\n    \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 restrict \u03bc (t i_1) = restrict \u03bd (t i_1)"}, {"tactic": "simp only [forall_eq_or_imp, iUnion_iUnion_eq_or_left, Finset.mem_insert]", "annotated_tactic": ["simp only [<a>forall_eq_or_imp</a>, <a>iUnion_iUnion_eq_or_left</a>, <a>Finset.mem_insert</a>]", [{"full_name": "forall_eq_or_imp", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [473, 17], "def_end_pos": [473, 33]}, {"full_name": "Set.iUnion_iUnion_eq_or_left", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [950, 9], "def_end_pos": [950, 33]}, {"full_name": "Finset.mem_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1087, 9], "def_end_pos": [1087, 19]}]], "state_before": "case insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nt : \u03b9 \u2192 Set \u03b1\ni : \u03b9\ns : Finset \u03b9\na\u271d : \u00aci \u2208 s\nhs : restrict \u03bc (\u22c3 i \u2208 s, t i) = restrict \u03bd (\u22c3 i \u2208 s, t i) \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 restrict \u03bc (t i) = restrict \u03bd (t i)\n\u22a2 restrict \u03bc (\u22c3 i_1 \u2208 insert i s, t i_1) = restrict \u03bd (\u22c3 i_1 \u2208 insert i s, t i_1) \u2194\n    \u2200 (i_1 : \u03b9), i_1 \u2208 insert i s \u2192 restrict \u03bc (t i_1) = restrict \u03bd (t i_1)", "state_after": "case insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nt : \u03b9 \u2192 Set \u03b1\ni : \u03b9\ns : Finset \u03b9\na\u271d : \u00aci \u2208 s\nhs : restrict \u03bc (\u22c3 i \u2208 s, t i) = restrict \u03bd (\u22c3 i \u2208 s, t i) \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 restrict \u03bc (t i) = restrict \u03bd (t i)\n\u22a2 restrict \u03bc (t i \u222a \u22c3 x \u2208 s, t x) = restrict \u03bd (t i \u222a \u22c3 x \u2208 s, t x) \u2194\n    restrict \u03bc (t i) = restrict \u03bd (t i) \u2227 \u2200 (a : \u03b9), a \u2208 s \u2192 restrict \u03bc (t a) = restrict \u03bd (t a)"}, {"tactic": "rw [restrict_union_congr, \u2190 hs]", "annotated_tactic": ["rw [<a>restrict_union_congr</a>, \u2190 hs]", [{"full_name": "MeasureTheory.Measure.restrict_union_congr", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1824, 9], "def_end_pos": [1824, 29]}]], "state_before": "case insert\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\nt : \u03b9 \u2192 Set \u03b1\ni : \u03b9\ns : Finset \u03b9\na\u271d : \u00aci \u2208 s\nhs : restrict \u03bc (\u22c3 i \u2208 s, t i) = restrict \u03bd (\u22c3 i \u2208 s, t i) \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 restrict \u03bc (t i) = restrict \u03bd (t i)\n\u22a2 restrict \u03bc (t i \u222a \u22c3 x \u2208 s, t x) = restrict \u03bd (t i \u222a \u22c3 x \u2208 s, t x) \u2194\n    restrict \u03bc (t i) = restrict \u03bd (t i) \u2227 \u2200 (a : \u03b9), a \u2208 s \u2192 restrict \u03bc (t a) = restrict \u03bd (t a)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t\u271d : Set \u03b1\nt : \u03b9 \u2192 Set \u03b1\n\u22a2 restrict \u03bc (\u22c3 i \u2208 \u2205, t i) = restrict \u03bd (\u22c3 i \u2208 \u2205, t i) \u2194 \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 restrict \u03bc (t i) = restrict \u03bd (t i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Dynamics/PeriodicPts.lean", "full_name": "Function.is_periodic_id", "start": [66, 1], "end": [67, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean", "full_name": "Real.continuousAt_rpow_of_pos", "start": [237, 1], "end": [252, 80], "traced_tactics": [{"tactic": "cases' p with x y", "annotated_tactic": ["cases' p with x y", []], "state_before": "p : \u211d \u00d7 \u211d\nhp : 0 < p.2\n\u22a2 ContinuousAt (fun p => p.1 ^ p.2) p", "state_after": "case mk\nx y : \u211d\nhp : 0 < (x, y).2\n\u22a2 ContinuousAt (fun p => p.1 ^ p.2) (x, y)"}, {"tactic": "dsimp only at hp", "annotated_tactic": ["dsimp only at hp", []], "state_before": "case mk\nx y : \u211d\nhp : 0 < (x, y).2\n\u22a2 ContinuousAt (fun p => p.1 ^ p.2) (x, y)", "state_after": "case mk\nx y : \u211d\nhp : 0 < y\n\u22a2 ContinuousAt (fun p => p.1 ^ p.2) (x, y)"}, {"tactic": "obtain hx | rfl := ne_or_eq x 0", "annotated_tactic": ["obtain hx | rfl := <a>ne_or_eq</a> x 0", [{"full_name": "ne_or_eq", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 17]}]], "state_before": "case mk\nx y : \u211d\nhp : 0 < y\n\u22a2 ContinuousAt (fun p => p.1 ^ p.2) (x, y)", "state_after": "case mk.inl\nx y : \u211d\nhp : 0 < y\nhx : x \u2260 0\n\u22a2 ContinuousAt (fun p => p.1 ^ p.2) (x, y)\n\ncase mk.inr\ny : \u211d\nhp : 0 < y\n\u22a2 ContinuousAt (fun p => p.1 ^ p.2) (0, y)"}, {"tactic": "have A : Tendsto (fun p : \u211d \u00d7 \u211d => exp (log p.1 * p.2)) (\ud835\udcdd[\u2260] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0) :=\n  tendsto_exp_atBot.comp\n    ((tendsto_log_nhdsWithin_zero.comp tendsto_fst).atBot_mul hp tendsto_snd)", "annotated_tactic": ["have A : <a>Tendsto</a> (fun p : \u211d \u00d7 \u211d => <a>exp</a> (<a>log</a> p.1 * p.2)) (\ud835\udcdd[\u2260] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0) :=\n    tendsto_exp_atBot.comp\n      ((tendsto_log_nhdsWithin_zero.comp <a>tendsto_fst</a>).<a>atBot_mul</a> hp <a>tendsto_snd</a>)", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [434, 12], "def_end_pos": [434, 15]}, {"full_name": "Real.log", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [43, 19], "def_end_pos": [43, 22]}, {"full_name": "Filter.tendsto_fst", "def_path": "Mathlib/Order/Filter/Prod.lean", "def_pos": [136, 9], "def_end_pos": [136, 20]}, {"full_name": "Filter.Tendsto.atBot_mul", "def_path": "Mathlib/Topology/Algebra/Order/Field.lean", "def_pos": [92, 9], "def_end_pos": [92, 33]}, {"full_name": "Filter.tendsto_snd", "def_path": "Mathlib/Order/Filter/Prod.lean", "def_pos": [140, 9], "def_end_pos": [140, 20]}]], "state_before": "case mk.inr\ny : \u211d\nhp : 0 < y\n\u22a2 ContinuousAt (fun p => p.1 ^ p.2) (0, y)", "state_after": "case mk.inr\ny : \u211d\nhp : 0 < y\nA : Tendsto (fun p => rexp (log p.1 * p.2)) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\n\u22a2 ContinuousAt (fun p => p.1 ^ p.2) (0, y)"}, {"tactic": "have B : Tendsto (fun p : \u211d \u00d7 \u211d => p.1 ^ p.2) (\ud835\udcdd[\u2260] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0) :=\n  squeeze_zero_norm (fun p => abs_rpow_le_exp_log_mul p.1 p.2) A", "annotated_tactic": ["have B : <a>Tendsto</a> (fun p : \u211d \u00d7 \u211d => p.1 ^ p.2) (\ud835\udcdd[\u2260] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0) :=\n    <a>squeeze_zero_norm</a> (fun p => <a>abs_rpow_le_exp_log_mul</a> p.1 p.2) A", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "squeeze_zero_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1118, 3], "def_end_pos": [1118, 14]}, {"full_name": "Real.abs_rpow_le_exp_log_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [159, 9], "def_end_pos": [159, 32]}]], "state_before": "case mk.inr\ny : \u211d\nhp : 0 < y\nA : Tendsto (fun p => rexp (log p.1 * p.2)) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\n\u22a2 ContinuousAt (fun p => p.1 ^ p.2) (0, y)", "state_after": "case mk.inr\ny : \u211d\nhp : 0 < y\nA : Tendsto (fun p => rexp (log p.1 * p.2)) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\nB : Tendsto (fun p => p.1 ^ p.2) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\n\u22a2 ContinuousAt (fun p => p.1 ^ p.2) (0, y)"}, {"tactic": "have C : Tendsto (fun p : \u211d \u00d7 \u211d => p.1 ^ p.2) (\ud835\udcdd[{0}] 0 \u00d7\u02e2 \ud835\udcdd y) (pure 0) := by\n  rw [nhdsWithin_singleton, tendsto_pure, pure_prod, eventually_map]\n  exact (lt_mem_nhds hp).mono fun y hy => zero_rpow hy.ne'", "annotated_tactic": ["have C : <a>Tendsto</a> (fun p : \u211d \u00d7 \u211d => p.1 ^ p.2) (\ud835\udcdd[{0}] 0 \u00d7\u02e2 \ud835\udcdd y) (<a>pure</a> 0) := by\n    rw [<a>nhdsWithin_singleton</a>, <a>tendsto_pure</a>, <a>pure_prod</a>, <a>eventually_map</a>]\n    exact (<a>lt_mem_nhds</a> hp).<a>mono</a> fun y hy => <a>zero_rpow</a> hy.ne'", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Pure.pure", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2670, 3], "def_end_pos": [2670, 7]}, {"full_name": "nhdsWithin_singleton", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [280, 9], "def_end_pos": [280, 29]}, {"full_name": "Filter.tendsto_pure", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3167, 17], "def_end_pos": [3167, 29]}, {"full_name": "Filter.pure_prod", "def_path": "Mathlib/Order/Filter/Prod.lean", "def_pos": [403, 9], "def_end_pos": [403, 18]}, {"full_name": "Filter.eventually_map", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 23]}, {"full_name": "lt_mem_nhds", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [897, 9], "def_end_pos": [897, 20]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}, {"full_name": "Real.zero_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [103, 9], "def_end_pos": [103, 18]}]], "state_before": "case mk.inr\ny : \u211d\nhp : 0 < y\nA : Tendsto (fun p => rexp (log p.1 * p.2)) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\nB : Tendsto (fun p => p.1 ^ p.2) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\n\u22a2 ContinuousAt (fun p => p.1 ^ p.2) (0, y)", "state_after": "case mk.inr\ny : \u211d\nhp : 0 < y\nA : Tendsto (fun p => rexp (log p.1 * p.2)) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\nB : Tendsto (fun p => p.1 ^ p.2) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\nC : Tendsto (fun p => p.1 ^ p.2) (\ud835\udcdd[{0}] 0 \u00d7\u02e2 \ud835\udcdd y) (pure 0)\n\u22a2 ContinuousAt (fun p => p.1 ^ p.2) (0, y)"}, {"tactic": "simpa only [\u2190 sup_prod, \u2190 nhdsWithin_union, compl_union_self, nhdsWithin_univ, nhds_prod_eq,\n  ContinuousAt, zero_rpow hp.ne'] using B.sup (C.mono_right (pure_le_nhds _))", "annotated_tactic": ["simpa only [\u2190 <a>sup_prod</a>, \u2190 <a>nhdsWithin_union</a>, <a>compl_union_self</a>, <a>nhdsWithin_univ</a>, <a>nhds_prod_eq</a>,\n    <a>ContinuousAt</a>, <a>zero_rpow</a> hp.ne'] using B.sup (C.mono_right (<a>pure_le_nhds</a> _))", [{"full_name": "Filter.sup_prod", "def_path": "Mathlib/Order/Filter/Prod.lean", "def_pos": [119, 9], "def_end_pos": [119, 17]}, {"full_name": "nhdsWithin_union", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [239, 9], "def_end_pos": [239, 25]}, {"full_name": "Set.compl_union_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1744, 9], "def_end_pos": [1744, 25]}, {"full_name": "nhdsWithin_univ", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [79, 9], "def_end_pos": [79, 24]}, {"full_name": "nhds_prod_eq", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [513, 9], "def_end_pos": [513, 21]}, {"full_name": "ContinuousAt", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1620, 5], "def_end_pos": [1620, 17]}, {"full_name": "Real.zero_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [103, 9], "def_end_pos": [103, 18]}, {"full_name": "pure_le_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1063, 9], "def_end_pos": [1063, 21]}]], "state_before": "case mk.inr\ny : \u211d\nhp : 0 < y\nA : Tendsto (fun p => rexp (log p.1 * p.2)) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\nB : Tendsto (fun p => p.1 ^ p.2) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\nC : Tendsto (fun p => p.1 ^ p.2) (\ud835\udcdd[{0}] 0 \u00d7\u02e2 \ud835\udcdd y) (pure 0)\n\u22a2 ContinuousAt (fun p => p.1 ^ p.2) (0, y)", "state_after": "no goals"}, {"tactic": "exact continuousAt_rpow_of_ne (x, y) hx", "annotated_tactic": ["exact <a>continuousAt_rpow_of_ne</a> (x, y) hx", [{"full_name": "Real.continuousAt_rpow_of_ne", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean", "def_pos": [219, 9], "def_end_pos": [219, 32]}]], "state_before": "case mk.inl\nx y : \u211d\nhp : 0 < y\nhx : x \u2260 0\n\u22a2 ContinuousAt (fun p => p.1 ^ p.2) (x, y)", "state_after": "no goals"}, {"tactic": "rw [nhdsWithin_singleton, tendsto_pure, pure_prod, eventually_map]", "annotated_tactic": ["rw [<a>nhdsWithin_singleton</a>, <a>tendsto_pure</a>, <a>pure_prod</a>, <a>eventually_map</a>]", [{"full_name": "nhdsWithin_singleton", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [280, 9], "def_end_pos": [280, 29]}, {"full_name": "Filter.tendsto_pure", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3167, 17], "def_end_pos": [3167, 29]}, {"full_name": "Filter.pure_prod", "def_path": "Mathlib/Order/Filter/Prod.lean", "def_pos": [403, 9], "def_end_pos": [403, 18]}, {"full_name": "Filter.eventually_map", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1845, 9], "def_end_pos": [1845, 23]}]], "state_before": "y : \u211d\nhp : 0 < y\nA : Tendsto (fun p => rexp (log p.1 * p.2)) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\nB : Tendsto (fun p => p.1 ^ p.2) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun p => p.1 ^ p.2) (\ud835\udcdd[{0}] 0 \u00d7\u02e2 \ud835\udcdd y) (pure 0)", "state_after": "y : \u211d\nhp : 0 < y\nA : Tendsto (fun p => rexp (log p.1 * p.2)) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\nB : Tendsto (fun p => p.1 ^ p.2) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\n\u22a2 \u2200\u1da0 (a : \u211d) in \ud835\udcdd y, (0, a).1 ^ (0, a).2 = 0"}, {"tactic": "exact (lt_mem_nhds hp).mono fun y hy => zero_rpow hy.ne'", "annotated_tactic": ["exact (<a>lt_mem_nhds</a> hp).<a>mono</a> fun y hy => <a>zero_rpow</a> hy.ne'", [{"full_name": "lt_mem_nhds", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [897, 9], "def_end_pos": [897, 20]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}, {"full_name": "Real.zero_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [103, 9], "def_end_pos": [103, 18]}]], "state_before": "y : \u211d\nhp : 0 < y\nA : Tendsto (fun p => rexp (log p.1 * p.2)) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\nB : Tendsto (fun p => p.1 ^ p.2) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\n\u22a2 \u2200\u1da0 (a : \u211d) in \ud835\udcdd y, (0, a).1 ^ (0, a).2 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Algebra/Bilinear.lean", "full_name": "LinearMap.mulRight_apply", "start": [83, 1], "end": [84, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Logic/Relation.lean", "full_name": "Relation.ReflTransGen.head_induction_on", "start": [298, 1], "end": [307, 46], "traced_tactics": [{"tactic": "induction h", "annotated_tactic": ["induction h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d b c d : \u03b1\nP : (a : \u03b1) \u2192 ReflTransGen r a b \u2192 Prop\na : \u03b1\nh : ReflTransGen r a b\nrefl : P b (_ : ReflTransGen r b b)\nhead : \u2200 {a c : \u03b1} (h' : r a c) (h : ReflTransGen r c b), P c h \u2192 P a (_ : ReflTransGen r a b)\n\u22a2 P a h", "state_after": "case refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d b c d a : \u03b1\nP : (a_1 : \u03b1) \u2192 ReflTransGen r a_1 a \u2192 Prop\nrefl : P a (_ : ReflTransGen r a a)\nhead : \u2200 {a_1 c : \u03b1} (h' : r a_1 c) (h : ReflTransGen r c a), P c h \u2192 P a_1 (_ : ReflTransGen r a_1 a)\n\u22a2 P a (_ : ReflTransGen r a a)\n\ncase tail\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d\u00b2 b c d a b\u271d c\u271d : \u03b1\na\u271d\u00b9 : ReflTransGen r a b\u271d\na\u271d : r b\u271d c\u271d\na_ih\u271d :\n  \u2200 {P : (a : \u03b1) \u2192 ReflTransGen r a b\u271d \u2192 Prop},\n    P b\u271d (_ : ReflTransGen r b\u271d b\u271d) \u2192\n      (\u2200 {a c : \u03b1} (h' : r a c) (h : ReflTransGen r c b\u271d), P c h \u2192 P a (_ : ReflTransGen r a b\u271d)) \u2192 P a a\u271d\u00b9\nP : (a : \u03b1) \u2192 ReflTransGen r a c\u271d \u2192 Prop\nrefl : P c\u271d (_ : ReflTransGen r c\u271d c\u271d)\nhead : \u2200 {a c : \u03b1} (h' : r a c) (h : ReflTransGen r c c\u271d), P c h \u2192 P a (_ : ReflTransGen r a c\u271d)\n\u22a2 P a (_ : ReflTransGen r a c\u271d)"}, {"tactic": "case refl => exact refl", "annotated_tactic": ["case refl => exact refl", []], "state_before": "case refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d b c d a : \u03b1\nP : (a_1 : \u03b1) \u2192 ReflTransGen r a_1 a \u2192 Prop\nrefl : P a (_ : ReflTransGen r a a)\nhead : \u2200 {a_1 c : \u03b1} (h' : r a_1 c) (h : ReflTransGen r c a), P c h \u2192 P a_1 (_ : ReflTransGen r a_1 a)\n\u22a2 P a (_ : ReflTransGen r a a)\n\ncase tail\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d\u00b2 b c d a b\u271d c\u271d : \u03b1\na\u271d\u00b9 : ReflTransGen r a b\u271d\na\u271d : r b\u271d c\u271d\na_ih\u271d :\n  \u2200 {P : (a : \u03b1) \u2192 ReflTransGen r a b\u271d \u2192 Prop},\n    P b\u271d (_ : ReflTransGen r b\u271d b\u271d) \u2192\n      (\u2200 {a c : \u03b1} (h' : r a c) (h : ReflTransGen r c b\u271d), P c h \u2192 P a (_ : ReflTransGen r a b\u271d)) \u2192 P a a\u271d\u00b9\nP : (a : \u03b1) \u2192 ReflTransGen r a c\u271d \u2192 Prop\nrefl : P c\u271d (_ : ReflTransGen r c\u271d c\u271d)\nhead : \u2200 {a c : \u03b1} (h' : r a c) (h : ReflTransGen r c c\u271d), P c h \u2192 P a (_ : ReflTransGen r a c\u271d)\n\u22a2 P a (_ : ReflTransGen r a c\u271d)", "state_after": "case tail\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d\u00b2 b c d a b\u271d c\u271d : \u03b1\na\u271d\u00b9 : ReflTransGen r a b\u271d\na\u271d : r b\u271d c\u271d\na_ih\u271d :\n  \u2200 {P : (a : \u03b1) \u2192 ReflTransGen r a b\u271d \u2192 Prop},\n    P b\u271d (_ : ReflTransGen r b\u271d b\u271d) \u2192\n      (\u2200 {a c : \u03b1} (h' : r a c) (h : ReflTransGen r c b\u271d), P c h \u2192 P a (_ : ReflTransGen r a b\u271d)) \u2192 P a a\u271d\u00b9\nP : (a : \u03b1) \u2192 ReflTransGen r a c\u271d \u2192 Prop\nrefl : P c\u271d (_ : ReflTransGen r c\u271d c\u271d)\nhead : \u2200 {a c : \u03b1} (h' : r a c) (h : ReflTransGen r c c\u271d), P c h \u2192 P a (_ : ReflTransGen r a c\u271d)\n\u22a2 P a (_ : ReflTransGen r a c\u271d)"}, {"tactic": "case tail b c _ hbc ih =>\nrefine @ih (\u03bb {a : \u03b1} (hab : ReflTransGen r a b) => P a (ReflTransGen.tail hab hbc)) ?_ ?_\n{ exact head hbc _ refl }\n{ exact fun h1 h2 \u21a6 head h1 (h2.tail hbc) }", "annotated_tactic": ["case tail b c _ hbc ih =>\n  -- Porting note: Lean 3 figured out the motive and `apply ih` worked\n  refine @ih (\u03bb {a : \u03b1} (hab : <a>ReflTransGen</a> r a b) => P a (<a>ReflTransGen.tail</a> hab hbc)) ?_ ?_\n  { exact head hbc _ refl }\n  { exact fun h1 h2 \u21a6 head h1 (h2.tail hbc) }", [{"full_name": "Relation.ReflTransGen", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [222, 11], "def_end_pos": [222, 23]}, {"full_name": "Relation.ReflTransGen.tail", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [224, 5], "def_end_pos": [224, 9]}]], "state_before": "case tail\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d\u00b2 b c d a b\u271d c\u271d : \u03b1\na\u271d\u00b9 : ReflTransGen r a b\u271d\na\u271d : r b\u271d c\u271d\na_ih\u271d :\n  \u2200 {P : (a : \u03b1) \u2192 ReflTransGen r a b\u271d \u2192 Prop},\n    P b\u271d (_ : ReflTransGen r b\u271d b\u271d) \u2192\n      (\u2200 {a c : \u03b1} (h' : r a c) (h : ReflTransGen r c b\u271d), P c h \u2192 P a (_ : ReflTransGen r a b\u271d)) \u2192 P a a\u271d\u00b9\nP : (a : \u03b1) \u2192 ReflTransGen r a c\u271d \u2192 Prop\nrefl : P c\u271d (_ : ReflTransGen r c\u271d c\u271d)\nhead : \u2200 {a c : \u03b1} (h' : r a c) (h : ReflTransGen r c c\u271d), P c h \u2192 P a (_ : ReflTransGen r a c\u271d)\n\u22a2 P a (_ : ReflTransGen r a c\u271d)", "state_after": "no goals"}, {"tactic": "exact refl", "annotated_tactic": ["exact refl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d b c d a : \u03b1\nP : (a_1 : \u03b1) \u2192 ReflTransGen r a_1 a \u2192 Prop\nrefl : P a (_ : ReflTransGen r a a)\nhead : \u2200 {a_1 c : \u03b1} (h' : r a_1 c) (h : ReflTransGen r c a), P c h \u2192 P a_1 (_ : ReflTransGen r a_1 a)\n\u22a2 P a (_ : ReflTransGen r a a)", "state_after": "no goals"}, {"tactic": "refine @ih (\u03bb {a : \u03b1} (hab : ReflTransGen r a b) => P a (ReflTransGen.tail hab hbc)) ?_ ?_", "annotated_tactic": ["refine @ih (\u03bb {a : \u03b1} (hab : <a>ReflTransGen</a> r a b) => P a (<a>ReflTransGen.tail</a> hab hbc)) ?_ ?_", [{"full_name": "Relation.ReflTransGen", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [222, 11], "def_end_pos": [222, 23]}, {"full_name": "Relation.ReflTransGen.tail", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [224, 5], "def_end_pos": [224, 9]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d\u00b9 b\u271d c\u271d d a b c : \u03b1\na\u271d : ReflTransGen r a b\nhbc : r b c\nih :\n  \u2200 {P : (a : \u03b1) \u2192 ReflTransGen r a b \u2192 Prop},\n    P b (_ : ReflTransGen r b b) \u2192\n      (\u2200 {a c : \u03b1} (h' : r a c) (h : ReflTransGen r c b), P c h \u2192 P a (_ : ReflTransGen r a b)) \u2192 P a a\u271d\nP : (a : \u03b1) \u2192 ReflTransGen r a c \u2192 Prop\nrefl : P c (_ : ReflTransGen r c c)\nhead : \u2200 {a c_1 : \u03b1} (h' : r a c_1) (h : ReflTransGen r c_1 c), P c_1 h \u2192 P a (_ : ReflTransGen r a c)\n\u22a2 P a (_ : ReflTransGen r a c)", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d\u00b9 b\u271d c\u271d d a b c : \u03b1\na\u271d : ReflTransGen r a b\nhbc : r b c\nih :\n  \u2200 {P : (a : \u03b1) \u2192 ReflTransGen r a b \u2192 Prop},\n    P b (_ : ReflTransGen r b b) \u2192\n      (\u2200 {a c : \u03b1} (h' : r a c) (h : ReflTransGen r c b), P c h \u2192 P a (_ : ReflTransGen r a b)) \u2192 P a a\u271d\nP : (a : \u03b1) \u2192 ReflTransGen r a c \u2192 Prop\nrefl : P c (_ : ReflTransGen r c c)\nhead : \u2200 {a c_1 : \u03b1} (h' : r a c_1) (h : ReflTransGen r c_1 c), P c_1 h \u2192 P a (_ : ReflTransGen r a c)\n\u22a2 (fun {a} hab => P a (_ : ReflTransGen r a c)) (_ : ReflTransGen r b b)\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d\u00b9 b\u271d c\u271d d a b c : \u03b1\na\u271d : ReflTransGen r a b\nhbc : r b c\nih :\n  \u2200 {P : (a : \u03b1) \u2192 ReflTransGen r a b \u2192 Prop},\n    P b (_ : ReflTransGen r b b) \u2192\n      (\u2200 {a c : \u03b1} (h' : r a c) (h : ReflTransGen r c b), P c h \u2192 P a (_ : ReflTransGen r a b)) \u2192 P a a\u271d\nP : (a : \u03b1) \u2192 ReflTransGen r a c \u2192 Prop\nrefl : P c (_ : ReflTransGen r c c)\nhead : \u2200 {a c_1 : \u03b1} (h' : r a c_1) (h : ReflTransGen r c_1 c), P c_1 h \u2192 P a (_ : ReflTransGen r a c)\n\u22a2 \u2200 {a c_1 : \u03b1} (h' : r a c_1) (h : ReflTransGen r c_1 b),\n    (fun {a} hab => P a (_ : ReflTransGen r a c)) h \u2192\n      (fun {a} hab => P a (_ : ReflTransGen r a c)) (_ : ReflTransGen r a b)"}, {"tactic": "{ exact head hbc _ refl }", "annotated_tactic": ["{ exact head hbc _ refl }", []], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d\u00b9 b\u271d c\u271d d a b c : \u03b1\na\u271d : ReflTransGen r a b\nhbc : r b c\nih :\n  \u2200 {P : (a : \u03b1) \u2192 ReflTransGen r a b \u2192 Prop},\n    P b (_ : ReflTransGen r b b) \u2192\n      (\u2200 {a c : \u03b1} (h' : r a c) (h : ReflTransGen r c b), P c h \u2192 P a (_ : ReflTransGen r a b)) \u2192 P a a\u271d\nP : (a : \u03b1) \u2192 ReflTransGen r a c \u2192 Prop\nrefl : P c (_ : ReflTransGen r c c)\nhead : \u2200 {a c_1 : \u03b1} (h' : r a c_1) (h : ReflTransGen r c_1 c), P c_1 h \u2192 P a (_ : ReflTransGen r a c)\n\u22a2 (fun {a} hab => P a (_ : ReflTransGen r a c)) (_ : ReflTransGen r b b)\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d\u00b9 b\u271d c\u271d d a b c : \u03b1\na\u271d : ReflTransGen r a b\nhbc : r b c\nih :\n  \u2200 {P : (a : \u03b1) \u2192 ReflTransGen r a b \u2192 Prop},\n    P b (_ : ReflTransGen r b b) \u2192\n      (\u2200 {a c : \u03b1} (h' : r a c) (h : ReflTransGen r c b), P c h \u2192 P a (_ : ReflTransGen r a b)) \u2192 P a a\u271d\nP : (a : \u03b1) \u2192 ReflTransGen r a c \u2192 Prop\nrefl : P c (_ : ReflTransGen r c c)\nhead : \u2200 {a c_1 : \u03b1} (h' : r a c_1) (h : ReflTransGen r c_1 c), P c_1 h \u2192 P a (_ : ReflTransGen r a c)\n\u22a2 \u2200 {a c_1 : \u03b1} (h' : r a c_1) (h : ReflTransGen r c_1 b),\n    (fun {a} hab => P a (_ : ReflTransGen r a c)) h \u2192\n      (fun {a} hab => P a (_ : ReflTransGen r a c)) (_ : ReflTransGen r a b)", "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d\u00b9 b\u271d c\u271d d a b c : \u03b1\na\u271d : ReflTransGen r a b\nhbc : r b c\nih :\n  \u2200 {P : (a : \u03b1) \u2192 ReflTransGen r a b \u2192 Prop},\n    P b (_ : ReflTransGen r b b) \u2192\n      (\u2200 {a c : \u03b1} (h' : r a c) (h : ReflTransGen r c b), P c h \u2192 P a (_ : ReflTransGen r a b)) \u2192 P a a\u271d\nP : (a : \u03b1) \u2192 ReflTransGen r a c \u2192 Prop\nrefl : P c (_ : ReflTransGen r c c)\nhead : \u2200 {a c_1 : \u03b1} (h' : r a c_1) (h : ReflTransGen r c_1 c), P c_1 h \u2192 P a (_ : ReflTransGen r a c)\n\u22a2 \u2200 {a c_1 : \u03b1} (h' : r a c_1) (h : ReflTransGen r c_1 b),\n    (fun {a} hab => P a (_ : ReflTransGen r a c)) h \u2192\n      (fun {a} hab => P a (_ : ReflTransGen r a c)) (_ : ReflTransGen r a b)"}, {"tactic": "{ exact fun h1 h2 \u21a6 head h1 (h2.tail hbc) }", "annotated_tactic": ["{ exact fun h1 h2 \u21a6 head h1 (h2.tail hbc) }", []], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d\u00b9 b\u271d c\u271d d a b c : \u03b1\na\u271d : ReflTransGen r a b\nhbc : r b c\nih :\n  \u2200 {P : (a : \u03b1) \u2192 ReflTransGen r a b \u2192 Prop},\n    P b (_ : ReflTransGen r b b) \u2192\n      (\u2200 {a c : \u03b1} (h' : r a c) (h : ReflTransGen r c b), P c h \u2192 P a (_ : ReflTransGen r a b)) \u2192 P a a\u271d\nP : (a : \u03b1) \u2192 ReflTransGen r a c \u2192 Prop\nrefl : P c (_ : ReflTransGen r c c)\nhead : \u2200 {a c_1 : \u03b1} (h' : r a c_1) (h : ReflTransGen r c_1 c), P c_1 h \u2192 P a (_ : ReflTransGen r a c)\n\u22a2 \u2200 {a c_1 : \u03b1} (h' : r a c_1) (h : ReflTransGen r c_1 b),\n    (fun {a} hab => P a (_ : ReflTransGen r a c)) h \u2192\n      (fun {a} hab => P a (_ : ReflTransGen r a c)) (_ : ReflTransGen r a b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/Chain.lean", "full_name": "List.Chain'.take", "start": [327, 1], "end": [328, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Group.lean", "full_name": "Set.pairwise_disjoint_Ico_mul_zpow", "start": [184, 1], "end": [195, 26], "traced_tactics": [{"tactic": "simp_rw [Function.onFun, Set.disjoint_iff]", "annotated_tactic": ["simp_rw [<a>Function.onFun</a>, <a>Set.disjoint_iff</a>]", [{"full_name": "Function.onFun", "def_path": "Mathlib/Init/Function.lean", "def_pos": [49, 5], "def_end_pos": [49, 10]}, {"full_name": "Set.disjoint_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1534, 19], "def_end_pos": [1534, 31]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\n\u22a2 Pairwise (Disjoint on fun n => Ico (a * b ^ n) (a * b ^ (n + 1)))", "state_after": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\n\u22a2 Pairwise fun x y => Ico (a * b ^ x) (a * b ^ (x + 1)) \u2229 Ico (a * b ^ y) (a * b ^ (y + 1)) \u2286 \u2205"}, {"tactic": "intro m n hmn x hx", "annotated_tactic": ["intro m n hmn x hx", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\n\u22a2 Pairwise fun x y => Ico (a * b ^ x) (a * b ^ (x + 1)) \u2229 Ico (a * b ^ y) (a * b ^ (y + 1)) \u2286 \u2205", "state_after": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1))\n\u22a2 x \u2208 \u2205"}, {"tactic": "apply hmn", "annotated_tactic": ["apply hmn", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1))\n\u22a2 x \u2208 \u2205", "state_after": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1))\n\u22a2 m = n"}, {"tactic": "have hb : 1 < b := by\n  have : a * b ^ m < a * b ^ (m + 1) := hx.1.1.trans_lt hx.1.2\n  rwa [mul_lt_mul_iff_left, \u2190 mul_one (b ^ m), zpow_add_one, mul_lt_mul_iff_left] at this", "annotated_tactic": ["have hb : 1 < b := by\n    have : a * b ^ m < a * b ^ (m + 1) := hx.1.1.<a>trans_lt</a> hx.1.2\n    rwa [<a>mul_lt_mul_iff_left</a>, \u2190 <a>mul_one</a> (b ^ m), <a>zpow_add_one</a>, <a>mul_lt_mul_iff_left</a>] at this", [{"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}, {"full_name": "mul_lt_mul_iff_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [105, 9], "def_end_pos": [105, 28]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "zpow_add_one", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [196, 9], "def_end_pos": [196, 21]}, {"full_name": "mul_lt_mul_iff_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [105, 9], "def_end_pos": [105, 28]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1))\n\u22a2 m = n", "state_after": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1))\nhb : 1 < b\n\u22a2 m = n"}, {"tactic": "have i1 := hx.1.1.trans_lt hx.2.2", "annotated_tactic": ["have i1 := hx.1.1.<a>trans_lt</a> hx.2.2", [{"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1))\nhb : 1 < b\n\u22a2 m = n", "state_after": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1))\nhb : 1 < b\ni1 : a * b ^ m < a * b ^ (n + 1)\n\u22a2 m = n"}, {"tactic": "have i2 := hx.2.1.trans_lt hx.1.2", "annotated_tactic": ["have i2 := hx.2.1.<a>trans_lt</a> hx.1.2", [{"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1))\nhb : 1 < b\ni1 : a * b ^ m < a * b ^ (n + 1)\n\u22a2 m = n", "state_after": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1))\nhb : 1 < b\ni1 : a * b ^ m < a * b ^ (n + 1)\ni2 : a * b ^ n < a * b ^ (m + 1)\n\u22a2 m = n"}, {"tactic": "rw [mul_lt_mul_iff_left, zpow_lt_zpow_iff hb, Int.lt_add_one_iff] at i1 i2", "annotated_tactic": ["rw [<a>mul_lt_mul_iff_left</a>, <a>zpow_lt_zpow_iff</a> hb, <a>Int.lt_add_one_iff</a>] at i1 i2", [{"full_name": "mul_lt_mul_iff_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [105, 9], "def_end_pos": [105, 28]}, {"full_name": "zpow_lt_zpow_iff", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [365, 9], "def_end_pos": [365, 25]}, {"full_name": "Int.lt_add_one_iff", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [851, 9], "def_end_pos": [851, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1))\nhb : 1 < b\ni1 : a * b ^ m < a * b ^ (n + 1)\ni2 : a * b ^ n < a * b ^ (m + 1)\n\u22a2 m = n", "state_after": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1))\nhb : 1 < b\ni1 : m \u2264 n\ni2 : n \u2264 m\n\u22a2 m = n"}, {"tactic": "exact le_antisymm i1 i2", "annotated_tactic": ["exact <a>le_antisymm</a> i1 i2", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1))\nhb : 1 < b\ni1 : m \u2264 n\ni2 : n \u2264 m\n\u22a2 m = n", "state_after": "no goals"}, {"tactic": "have : a * b ^ m < a * b ^ (m + 1) := hx.1.1.trans_lt hx.1.2", "annotated_tactic": ["have : a * b ^ m < a * b ^ (m + 1) := hx.1.1.<a>trans_lt</a> hx.1.2", [{"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [124, 7], "def_end_pos": [124, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1))\n\u22a2 1 < b", "state_after": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1))\nthis : a * b ^ m < a * b ^ (m + 1)\n\u22a2 1 < b"}, {"tactic": "rwa [mul_lt_mul_iff_left, \u2190 mul_one (b ^ m), zpow_add_one, mul_lt_mul_iff_left] at this", "annotated_tactic": ["rwa [<a>mul_lt_mul_iff_left</a>, \u2190 <a>mul_one</a> (b ^ m), <a>zpow_add_one</a>, <a>mul_lt_mul_iff_left</a>] at this", [{"full_name": "mul_lt_mul_iff_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [105, 9], "def_end_pos": [105, 28]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "zpow_add_one", "def_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "def_pos": [196, 9], "def_end_pos": [196, 21]}, {"full_name": "mul_lt_mul_iff_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [105, 9], "def_end_pos": [105, 28]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : OrderedCommGroup \u03b1\na b : \u03b1\nm n : \u2124\nhmn : m \u2260 n\nx : \u03b1\nhx : x \u2208 Ico (a * b ^ m) (a * b ^ (m + 1)) \u2229 Ico (a * b ^ n) (a * b ^ (n + 1))\nthis : a * b ^ m < a * b ^ (m + 1)\n\u22a2 1 < b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.symmDiff_def", "start": [2120, 11], "end": [2121, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/NormedSpace/lpSpace.lean", "full_name": "lp.uniformContinuous_coe", "start": [1116, 1], "end": [1127, 26], "traced_tactics": [{"tactic": "have hp : p \u2260 0 := (zero_lt_one.trans_le _i.elim).ne'", "annotated_tactic": ["have hp : p \u2260 0 := (zero_lt_one.trans_le _i.elim).<a>ne'</a>", [{"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\n\u22a2 UniformContinuous Subtype.val", "state_after": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\nhp : p \u2260 0\n\u22a2 UniformContinuous Subtype.val"}, {"tactic": "rw [uniformContinuous_pi]", "annotated_tactic": ["rw [<a>uniformContinuous_pi</a>]", [{"full_name": "uniformContinuous_pi", "def_path": "Mathlib/Topology/UniformSpace/Pi.lean", "def_pos": [45, 9], "def_end_pos": [45, 29]}]], "state_before": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\nhp : p \u2260 0\n\u22a2 UniformContinuous Subtype.val", "state_after": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\nhp : p \u2260 0\n\u22a2 \u2200 (i : \u03b1), UniformContinuous fun x => \u2191x i"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\nhp : p \u2260 0\n\u22a2 \u2200 (i : \u03b1), UniformContinuous fun x => \u2191x i", "state_after": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\nhp : p \u2260 0\ni : \u03b1\n\u22a2 UniformContinuous fun x => \u2191x i"}, {"tactic": "rw [NormedAddCommGroup.uniformity_basis_dist.uniformContinuous_iff\n  NormedAddCommGroup.uniformity_basis_dist]", "annotated_tactic": ["rw [NormedAddCommGroup.uniformity_basis_dist.uniformContinuous_iff\n    <a>NormedAddCommGroup.uniformity_basis_dist</a>]", [{"full_name": "NormedAddCommGroup.uniformity_basis_dist", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [792, 3], "def_end_pos": [792, 14]}]], "state_before": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\nhp : p \u2260 0\ni : \u03b1\n\u22a2 UniformContinuous fun x => \u2191x i", "state_after": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\nhp : p \u2260 0\ni : \u03b1\n\u22a2 \u2200 (i_1 : \u211d),\n    0 < i_1 \u2192\n      \u2203 j,\n        0 < j \u2227\n          \u2200 (x y : { x // x \u2208 lp E p }), (x, y) \u2208 {p_1 | \u2016p_1.1 - p_1.2\u2016 < j} \u2192 (\u2191x i, \u2191y i) \u2208 {p | \u2016p.1 - p.2\u2016 < i_1}"}, {"tactic": "intro \u03b5 h\u03b5", "annotated_tactic": ["intro \u03b5 h\u03b5", []], "state_before": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\nhp : p \u2260 0\ni : \u03b1\n\u22a2 \u2200 (i_1 : \u211d),\n    0 < i_1 \u2192\n      \u2203 j,\n        0 < j \u2227\n          \u2200 (x y : { x // x \u2208 lp E p }), (x, y) \u2208 {p_1 | \u2016p_1.1 - p_1.2\u2016 < j} \u2192 (\u2191x i, \u2191y i) \u2208 {p | \u2016p.1 - p.2\u2016 < i_1}", "state_after": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\nhp : p \u2260 0\ni : \u03b1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 j,\n    0 < j \u2227 \u2200 (x y : { x // x \u2208 lp E p }), (x, y) \u2208 {p_1 | \u2016p_1.1 - p_1.2\u2016 < j} \u2192 (\u2191x i, \u2191y i) \u2208 {p | \u2016p.1 - p.2\u2016 < \u03b5}"}, {"tactic": "refine' \u27e8\u03b5, h\u03b5, _\u27e9", "annotated_tactic": ["refine' \u27e8\u03b5, h\u03b5, _\u27e9", []], "state_before": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\nhp : p \u2260 0\ni : \u03b1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 j,\n    0 < j \u2227 \u2200 (x y : { x // x \u2208 lp E p }), (x, y) \u2208 {p_1 | \u2016p_1.1 - p_1.2\u2016 < j} \u2192 (\u2191x i, \u2191y i) \u2208 {p | \u2016p.1 - p.2\u2016 < \u03b5}", "state_after": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\nhp : p \u2260 0\ni : \u03b1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2200 (x y : { x // x \u2208 lp E p }), (x, y) \u2208 {p_1 | \u2016p_1.1 - p_1.2\u2016 < \u03b5} \u2192 (\u2191x i, \u2191y i) \u2208 {p | \u2016p.1 - p.2\u2016 < \u03b5}"}, {"tactic": "rintro f g (hfg : \u2016f - g\u2016 < \u03b5)", "annotated_tactic": ["rintro f g (hfg : \u2016f - g\u2016 < \u03b5)", []], "state_before": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\nhp : p \u2260 0\ni : \u03b1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2200 (x y : { x // x \u2208 lp E p }), (x, y) \u2208 {p_1 | \u2016p_1.1 - p_1.2\u2016 < \u03b5} \u2192 (\u2191x i, \u2191y i) \u2208 {p | \u2016p.1 - p.2\u2016 < \u03b5}", "state_after": "\u03b1 : Type u_1\nE : \u03b1 \u2192 Type u_2\np q : \u211d\u22650\u221e\ninst\u271d : (i : \u03b1) \u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\nhp : p \u2260 0\ni : \u03b1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nf g : { x // x \u2208 lp E p }\nhfg : \u2016f - g\u2016 < \u03b5\n\u22a2 (\u2191f i, \u2191g i) \u2208 {p | \u2016p.1 - p.2\u2016 < \u03b5}"}, {"tactic": "have : \u2016f i - g i\u2016 \u2264 \u2016f - g\u2016 := norm_apply_le_norm hp (f - g) i", "annotated_tactic": ["have : \u2016f i - g i\u2016 \u2264 \u2016f - g\u2016 := <a>norm_apply_le_norm</a> hp (f - g) i", [{"full_name": "lp.norm_apply_le_norm", "def_path": "Mathlib/Analysis/NormedSpace/lpSpace.lean", "def_pos": [559, 9], "def_end_pos": [559, 27]}]], "state_before": "\u03b1 : Type u_1\nE : \u03b1 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\u2192 NormedAddCommGroup (E i)\n_i : Fact (1 \u2264 p)\nhp : p \u2260 0\ni : \u03b1\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nf g : { x // x \u2208 lp E p }\nhfg : \u2016f - g\u2016 < \u03b5\nthis : \u2016\u2191f i - \u2191g i\u2016 \u2264 \u2016f - g\u2016\n\u22a2 (\u2191f i, \u2191g i) \u2208 {p | \u2016p.1 - p.2\u2016 < \u03b5}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Manifold/ContMDiff.lean", "full_name": "Smooth.smul", "start": [2028, 8], "end": [2030, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/WittVector/Basic.lean", "full_name": "WittVector.ghostFun_nat_cast", "start": [195, 9], "end": [198, 86], "traced_tactics": [{"tactic": "induction i <;>\n  simp [*, Nat.unaryCast, ghostFun_zero, ghostFun_one, ghostFun_add, 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"annotated_tactic": ["rw [<a>image_subset_iff</a>, e.image_eq_preimage]", [{"full_name": "Set.image_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [497, 9], "def_end_pos": [497, 25]}]], "state_before": "\u03b1\u271d : Sort u\n\u03b2\u271d : Sort v\n\u03b3 : Sort w\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ne : \u03b1 \u2243 \u03b2\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 \u2191e.symm '' t \u2286 s \u2194 t \u2286 \u2191e '' s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.countP_eq_countP_filter_add", "start": [2279, 1], "end": [2283, 25], "traced_tactics": [{"tactic": "simp [countP_filter]", "annotated_tactic": ["simp [<a>countP_filter</a>]", [{"full_name": "Multiset.countP_filter", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2275, 9], "def_end_pos": [2275, 22]}]], "state_before": "case 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[555, 98], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "R : Type u\nS : Type v\na b c d : R\nn\u271d m : \u2115\ninst\u271d : Ring R\nn : \u2124\n\u22a2 natDegree \u2191n \u2264 Zero.zero", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/PDeriv.lean", "full_name": "MvPolynomial.pderiv_C_mul", "start": [123, 1], "end": [124, 43], "traced_tactics": [{"tactic": "rw [C_mul', Derivation.map_smul, C_mul']", "annotated_tactic": ["rw [<a>C_mul'</a>, <a>Derivation.map_smul</a>, <a>C_mul'</a>]", [{"full_name": "MvPolynomial.C_mul'", "def_path": "Mathlib/Data/MvPolynomial/Basic.lean", "def_pos": [276, 9], "def_end_pos": [276, 15]}, {"full_name": "Derivation.map_smul", "def_path": "Mathlib/RingTheory/Derivation/Basic.lean", "def_pos": [120, 9], "def_end_pos": [120, 17]}, {"full_name": "MvPolynomial.C_mul'", "def_path": 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"state_after": "\u03b1 : Type u\n\u03b2 : Type v\na : \u03b1\nf : Filter \u03b1\nt : TopologicalSpace \u03b1\n\u22a2 nhdsAdjoint a f \u2264 t \u2194 \u2200 (s : Set \u03b1), a \u2208 s \u2192 IsOpen s \u2192 s \u2208 f"}, {"tactic": "exact \u27e8fun H s hs has => H _ has hs, fun H s has hs => H _ hs has\u27e9", "annotated_tactic": ["exact \u27e8fun H s hs has => H _ has hs, fun H s has hs => H _ hs has\u27e9", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\na : \u03b1\nf : Filter \u03b1\nt : TopologicalSpace \u03b1\n\u22a2 nhdsAdjoint a f \u2264 t \u2194 \u2200 (s : Set \u03b1), a \u2208 s \u2192 IsOpen s \u2192 s \u2208 f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/AlgebraicGeometry/Morphisms/OpenImmersion.lean", "full_name": "AlgebraicGeometry.isOpenImmersion_iff_stalk", "start": [34, 1], "end": [38, 61], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "X Y Z : Scheme\nf\u271d : X \u27f6 Y\ng : Y \u27f6 Z\nf : X \u27f6 Y\n\u22a2 IsOpenImmersion f \u2194\n    OpenEmbedding \u2191f.val.base \u2227 \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x)", "state_after": "case mp\nX Y Z : Scheme\nf\u271d : X \u27f6 Y\ng : Y \u27f6 Z\nf : X \u27f6 Y\n\u22a2 IsOpenImmersion f \u2192\n    OpenEmbedding \u2191f.val.base \u2227 \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x)\n\ncase mpr\nX Y Z : Scheme\nf\u271d : X \u27f6 Y\ng : Y \u27f6 Z\nf : X \u27f6 Y\n\u22a2 (OpenEmbedding \u2191f.val.base \u2227 \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x)) \u2192\n    IsOpenImmersion f"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mp\nX Y Z : Scheme\nf\u271d : X \u27f6 Y\ng : Y \u27f6 Z\nf : X \u27f6 Y\n\u22a2 IsOpenImmersion f \u2192\n    OpenEmbedding \u2191f.val.base \u2227 \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x)", "state_after": "case mp\nX Y Z : Scheme\nf\u271d : X \u27f6 Y\ng : Y \u27f6 Z\nf : X \u27f6 Y\nh : IsOpenImmersion f\n\u22a2 OpenEmbedding \u2191f.val.base \u2227 \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x)"}, {"tactic": "exact \u27e8h.1, inferInstance\u27e9", "annotated_tactic": ["exact \u27e8h.1, <a>inferInstance</a>\u27e9", [{"full_name": "inferInstance", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [86, 8], "def_end_pos": [86, 21]}]], "state_before": "case mp\nX Y Z : Scheme\nf\u271d : X \u27f6 Y\ng : Y \u27f6 Z\nf : X \u27f6 Y\nh : IsOpenImmersion f\n\u22a2 OpenEmbedding \u2191f.val.base \u2227 \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x)", "state_after": "no goals"}, {"tactic": "rintro \u27e8h\u2081, h\u2082\u27e9", "annotated_tactic": ["rintro \u27e8h\u2081, h\u2082\u27e9", []], "state_before": "case mpr\nX Y Z : Scheme\nf\u271d : X \u27f6 Y\ng : Y \u27f6 Z\nf : X \u27f6 Y\n\u22a2 (OpenEmbedding \u2191f.val.base \u2227 \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x)) \u2192\n    IsOpenImmersion f", "state_after": "case mpr.intro\nX Y Z : Scheme\nf\u271d : X \u27f6 Y\ng : Y \u27f6 Z\nf : X \u27f6 Y\nh\u2081 : OpenEmbedding \u2191f.val.base\nh\u2082 : \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x)\n\u22a2 IsOpenImmersion f"}, {"tactic": "exact IsOpenImmersion.of_stalk_iso f h\u2081", "annotated_tactic": ["exact <a>IsOpenImmersion.of_stalk_iso</a> f h\u2081", [{"full_name": "AlgebraicGeometry.IsOpenImmersion.of_stalk_iso", "def_path": "Mathlib/AlgebraicGeometry/OpenImmersion.lean", "def_pos": [417, 9], "def_end_pos": [417, 21]}]], "state_before": "case mpr.intro\nX Y Z : Scheme\nf\u271d : X \u27f6 Y\ng : Y \u27f6 Z\nf : X \u27f6 Y\nh\u2081 : OpenEmbedding \u2191f.val.base\nh\u2082 : \u2200 (x : \u2191\u2191X.toPresheafedSpace), IsIso (PresheafedSpace.stalkMap f.val x)\n\u22a2 IsOpenImmersion f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Quaternion.lean", "full_name": "Quaternion.mul_coe_eq_smul", "start": [1098, 1], "end": [1099, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Sum/Lemmas.lean", "full_name": "Sum.isRight_eq_false", "start": [79, 9], "end": [79, 99], "traced_tactics": [{"tactic": "cases x <;> simp", "annotated_tactic": ["cases x <;> simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nx : \u03b1 \u2295 \u03b2\n\u22a2 isRight x = false \u2194 isLeft x = true", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Compactness/Compact.lean", "full_name": "LocallyFinite.finite_nonempty_inter_compact", "start": [235, 1], "end": [242, 39], "traced_tactics": [{"tactic": "choose U hxU hUf using hf", "annotated_tactic": ["choose U hxU hUf using hf", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 Set \u03b1\nhf : LocallyFinite f\ns : Set \u03b1\nhs : IsCompact s\n\u22a2 Set.Finite {i | Set.Nonempty (f i \u2229 s)}", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsCompact s\nU : \u03b1 \u2192 Set \u03b1\nhxU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x\nhUf : \u2200 (x : \u03b1), Set.Finite {i | Set.Nonempty (f i \u2229 U x)}\n\u22a2 Set.Finite {i | Set.Nonempty (f i \u2229 s)}"}, {"tactic": "rcases hs.elim_nhds_subcover U fun x _ => hxU x with \u27e8t, -, hsU\u27e9", "annotated_tactic": ["rcases hs.elim_nhds_subcover U fun x _ => hxU x with \u27e8t, -, hsU\u27e9", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsCompact s\nU : \u03b1 \u2192 Set \u03b1\nhxU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x\nhUf : \u2200 (x : \u03b1), Set.Finite {i | Set.Nonempty (f i \u2229 U x)}\n\u22a2 Set.Finite {i | Set.Nonempty (f i \u2229 s)}", "state_after": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsCompact s\nU : \u03b1 \u2192 Set \u03b1\nhxU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x\nhUf : \u2200 (x : \u03b1), Set.Finite {i | Set.Nonempty (f i \u2229 U x)}\nt : Finset \u03b1\nhsU : s \u2286 \u22c3 x \u2208 t, U x\n\u22a2 Set.Finite {i | Set.Nonempty (f i \u2229 s)}"}, {"tactic": "refine' (t.finite_toSet.biUnion fun x _ => hUf x).subset _", "annotated_tactic": ["refine' (t.finite_toSet.biUnion fun x _ => hUf x).<a>subset</a> _", [{"full_name": "Set.Finite.subset", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [771, 9], "def_end_pos": [771, 22]}]], "state_before": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsCompact s\nU : \u03b1 \u2192 Set \u03b1\nhxU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x\nhUf : \u2200 (x : \u03b1), Set.Finite {i | Set.Nonempty (f i \u2229 U x)}\nt : Finset \u03b1\nhsU : s \u2286 \u22c3 x \u2208 t, U x\n\u22a2 Set.Finite {i | Set.Nonempty (f i \u2229 s)}", "state_after": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsCompact s\nU : \u03b1 \u2192 Set \u03b1\nhxU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x\nhUf : \u2200 (x : \u03b1), Set.Finite {i | Set.Nonempty (f i \u2229 U x)}\nt : Finset \u03b1\nhsU : s \u2286 \u22c3 x \u2208 t, U x\n\u22a2 {i | Set.Nonempty (f i \u2229 s)} \u2286 \u22c3 i \u2208 \u2191t, {i_1 | Set.Nonempty (f i_1 \u2229 U i)}"}, {"tactic": "rintro i \u27e8x, hx\u27e9", "annotated_tactic": ["rintro i \u27e8x, hx\u27e9", []], "state_before": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsCompact s\nU : \u03b1 \u2192 Set \u03b1\nhxU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x\nhUf : \u2200 (x : \u03b1), Set.Finite {i | Set.Nonempty (f i \u2229 U x)}\nt : Finset \u03b1\nhsU : s \u2286 \u22c3 x \u2208 t, U x\n\u22a2 {i | Set.Nonempty (f i \u2229 s)} \u2286 \u22c3 i \u2208 \u2191t, {i_1 | Set.Nonempty (f i_1 \u2229 U i)}", "state_after": "case intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsCompact s\nU : \u03b1 \u2192 Set \u03b1\nhxU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x\nhUf : \u2200 (x : \u03b1), Set.Finite {i | Set.Nonempty (f i \u2229 U x)}\nt : Finset \u03b1\nhsU : s \u2286 \u22c3 x \u2208 t, U x\ni : \u03b9\nx : \u03b1\nhx : x \u2208 f i \u2229 s\n\u22a2 i \u2208 \u22c3 i \u2208 \u2191t, {i_1 | Set.Nonempty (f i_1 \u2229 U i)}"}, {"tactic": "rcases mem_iUnion\u2082.1 (hsU hx.2) with \u27e8c, hct, hcx\u27e9", "annotated_tactic": ["rcases <a>mem_iUnion\u2082</a>.1 (hsU hx.2) with \u27e8c, hct, hcx\u27e9", [{"full_name": "Set.mem_iUnion\u2082", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [212, 9], "def_end_pos": [212, 20]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsCompact s\nU : \u03b1 \u2192 Set \u03b1\nhxU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x\nhUf : \u2200 (x : \u03b1), Set.Finite {i | Set.Nonempty (f i \u2229 U x)}\nt : Finset \u03b1\nhsU : s \u2286 \u22c3 x \u2208 t, U x\ni : \u03b9\nx : \u03b1\nhx : x \u2208 f i \u2229 s\n\u22a2 i \u2208 \u22c3 i \u2208 \u2191t, {i_1 | Set.Nonempty (f i_1 \u2229 U i)}", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsCompact s\nU : \u03b1 \u2192 Set \u03b1\nhxU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x\nhUf : \u2200 (x : \u03b1), Set.Finite {i | Set.Nonempty (f i \u2229 U x)}\nt : Finset \u03b1\nhsU : s \u2286 \u22c3 x \u2208 t, U x\ni : \u03b9\nx : \u03b1\nhx : x \u2208 f i \u2229 s\nc : \u03b1\nhct : c \u2208 t\nhcx : x \u2208 U c\n\u22a2 i \u2208 \u22c3 i \u2208 \u2191t, {i_1 | Set.Nonempty (f i_1 \u2229 U i)}"}, {"tactic": "exact mem_biUnion hct \u27e8x, hx.1, hcx\u27e9", "annotated_tactic": ["exact <a>mem_biUnion</a> hct \u27e8x, hx.1, hcx\u27e9", [{"full_name": "Set.mem_biUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [966, 9], "def_end_pos": [966, 20]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\n\u03b9 : Type u_3\nf : \u03b9 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsCompact s\nU : \u03b1 \u2192 Set \u03b1\nhxU : \u2200 (x : \u03b1), U x \u2208 \ud835\udcdd x\nhUf : \u2200 (x : \u03b1), Set.Finite {i | Set.Nonempty (f i \u2229 U x)}\nt : Finset \u03b1\nhsU : s \u2286 \u22c3 x \u2208 t, U x\ni : \u03b9\nx : \u03b1\nhx : x \u2208 f i \u2229 s\nc : \u03b1\nhct : c \u2208 t\nhcx : x \u2208 U c\n\u22a2 i \u2208 \u22c3 i \u2208 \u2191t, {i_1 | Set.Nonempty (f i_1 \u2229 U i)}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Category/ModuleCat/ChangeOfRings.lean", "full_name": "ModuleCat.ExtendScalars.map'_id", "start": [240, 1], "end": [249, 29], "traced_tactics": [{"tactic": "dsimp only [map']", "annotated_tactic": ["dsimp only [<a>map'</a>]", [{"full_name": "ModuleCat.ExtendScalars.map'", "def_path": "Mathlib/Algebra/Category/ModuleCat/ChangeOfRings.lean", "def_pos": [235, 5], "def_end_pos": [235, 9]}]], "state_before": "R : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM\u271d M : ModuleCat R\nx : \u2191(obj' f M)\n\u22a2 \u2191(map' f (\ud835\udfd9 M)) x = \u2191(\ud835\udfd9 (obj' f M)) x", "state_after": "R : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM\u271d M : ModuleCat R\nx : \u2191(obj' f M)\n\u22a2 \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) x = \u2191(\ud835\udfd9 (obj' f M)) x"}, {"tactic": "rw [ModuleCat.id_apply]", "annotated_tactic": ["rw [<a>ModuleCat.id_apply</a>]", [{"full_name": "ModuleCat.id_apply", "def_path": "Mathlib/Algebra/Category/ModuleCat/Basic.lean", "def_pos": [218, 9], "def_end_pos": [218, 17]}]], "state_before": "R : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM\u271d M : ModuleCat R\nx : \u2191(obj' f M)\n\u22a2 \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) x = \u2191(\ud835\udfd9 (obj' f M)) x", "state_after": "R : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM\u271d M : ModuleCat R\nx : \u2191(obj' f M)\n\u22a2 \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) x = x"}, {"tactic": "induction' x using TensorProduct.induction_on with _ _ m s ihx ihy", "annotated_tactic": ["induction' x using <a>TensorProduct.induction_on</a> with _ _ m s ihx ihy", [{"full_name": "TensorProduct.induction_on", "def_path": "Mathlib/LinearAlgebra/TensorProduct.lean", "def_pos": [136, 19], "def_end_pos": [136, 31]}]], "state_before": "R : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM\u271d M : ModuleCat R\nx : \u2191(obj' f M)\n\u22a2 \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) x = x", "state_after": "case zero\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM\u271d M : ModuleCat R\n\u22a2 \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) 0 = 0\n\ncase tmul\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM\u271d M : ModuleCat R\nx\u271d : \u2191((restrictScalars f).obj (mk S))\ny\u271d : \u2191M\n\u22a2 \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) (x\u271d \u2297\u209c[R] y\u271d) = x\u271d \u2297\u209c[R] y\u271d\n\ncase add\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM\u271d M : ModuleCat R\nm s : \u2191((restrictScalars f).obj (mk S)) \u2297[R] \u2191M\nihx : \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) m = m\nihy : \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) s = s\n\u22a2 \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) (m + s) = m + s"}, {"tactic": "rw [map_zero]", "annotated_tactic": ["rw [<a>map_zero</a>]", [{"full_name": "map_zero", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [201, 3], "def_end_pos": [201, 14]}]], "state_before": "case zero\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM\u271d M : ModuleCat R\n\u22a2 \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) 0 = 0", "state_after": "no goals"}, {"tactic": "erw [@LinearMap.baseChange_tmul R S M M _ _ (_), ModuleCat.id_apply]", "annotated_tactic": ["erw [@<a>LinearMap.baseChange_tmul</a> R S M M _ _ (_), <a>ModuleCat.id_apply</a>]", [{"full_name": "LinearMap.baseChange_tmul", "def_path": "Mathlib/RingTheory/TensorProduct.lean", "def_pos": [75, 9], "def_end_pos": [75, 24]}, {"full_name": "ModuleCat.id_apply", "def_path": "Mathlib/Algebra/Category/ModuleCat/Basic.lean", "def_pos": [218, 9], "def_end_pos": [218, 17]}]], "state_before": "case tmul\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM\u271d M : ModuleCat R\nx\u271d : \u2191((restrictScalars f).obj (mk S))\ny\u271d : \u2191M\n\u22a2 \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) (x\u271d \u2297\u209c[R] y\u271d) = x\u271d \u2297\u209c[R] y\u271d", "state_after": "no goals"}, {"tactic": "rw [map_add, ihx, ihy]", "annotated_tactic": ["rw [<a>map_add</a>, ihx, ihy]", [{"full_name": "map_add", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [298, 3], "def_end_pos": [298, 14]}]], "state_before": "case add\nR : Type u\u2081\nS : Type u\u2082\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nM\u271d M : ModuleCat R\nm s : \u2191((restrictScalars f).obj (mk S)) \u2297[R] \u2191M\nihx : \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) m = m\nihy : \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) s = s\n\u22a2 \u2191(LinearMap.baseChange S (\ud835\udfd9 M)) (m + s) = m + s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecificLimits/Normed.lean", "full_name": "tendsto_pow_atTop_nhds_0_of_abs_lt_1", "start": [270, 1], "end": [272, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "full_name": "Subgroup.sup_eq_closure", "start": [1268, 1], "end": [1269, 41], "traced_tactics": [{"tactic": "simp_rw [closure_union, closure_eq]", "annotated_tactic": ["simp_rw [<a>closure_union</a>, <a>closure_eq</a>]", [{"full_name": "Subgroup.closure_union", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [1262, 9], "def_end_pos": [1262, 22]}, {"full_name": "Subgroup.closure_eq", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [1244, 9], "def_end_pos": [1244, 19]}]], "state_before": "G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group G'\ninst\u271d\u00b9 : Group G''\nA : Type u_4\ninst\u271d : AddGroup A\nH\u271d K : Subgroup G\nk : Set G\nH H' : Subgroup G\n\u22a2 H \u2294 H' = closure (\u2191H \u222a \u2191H')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/GroupAction/ConjAct.lean", "full_name": "ConjAct.toConjAct_mul", "start": [140, 1], "end": [141, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Subobject/Comma.lean", "full_name": "CategoryTheory.StructuredArrow.projectSubobject_factors", "start": [63, 1], "end": [68, 27], "traced_tactics": [{"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b3 : Category.{v, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v, u\u2082} D\nS : D\nT : C \u2964 D\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : PreservesLimits T\nA P : StructuredArrow S T\nf : P \u27f6 A\nhf : Mono f\n\u22a2 (P.hom \u226b T.map (Subobject.underlyingIso (MonoOver.arrow (MonoOver.mk' f)).right).inv) \u226b\n      T.map (Subobject.arrow (projectSubobject (Subobject.mk f))) =\n    A.hom", "state_after": "C : Type u\u2081\ninst\u271d\u00b3 : Category.{v, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v, u\u2082} D\nS : D\nT : C \u2964 D\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : PreservesLimits T\nA P : StructuredArrow S T\nf : P \u27f6 A\nhf : Mono f\n\u22a2 (P.hom \u226b T.map (Subobject.underlyingIso f.right).inv) \u226b T.map (Subobject.arrow (Subobject.mk f.right)) = A.hom"}, {"tactic": "simp [\u2190 T.map_comp]", "annotated_tactic": ["simp [\u2190 T.map_comp]", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b3 : Category.{v, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v, u\u2082} D\nS : D\nT : C \u2964 D\ninst\u271d\u00b9 : HasLimits C\ninst\u271d : PreservesLimits T\nA P : StructuredArrow S T\nf : P \u27f6 A\nhf : Mono f\n\u22a2 (P.hom \u226b T.map (Subobject.underlyingIso f.right).inv) \u226b T.map (Subobject.arrow (Subobject.mk f.right)) = A.hom", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/UniformSpace/Basic.lean", "full_name": "uniformContinuous_sInf_dom", "start": [1399, 1], "end": [1404, 33], "traced_tactics": [{"tactic": "delta UniformContinuous", "annotated_tactic": ["delta <a>UniformContinuous</a>", [{"full_name": "UniformContinuous", "def_path": "Mathlib/Topology/UniformSpace/Basic.lean", "def_pos": [1094, 5], "def_end_pos": [1094, 22]}]], "state_before": "\u03b1 : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort u_1\nf : \u03b1 \u2192 \u03b2\nu\u2081 : Set (UniformSpace \u03b1)\nu\u2082 : UniformSpace \u03b2\nu : UniformSpace \u03b1\nh\u2081 : u \u2208 u\u2081\nhf : UniformContinuous f\n\u22a2 UniformContinuous f", "state_after": "\u03b1 : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort u_1\nf : \u03b1 \u2192 \u03b2\nu\u2081 : Set (UniformSpace \u03b1)\nu\u2082 : UniformSpace \u03b2\nu : UniformSpace \u03b1\nh\u2081 : u \u2208 u\u2081\nhf : UniformContinuous f\n\u22a2 Tendsto (fun x => (f x.1, f x.2)) (\ud835\udce4 \u03b1) (\ud835\udce4 \u03b2)"}, {"tactic": "rw [sInf_eq_iInf', iInf_uniformity]", "annotated_tactic": ["rw [<a>sInf_eq_iInf'</a>, <a>iInf_uniformity</a>]", [{"full_name": "sInf_eq_iInf'", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [753, 9], "def_end_pos": [753, 22]}, {"full_name": "iInf_uniformity", "def_path": "Mathlib/Topology/UniformSpace/Basic.lean", "def_pos": [1235, 9], "def_end_pos": [1235, 24]}]], "state_before": "\u03b1 : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort u_1\nf : \u03b1 \u2192 \u03b2\nu\u2081 : Set (UniformSpace \u03b1)\nu\u2082 : UniformSpace \u03b2\nu : UniformSpace \u03b1\nh\u2081 : u \u2208 u\u2081\nhf : UniformContinuous f\n\u22a2 Tendsto (fun x => (f x.1, f x.2)) (\ud835\udce4 \u03b1) (\ud835\udce4 \u03b2)", "state_after": "\u03b1 : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort u_1\nf : \u03b1 \u2192 \u03b2\nu\u2081 : Set (UniformSpace \u03b1)\nu\u2082 : UniformSpace \u03b2\nu : UniformSpace \u03b1\nh\u2081 : u \u2208 u\u2081\nhf : UniformContinuous f\n\u22a2 Tendsto (fun x => (f x.1, f x.2)) (\u2a05 i, \ud835\udce4 \u03b1) (\ud835\udce4 \u03b2)"}, {"tactic": "exact tendsto_iInf' \u27e8u, h\u2081\u27e9 hf", "annotated_tactic": ["exact <a>tendsto_iInf'</a> \u27e8u, h\u2081\u27e9 hf", [{"full_name": "Filter.tendsto_iInf'", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3126, 9], "def_end_pos": [3126, 22]}]], "state_before": "\u03b1 : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort u_1\nf : \u03b1 \u2192 \u03b2\nu\u2081 : Set (UniformSpace \u03b1)\nu\u2082 : UniformSpace \u03b2\nu : UniformSpace \u03b1\nh\u2081 : u \u2208 u\u2081\nhf : UniformContinuous f\n\u22a2 Tendsto (fun x => (f x.1, f x.2)) (\u2a05 i, \ud835\udce4 \u03b1) (\ud835\udce4 \u03b2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Periodic.lean", "full_name": "Function.Antiperiodic.int_odd_mul_antiperiodic", "start": [399, 1], "end": [401, 42], "traced_tactics": [{"tactic": "rw [\u2190 add_assoc, h, h.periodic.int_mul]", "annotated_tactic": ["rw [\u2190 <a>add_assoc</a>, h, h.periodic.int_mul]", [{"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [263, 3], "def_end_pos": [263, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf g : \u03b1 \u2192 \u03b2\nc c\u2081 c\u2082 x\u271d : \u03b1\ninst\u271d\u00b9 : Ring \u03b1\ninst\u271d : InvolutiveNeg \u03b2\nh : Antiperiodic f c\nn : \u2124\nx : \u03b1\n\u22a2 f (x + (\u2191n * (2 * c) + c)) = -f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Subobject/FactorThru.lean", "full_name": "CategoryTheory.MonoOver.factors_congr", "start": [42, 1], "end": [45, 62], "traced_tactics": [{"tactic": "simp [hu]", "annotated_tactic": ["simp [hu]", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX : C\nf g : MonoOver X\nY : C\nh : Y \u27f6 X\ne : f \u2245 g\nx\u271d : Factors f h\nu : Y \u27f6 f.obj.left\nhu : u \u226b arrow f = h\n\u22a2 (u \u226b ((forget X).map e.hom).left) \u226b arrow g = h", "state_after": "no goals"}, {"tactic": "simp [hu]", "annotated_tactic": ["simp [hu]", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nX\u271d Y\u271d Z : C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nX : C\nf g : MonoOver X\nY : C\nh : Y \u27f6 X\ne : f \u2245 g\nx\u271d : Factors g h\nu : Y \u27f6 g.obj.left\nhu : u \u226b arrow g = h\n\u22a2 (u \u226b ((forget X).map e.inv).left) \u226b arrow f = h", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Logic/Function/Conjugate.lean", "full_name": "Function.Commute.comp_right", "start": [112, 1], "end": [113, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Germ.lean", "full_name": "Filter.Germ.le_def", "start": [720, 1], "end": [721, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Process/HittingTime.lean", "full_name": "MeasureTheory.hitting_eq_hitting_of_exists", "start": [195, 1], "end": [208, 41], "traced_tactics": [{"tactic": "simp only [hitting, if_pos h']", "annotated_tactic": ["simp only [<a>hitting</a>, <a>if_pos</a> h']", [{"full_name": "MeasureTheory.hitting", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [51, 19], "def_end_pos": [51, 26]}, {"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}]], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nh' : \u2203 j, j \u2208 Set.Icc n m\u2081 \u2227 u j \u03c9 \u2208 s\n\u22a2 hitting u s n m\u2081 \u03c9 = hitting u s n m\u2082 \u03c9", "state_after": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nh' : \u2203 j, j \u2208 Set.Icc n m\u2081 \u2227 u j \u03c9 \u2208 s\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) =\n    if \u2203 j, j \u2208 Set.Icc n m\u2082 \u2227 u j \u03c9 \u2208 s then sInf (Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s}) else m\u2082"}, {"tactic": "obtain \u27e8j, hj\u2081, hj\u2082\u27e9 := h'", "annotated_tactic": ["obtain \u27e8j, hj\u2081, hj\u2082\u27e9 := h'", []], "state_before": "\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nh' : \u2203 j, j \u2208 Set.Icc n m\u2081 \u2227 u j \u03c9 \u2208 s\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) =\n    if \u2203 j, j \u2208 Set.Icc n m\u2082 \u2227 u j \u03c9 \u2208 s then sInf (Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s}) else m\u2082", "state_after": "case intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\nhj\u2082 : u j \u03c9 \u2208 s\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) =\n    if \u2203 j, j \u2208 Set.Icc n m\u2082 \u2227 u j \u03c9 \u2208 s then sInf (Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s}) else m\u2082"}, {"tactic": "rw [if_pos]", "annotated_tactic": ["rw [<a>if_pos</a>]", [{"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}]], "state_before": "case intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\nhj\u2082 : u j \u03c9 \u2208 s\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) =\n    if \u2203 j, j \u2208 Set.Icc n m\u2082 \u2227 u j \u03c9 \u2208 s then sInf (Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s}) else m\u2082", "state_after": "case intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\nhj\u2082 : u j \u03c9 \u2208 s\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) = sInf (Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s})\n\ncase intro.intro.hc\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\nhj\u2082 : u j \u03c9 \u2208 s\n\u22a2 \u2203 j, j \u2208 Set.Icc n m\u2082 \u2227 u j \u03c9 \u2208 s"}, {"tactic": "exact \u27e8j, \u27e8hj\u2081.1, hj\u2081.2.trans h\u27e9, hj\u2082\u27e9", "annotated_tactic": ["exact \u27e8j, \u27e8hj\u2081.1, hj\u2081.2.<a>trans</a> h\u27e9, hj\u2082\u27e9", [{"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "case intro.intro.hc\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\nhj\u2082 : u j \u03c9 \u2208 s\n\u22a2 \u2203 j, j \u2208 Set.Icc n m\u2082 \u2227 u j \u03c9 \u2208 s", "state_after": "no goals"}, {"tactic": "refine' le_antisymm _ (csInf_le_csInf bddBelow_Icc.inter_of_left \u27e8j, hj\u2081, hj\u2082\u27e9\n  (Set.inter_subset_inter_left _ (Set.Icc_subset_Icc_right h)))", "annotated_tactic": ["refine' <a>le_antisymm</a> _ (<a>csInf_le_csInf</a> bddBelow_Icc.inter_of_left \u27e8j, hj\u2081, hj\u2082\u27e9\n      (<a>Set.inter_subset_inter_left</a> _ (<a>Set.Icc_subset_Icc_right</a> h)))", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "csInf_le_csInf", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [485, 9], "def_end_pos": [485, 23]}, {"full_name": "Set.inter_subset_inter_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1027, 9], "def_end_pos": [1027, 32]}, {"full_name": "Set.Icc_subset_Icc_right", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [463, 9], "def_end_pos": [463, 29]}]], "state_before": "case intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\nhj\u2082 : u j \u03c9 \u2208 s\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) = sInf (Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s})", "state_after": "case intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\nhj\u2082 : u j \u03c9 \u2208 s\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) \u2264 sInf (Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s})"}, {"tactic": "refine' le_csInf \u27e8j, Set.Icc_subset_Icc_right h hj\u2081, hj\u2082\u27e9 fun i hi => _", "annotated_tactic": ["refine' <a>le_csInf</a> \u27e8j, <a>Set.Icc_subset_Icc_right</a> h hj\u2081, hj\u2082\u27e9 fun i hi => _", [{"full_name": "le_csInf", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [469, 9], "def_end_pos": [469, 17]}, {"full_name": "Set.Icc_subset_Icc_right", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [463, 9], "def_end_pos": [463, 29]}]], "state_before": "case intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\nhj\u2082 : u j \u03c9 \u2208 s\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) \u2264 sInf (Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s})", "state_after": "case intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\nhj\u2082 : u j \u03c9 \u2208 s\ni : \u03b9\nhi : i \u2208 Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s}\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) \u2264 i"}, {"tactic": "by_cases hi' : i \u2264 m\u2081", "annotated_tactic": ["by_cases hi' : i \u2264 m\u2081", []], "state_before": "case intro.intro\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\nhj\u2082 : u j \u03c9 \u2208 s\ni : \u03b9\nhi : i \u2208 Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s}\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) \u2264 i", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\nhj\u2082 : u j \u03c9 \u2208 s\ni : \u03b9\nhi : i \u2208 Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s}\nhi' : i \u2264 m\u2081\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) \u2264 i\n\ncase neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\nhj\u2082 : u j \u03c9 \u2208 s\ni : \u03b9\nhi : i \u2208 Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s}\nhi' : \u00aci \u2264 m\u2081\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) \u2264 i"}, {"tactic": "exact csInf_le bddBelow_Icc.inter_of_left \u27e8\u27e8hi.1.1, hi'\u27e9, hi.2\u27e9", "annotated_tactic": ["exact <a>csInf_le</a> bddBelow_Icc.inter_of_left \u27e8\u27e8hi.1.1, hi'\u27e9, hi.2\u27e9", [{"full_name": "csInf_le", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [465, 9], "def_end_pos": [465, 17]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\nhj\u2082 : u j \u03c9 \u2208 s\ni : \u03b9\nhi : i \u2208 Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s}\nhi' : i \u2264 m\u2081\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) \u2264 i", "state_after": "no goals"}, {"tactic": "change j \u2208 {i | u i \u03c9 \u2208 s} at hj\u2082", "annotated_tactic": ["change j \u2208 {i | u i \u03c9 \u2208 s} at hj\u2082", []], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\nhj\u2082 : u j \u03c9 \u2208 s\ni : \u03b9\nhi : i \u2208 Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s}\nhi' : \u00aci \u2264 m\u2081\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) \u2264 i", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\ni : \u03b9\nhi : i \u2208 Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s}\nhi' : \u00aci \u2264 m\u2081\nhj\u2082 : j \u2208 {i | u i \u03c9 \u2208 s}\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) \u2264 i"}, {"tactic": "exact ((csInf_le bddBelow_Icc.inter_of_left \u27e8hj\u2081, hj\u2082\u27e9).trans (hj\u2081.2.trans le_rfl)).trans\n  (le_of_lt (not_le.1 hi'))", "annotated_tactic": ["exact ((<a>csInf_le</a> bddBelow_Icc.inter_of_left \u27e8hj\u2081, hj\u2082\u27e9).<a>trans</a> (hj\u2081.2.<a>trans</a> <a>le_rfl</a>)).<a>trans</a>\n        (<a>le_of_lt</a> (<a>not_le</a>.1 hi'))", [{"full_name": "csInf_le", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [465, 9], "def_end_pos": [465, 17]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03a9\ninst\u271d : ConditionallyCompleteLinearOrder \u03b9\nu : \u03b9 \u2192 \u03a9 \u2192 \u03b2\ns : Set \u03b2\nn i\u271d : \u03b9\n\u03c9 : \u03a9\nm\u2081 m\u2082 : \u03b9\nh : m\u2081 \u2264 m\u2082\nj : \u03b9\nhj\u2081 : j \u2208 Set.Icc n m\u2081\ni : \u03b9\nhi : i \u2208 Set.Icc n m\u2082 \u2229 {i | u i \u03c9 \u2208 s}\nhi' : \u00aci \u2264 m\u2081\nhj\u2082 : j \u2208 {i | u i \u03c9 \u2208 s}\n\u22a2 sInf (Set.Icc n m\u2081 \u2229 {i | u i \u03c9 \u2208 s}) \u2264 i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Bases.lean", "full_name": "Filter.HasBasis.inf_basis_neBot_iff", "start": [629, 1], "end": [631, 63], "traced_tactics": [{"tactic": "simp [@forall_swap _ \u03b9']", "annotated_tactic": ["simp [@<a>forall_swap</a> _ \u03b9']", [{"full_name": "forall_swap", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [667, 9], "def_end_pos": [667, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\n\u22a2 (\u2200 {i : PProd \u03b9 \u03b9'}, p i.fst \u2227 p' i.snd \u2192 Set.Nonempty (s i.fst \u2229 s' i.snd)) \u2194\n    \u2200 \u2983i : \u03b9\u2984, p i \u2192 \u2200 \u2983i' : \u03b9'\u2984, p' i' \u2192 Set.Nonempty (s i \u2229 s' i')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/GaloisConnection.lean", "full_name": "GaloisConnection.l_iSup", "start": [285, 1], "end": [289, 77], "traced_tactics": [{"tactic": "rw [range_comp, \u2190 sSup_range]", "annotated_tactic": ["rw [<a>range_comp</a>, \u2190 <a>sSup_range</a>]", [{"full_name": "Set.range_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [744, 9], "def_end_pos": [744, 19]}, {"full_name": "sSup_range", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [685, 9], "def_end_pos": [685, 19]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : CompleteLattice \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\nf : \u03b9 \u2192 \u03b1\n\u22a2 IsLUB (range (l \u2218 f)) (l (iSup f))", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : CompleteLattice \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\nf : \u03b9 \u2192 \u03b1\n\u22a2 IsLUB (l '' range f) (l (sSup (range f)))"}, {"tactic": "exact gc.isLUB_l_image (isLUB_sSup _)", "annotated_tactic": ["exact gc.isLUB_l_image (<a>isLUB_sSup</a> _)", [{"full_name": "isLUB_sSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [204, 9], "def_end_pos": [204, 19]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : CompleteLattice \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\nf : \u03b9 \u2192 \u03b1\n\u22a2 IsLUB (l '' range f) (l (sSup (range f)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/HausdorffDimension.lean", "full_name": "Real.dimH_ball_pi_fin", "start": [529, 1], "end": [530, 78], "traced_tactics": [{"tactic": "rw [dimH_ball_pi x hr, Fintype.card_fin]", "annotated_tactic": ["rw [<a>dimH_ball_pi</a> x hr, <a>Fintype.card_fin</a>]", [{"full_name": "Real.dimH_ball_pi", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDimension.lean", "def_pos": [515, 9], "def_end_pos": [515, 21]}, {"full_name": "Fintype.card_fin", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [308, 9], "def_end_pos": [308, 25]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\nC K r\u271d : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nE : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nn : \u2115\nx : Fin n \u2192 \u211d\nr : \u211d\nhr : 0 < r\n\u22a2 dimH (Metric.ball x r) = \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.eqOn_comp_right_iff", "start": [910, 1], "end": [911, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Field/Basic.lean", "full_name": "one_le_one_div", "start": [483, 1], "end": [484, 62], "traced_tactics": [{"tactic": "rwa [le_one_div (@zero_lt_one \u03b1 _ _ _ _ _) h1, one_div_one]", "annotated_tactic": ["rwa [<a>le_one_div</a> (@<a>zero_lt_one</a> \u03b1 _ _ _ _ _) h1, <a>one_div_one</a>]", [{"full_name": "le_one_div", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [440, 9], "def_end_pos": [440, 19]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}, {"full_name": "one_div_one", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [345, 9], "def_end_pos": [345, 20]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nh1 : 0 < a\nh2 : a \u2264 1\n\u22a2 1 \u2264 1 / a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Range/Lemmas.lean", "full_name": "Std.Range.numElems_stop_le_start", "start": [14, 1], "end": [18, 26], "traced_tactics": [{"tactic": "simp [numElems]", "annotated_tactic": ["simp [<a>numElems</a>]", [{"full_name": "Std.Range.numElems", "def_path": "lake-packages/std/Std/Data/Range/Lemmas.lean", "def_pos": [7, 5], "def_end_pos": [7, 13]}]], "state_before": "start stop step : Nat\nh : { start := start, stop := stop, step := step }.stop \u2264 { start := start, stop := stop, step := step }.start\n\u22a2 numElems { start := start, stop := stop, step := step } = 0", "state_after": "start stop step : Nat\nh : { start := start, stop := stop, step := step }.stop \u2264 { start := start, stop := stop, step := step }.start\n\u22a2 (if step = 0 then if stop \u2264 start then 0 else stop else (stop - start + step - 1) / step) = 0"}, {"tactic": "split <;> simp_all", "annotated_tactic": ["split <;> simp_all", []], "state_before": "start stop step : Nat\nh : { start := start, stop := stop, step := step }.stop \u2264 { start := start, stop := stop, step := step }.start\n\u22a2 (if step = 0 then if stop \u2264 start then 0 else stop else (stop - start + step - 1) / step) = 0", "state_after": "case inr\nstart stop step : Nat\nh : stop \u2264 start\nh\u271d : \u00acstep = 0\n\u22a2 (stop - start + step - 1) / step = 0"}, {"tactic": "apply Nat.div_eq_of_lt", "annotated_tactic": ["apply <a>Nat.div_eq_of_lt</a>", [{"full_name": "Nat.div_eq_of_lt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [590, 9], "def_end_pos": [590, 21]}]], "state_before": "case inr\nstart stop step : Nat\nh : stop \u2264 start\nh\u271d : \u00acstep = 0\n\u22a2 (stop - start + step - 1) / step = 0", "state_after": "case inr.h\u2080\nstart stop step : Nat\nh : stop \u2264 start\nh\u271d : \u00acstep = 0\n\u22a2 stop - start + step - 1 < step"}, {"tactic": "simp [Nat.sub_eq_zero_of_le h]", "annotated_tactic": ["simp [<a>Nat.sub_eq_zero_of_le</a> h]", [{"full_name": "Nat.sub_eq_zero_of_le", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [638, 19], "def_end_pos": [638, 36]}]], "state_before": "case inr.h\u2080\nstart stop step : Nat\nh : stop \u2264 start\nh\u271d : \u00acstep = 0\n\u22a2 stop - start + step - 1 < step", "state_after": "case inr.h\u2080\nstart stop step : Nat\nh : stop \u2264 start\nh\u271d : \u00acstep = 0\n\u22a2 step - 1 < step"}, {"tactic": "exact Nat.pred_lt \u2039_\u203a", "annotated_tactic": ["exact <a>Nat.pred_lt</a> \u2039_\u203a", [{"full_name": "Nat.pred_lt", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [227, 9], "def_end_pos": [227, 16]}]], "state_before": "case inr.h\u2080\nstart stop step : Nat\nh : stop \u2264 start\nh\u271d : \u00acstep = 0\n\u22a2 step - 1 < step", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean", "full_name": "one_add_mul_self_lt_rpow_one_add", "start": [188, 1], "end": [217, 16], "traced_tactics": [{"tactic": "rcases eq_or_lt_of_le hs with (rfl | hs)", "annotated_tactic": ["rcases <a>eq_or_lt_of_le</a> hs with (rfl | hs)", [{"full_name": "eq_or_lt_of_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [414, 9], "def_end_pos": [414, 23]}]], "state_before": "s : \u211d\nhs : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\n\u22a2 1 + p * s < (1 + s) ^ p", "state_after": "case inl\np : \u211d\nhp : 1 < p\nhs : -1 \u2264 -1\nhs' : -1 \u2260 0\n\u22a2 1 + p * -1 < (1 + -1) ^ p\n\ncase inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\nhs : -1 < s\n\u22a2 1 + p * s < (1 + s) ^ p"}, {"tactic": "have hs1 : 0 < 1 + s := by linarith", "annotated_tactic": ["have hs1 : 0 < 1 + s := by linarith", []], "state_before": "case inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\nhs : -1 < s\n\u22a2 1 + p * s < (1 + s) ^ p", "state_after": "case inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\n\u22a2 1 + p * s < (1 + s) ^ p"}, {"tactic": "cases' le_or_lt (1 + p * s) 0 with hs2 hs2", "annotated_tactic": ["cases' <a>le_or_lt</a> (1 + p * s) 0 with hs2 hs2", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [340, 9], "def_end_pos": [340, 17]}]], "state_before": "case inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\n\u22a2 1 + p * s < (1 + s) ^ p", "state_after": "case inr.inl\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 1 + p * s \u2264 0\n\u22a2 1 + p * s < (1 + s) ^ p\n\ncase inr.inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\n\u22a2 1 + p * s < (1 + s) ^ p"}, {"tactic": "rw [rpow_def_of_pos hs1, \u2190 exp_log hs2]", "annotated_tactic": ["rw [<a>rpow_def_of_pos</a> hs1, \u2190 <a>exp_log</a> hs2]", [{"full_name": "Real.rpow_def_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [54, 9], "def_end_pos": [54, 24]}, {"full_name": "Real.exp_log", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [61, 9], "def_end_pos": [61, 16]}]], "state_before": "case inr.inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\n\u22a2 1 + p * s < (1 + s) ^ p", "state_after": "case inr.inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\n\u22a2 rexp (log (1 + p * s)) < rexp (log (1 + s) * p)"}, {"tactic": "apply exp_strictMono", "annotated_tactic": ["apply <a>exp_strictMono</a>", [{"full_name": "Real.exp_strictMono", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1511, 9], "def_end_pos": [1511, 23]}]], "state_before": "case inr.inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\n\u22a2 rexp (log (1 + p * s)) < rexp (log (1 + s) * p)", "state_after": "case inr.inr.a\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\n\u22a2 log (1 + p * s) < log (1 + s) * p"}, {"tactic": "have hp : 0 < p := by positivity", "annotated_tactic": ["have hp : 0 < p := by positivity", []], "state_before": "case inr.inr.a\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\n\u22a2 log (1 + p * s) < log (1 + s) * p", "state_after": "case inr.inr.a\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\n\u22a2 log (1 + p * s) < log (1 + s) * p"}, {"tactic": "have hs3 : 1 + s \u2260 1 := by contrapose! hs'; linarith", "annotated_tactic": ["have hs3 : 1 + s \u2260 1 := by contrapose! hs'; linarith", []], "state_before": "case inr.inr.a\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\n\u22a2 log (1 + p * s) < log (1 + s) * p", "state_after": "case inr.inr.a\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\n\u22a2 log (1 + p * s) < log (1 + s) * p"}, {"tactic": "have hs4 : 1 + p * s \u2260 1 := by contrapose! hs'; nlinarith", "annotated_tactic": ["have hs4 : 1 + p * s \u2260 1 := by contrapose! hs'; nlinarith", []], "state_before": "case inr.inr.a\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\n\u22a2 log (1 + p * s) < log (1 + s) * p", "state_after": "case inr.inr.a\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\n\u22a2 log (1 + p * s) < log (1 + s) * p"}, {"tactic": "cases' lt_or_gt_of_ne hs' with hs' hs'", "annotated_tactic": ["cases' <a>lt_or_gt_of_ne</a> hs' with hs' hs'", [{"full_name": "lt_or_gt_of_ne", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [352, 9], "def_end_pos": [352, 23]}]], "state_before": "case inr.inr.a\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\n\u22a2 log (1 + p * s) < log (1 + s) * p", "state_after": "case inr.inr.a.inl\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s < 0\n\u22a2 log (1 + p * s) < log (1 + s) * p\n\ncase inr.inr.a.inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s > 0\n\u22a2 log (1 + p * s) < log (1 + s) * p"}, {"tactic": "have : p \u2260 0 := by positivity", "annotated_tactic": ["have : p \u2260 0 := by positivity", []], "state_before": "case inl\np : \u211d\nhp : 1 < p\nhs : -1 \u2264 -1\nhs' : -1 \u2260 0\n\u22a2 1 + p * -1 < (1 + -1) ^ p", "state_after": "case inl\np : \u211d\nhp : 1 < p\nhs : -1 \u2264 -1\nhs' : -1 \u2260 0\nthis : p \u2260 0\n\u22a2 1 + p * -1 < (1 + -1) ^ p"}, {"tactic": "simpa [zero_rpow this]", "annotated_tactic": ["simpa [<a>zero_rpow</a> this]", [{"full_name": "Real.zero_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [103, 9], "def_end_pos": [103, 18]}]], "state_before": "case inl\np : \u211d\nhp : 1 < p\nhs : -1 \u2264 -1\nhs' : -1 \u2260 0\nthis : p \u2260 0\n\u22a2 1 + p * -1 < (1 + -1) ^ p", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "p : \u211d\nhp : 1 < p\nhs : -1 \u2264 -1\nhs' : -1 \u2260 0\n\u22a2 p \u2260 0", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "s : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\nhs : -1 < s\n\u22a2 0 < 1 + s", "state_after": "no goals"}, {"tactic": "exact hs2.trans_lt (rpow_pos_of_pos hs1 _)", "annotated_tactic": ["exact hs2.trans_lt (<a>rpow_pos_of_pos</a> hs1 _)", [{"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}]], "state_before": "case inr.inl\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 1 + p * s \u2264 0\n\u22a2 1 + p * s < (1 + s) ^ p", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "s : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\n\u22a2 0 < p", "state_after": "no goals"}, {"tactic": "contrapose! hs'", "annotated_tactic": ["contrapose! hs'", []], "state_before": "s : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\n\u22a2 1 + s \u2260 1", "state_after": "s : \u211d\nhs\u271d : -1 \u2264 s\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs' : 1 + s = 1\n\u22a2 s = 0"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "s : \u211d\nhs\u271d : -1 \u2264 s\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs' : 1 + s = 1\n\u22a2 s = 0", "state_after": "no goals"}, {"tactic": "contrapose! hs'", "annotated_tactic": ["contrapose! hs'", []], "state_before": "s : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\n\u22a2 1 + p * s \u2260 1", "state_after": "s : \u211d\nhs\u271d : -1 \u2264 s\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs' : 1 + p * s = 1\n\u22a2 s = 0"}, {"tactic": "nlinarith", "annotated_tactic": ["nlinarith", []], "state_before": "s : \u211d\nhs\u271d : -1 \u2264 s\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs' : 1 + p * s = 1\n\u22a2 s = 0", "state_after": "no goals"}, {"tactic": "rw [\u2190 div_lt_iff hp, \u2190 div_lt_div_right_of_neg hs']", "annotated_tactic": ["rw [\u2190 <a>div_lt_iff</a> hp, \u2190 <a>div_lt_div_right_of_neg</a> hs']", [{"full_name": "div_lt_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}, {"full_name": "div_lt_div_right_of_neg", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [798, 9], "def_end_pos": [798, 32]}]], "state_before": "case inr.inr.a.inl\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s < 0\n\u22a2 log (1 + p * s) < log (1 + s) * p", "state_after": "case inr.inr.a.inl\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s < 0\n\u22a2 log (1 + s) / s < log (1 + p * s) / p / s"}, {"tactic": "haveI : (1 : \u211d) \u2208 Ioi 0 := zero_lt_one", "annotated_tactic": ["haveI : (1 : \u211d) \u2208 <a>Ioi</a> 0 := <a>zero_lt_one</a>", [{"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "case inr.inr.a.inl\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s < 0\n\u22a2 log (1 + s) / s < log (1 + p * s) / p / s", "state_after": "case inr.inr.a.inl\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s < 0\nthis : 1 \u2208 Ioi 0\n\u22a2 log (1 + s) / s < log (1 + p * s) / p / s"}, {"tactic": "convert strictConcaveOn_log_Ioi.secant_strict_mono this hs2 hs1 hs4 hs3 _ using 1", "annotated_tactic": ["convert strictConcaveOn_log_Ioi.secant_strict_mono this hs2 hs1 hs4 hs3 _ using 1", []], "state_before": "case inr.inr.a.inl\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s < 0\nthis : 1 \u2208 Ioi 0\n\u22a2 log (1 + s) / s < log (1 + p * s) / p / s", "state_after": "case h.e'_3\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s < 0\nthis : 1 \u2208 Ioi 0\n\u22a2 log (1 + s) / s = (log (1 + s) - log 1) / (1 + s - 1)\n\ncase h.e'_4\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s < 0\nthis : 1 \u2208 Ioi 0\n\u22a2 log (1 + p * s) / p / s = (log (1 + p * s) - log 1) / (1 + p * s - 1)\n\ncase inr.inr.a.inl\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s < 0\nthis : 1 \u2208 Ioi 0\n\u22a2 1 + p * s < 1 + s"}, {"tactic": "field_simp", "annotated_tactic": ["field_simp", []], "state_before": "case h.e'_3\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s < 0\nthis : 1 \u2208 Ioi 0\n\u22a2 log (1 + s) / s = (log (1 + s) - log 1) / (1 + s - 1)", "state_after": "no goals"}, {"tactic": "field_simp", "annotated_tactic": ["field_simp", []], "state_before": "case h.e'_4\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s < 0\nthis : 1 \u2208 Ioi 0\n\u22a2 log (1 + p * s) / p / s = (log (1 + p * s) - log 1) / (1 + p * s - 1)", "state_after": "no goals"}, {"tactic": "nlinarith", "annotated_tactic": ["nlinarith", []], "state_before": "case inr.inr.a.inl\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s < 0\nthis : 1 \u2208 Ioi 0\n\u22a2 1 + p * s < 1 + s", "state_after": "no goals"}, {"tactic": "rw [\u2190 div_lt_iff hp, \u2190 div_lt_div_right hs']", "annotated_tactic": ["rw [\u2190 <a>div_lt_iff</a> hp, \u2190 <a>div_lt_div_right</a> hs']", [{"full_name": "div_lt_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}, {"full_name": "div_lt_div_right", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [367, 9], "def_end_pos": [367, 25]}]], "state_before": "case inr.inr.a.inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s > 0\n\u22a2 log (1 + p * s) < log (1 + s) * p", "state_after": "case inr.inr.a.inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s > 0\n\u22a2 log (1 + p * s) / p / s < log (1 + s) / s"}, {"tactic": "haveI : (1 : \u211d) \u2208 Ioi 0 := zero_lt_one", "annotated_tactic": ["haveI : (1 : \u211d) \u2208 <a>Ioi</a> 0 := <a>zero_lt_one</a>", [{"full_name": "Set.Ioi", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "case inr.inr.a.inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s > 0\n\u22a2 log (1 + p * s) / p / s < log (1 + s) / s", "state_after": "case inr.inr.a.inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s > 0\nthis : 1 \u2208 Ioi 0\n\u22a2 log (1 + p * s) / p / s < log (1 + s) / s"}, {"tactic": "convert strictConcaveOn_log_Ioi.secant_strict_mono this hs1 hs2 hs3 hs4 _ using 1", "annotated_tactic": ["convert strictConcaveOn_log_Ioi.secant_strict_mono this hs1 hs2 hs3 hs4 _ using 1", []], "state_before": "case inr.inr.a.inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s > 0\nthis : 1 \u2208 Ioi 0\n\u22a2 log (1 + p * s) / p / s < log (1 + s) / s", "state_after": "case h.e'_3\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s > 0\nthis : 1 \u2208 Ioi 0\n\u22a2 log (1 + p * s) / p / s = (log (1 + p * s) - log 1) / (1 + p * s - 1)\n\ncase h.e'_4\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s > 0\nthis : 1 \u2208 Ioi 0\n\u22a2 log (1 + s) / s = (log (1 + s) - log 1) / (1 + s - 1)\n\ncase inr.inr.a.inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s > 0\nthis : 1 \u2208 Ioi 0\n\u22a2 1 + s < 1 + p * s"}, {"tactic": "field_simp", "annotated_tactic": ["field_simp", []], "state_before": "case h.e'_3\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s > 0\nthis : 1 \u2208 Ioi 0\n\u22a2 log (1 + p * s) / p / s = (log (1 + p * s) - log 1) / (1 + p * s - 1)", "state_after": "no goals"}, {"tactic": "field_simp", "annotated_tactic": ["field_simp", []], "state_before": "case h.e'_4\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s > 0\nthis : 1 \u2208 Ioi 0\n\u22a2 log (1 + s) / s = (log (1 + s) - log 1) / (1 + s - 1)", "state_after": "no goals"}, {"tactic": "nlinarith", "annotated_tactic": ["nlinarith", []], "state_before": "case inr.inr.a.inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s > 0\nthis : 1 \u2208 Ioi 0\n\u22a2 1 + s < 1 + p * s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Rearrangement.lean", "full_name": "Monovary.sum_comp_perm_smul_le_sum_smul", "start": [267, 1], "end": [269, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Algebra/Spectrum.lean", "full_name": "spectrum.smul_mem_smul_iff", "start": [211, 1], "end": [213, 21], "traced_tactics": [{"tactic": "simp only [mem_iff, not_iff_not, Algebra.algebraMap_eq_smul_one, smul_assoc, \u2190 smul_sub,\n  isUnit_smul_iff]", "annotated_tactic": ["simp only [<a>mem_iff</a>, <a>not_iff_not</a>, <a>Algebra.algebraMap_eq_smul_one</a>, <a>smul_assoc</a>, \u2190 <a>smul_sub</a>,\n    <a>isUnit_smul_iff</a>]", [{"full_name": "spectrum.mem_iff", "def_path": "Mathlib/Algebra/Algebra/Spectrum.lean", "def_pos": [107, 9], "def_end_pos": [107, 16]}, {"full_name": "not_iff_not", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [439, 9], "def_end_pos": [439, 20]}, {"full_name": "Algebra.algebraMap_eq_smul_one", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [351, 9], "def_end_pos": [351, 31]}, {"full_name": "smul_assoc", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 19]}, {"full_name": "smul_sub", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [988, 9], "def_end_pos": [988, 17]}, {"full_name": "isUnit_smul_iff", "def_path": "Mathlib/GroupTheory/GroupAction/Group.lean", "def_pos": [412, 9], "def_end_pos": [412, 24]}]], "state_before": "R : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\ns : R\nr : R\u02e3\n\u22a2 r \u2022 s \u2208 \u03c3 (r \u2022 a) \u2194 s \u2208 \u03c3 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/ZFC/Basic.lean", "full_name": "Class.sInter_empty", "start": [1689, 1], "end": [1690, 57], "traced_tactics": [{"tactic": "rw [sInter, classToCong_empty, Set.sInter_empty, univ]", "annotated_tactic": ["rw [<a>sInter</a>, <a>classToCong_empty</a>, <a>Set.sInter_empty</a>, <a>univ</a>]", [{"full_name": "Class.sInter", "def_path": "Mathlib/SetTheory/ZFC/Basic.lean", "def_pos": [1561, 5], "def_end_pos": [1561, 11]}, {"full_name": "Class.classToCong_empty", "def_path": "Mathlib/SetTheory/ZFC/Basic.lean", "def_pos": [1542, 9], "def_end_pos": [1542, 26]}, {"full_name": "Set.sInter_empty", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1211, 9], "def_end_pos": [1211, 21]}, {"full_name": "Class.univ", "def_path": "Mathlib/SetTheory/ZFC/Basic.lean", "def_pos": [1440, 5], "def_end_pos": [1440, 9]}]], "state_before": "\u22a2 \u22c2\u2080 \u2205 = univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.constantCoeff_smul", "start": [909, 1], "end": [911, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "full_name": "AlgHom.range_comp", "start": [630, 1], "end": [631, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Functor/Const.lean", "full_name": "CategoryTheory.Functor.const.opObjUnop_hom_app", "start": [72, 1], "end": [73, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/LineDeriv/Basic.lean", "full_name": "HasLineDerivAt.smul", "start": [508, 1], "end": [511, 38], "traced_tactics": [{"tactic": "simp only [\u2190 hasLineDerivWithinAt_univ] at h \u22a2", "annotated_tactic": ["simp only [\u2190 <a>hasLineDerivWithinAt_univ</a>] at h \u22a2", [{"full_name": "hasLineDerivWithinAt_univ", "def_path": "Mathlib/Analysis/Calculus/LineDeriv/Basic.lean", "def_pos": [134, 15], "def_end_pos": [134, 40]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type u_3\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nf : E \u2192 F\ns : Set E\nx v : E\nf' : F\nh : HasLineDerivAt \ud835\udd5c f f' x v\nc : \ud835\udd5c\n\u22a2 HasLineDerivAt \ud835\udd5c f (c \u2022 f') x (c \u2022 v)", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type u_3\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nf : E \u2192 F\ns : Set E\nx v : E\nf' : F\nc : \ud835\udd5c\nh : HasLineDerivWithinAt \ud835\udd5c f f' univ x v\n\u22a2 HasLineDerivWithinAt \ud835\udd5c f (c \u2022 f') univ x (c \u2022 v)"}, {"tactic": "exact HasLineDerivWithinAt.smul h c", "annotated_tactic": ["exact <a>HasLineDerivWithinAt.smul</a> h c", [{"full_name": "HasLineDerivWithinAt.smul", "def_path": "Mathlib/Analysis/Calculus/LineDeriv/Basic.lean", "def_pos": [491, 9], "def_end_pos": [491, 34]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nE : Type u_3\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nf : E \u2192 F\ns : Set E\nx v : E\nf' : F\nc : \ud835\udd5c\nh : HasLineDerivWithinAt \ud835\udd5c f f' univ x v\n\u22a2 HasLineDerivWithinAt \ud835\udd5c f (c \u2022 f') univ x (c \u2022 v)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/AlgebraicTopology/SimplexCategory.lean", "full_name": "SimplexCategory.skeletal", "start": [369, 1], "end": [374, 72], "traced_tactics": [{"tactic": "suffices Fintype.card (Fin (X.len + 1)) = Fintype.card (Fin (Y.len + 1)) by\n  ext\n  simpa", "annotated_tactic": ["suffices <a>Fintype.card</a> (<a>Fin</a> (X.len + 1)) = <a>Fintype.card</a> (<a>Fin</a> (Y.len + 1)) by\n    ext\n    simpa", [{"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "Fin", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1745, 11], "def_end_pos": [1745, 14]}]], "state_before": "X Y : SimplexCategory\nx\u271d : IsIsomorphic X Y\nI : X \u2245 Y\n\u22a2 X = Y", "state_after": "X Y : SimplexCategory\nx\u271d : IsIsomorphic X Y\nI : X \u2245 Y\n\u22a2 Fintype.card (Fin (len X + 1)) = Fintype.card (Fin (len Y + 1))"}, {"tactic": "apply Fintype.card_congr", "annotated_tactic": ["apply <a>Fintype.card_congr</a>", [{"full_name": "Fintype.card_congr", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [149, 9], "def_end_pos": [149, 19]}]], "state_before": "X Y : SimplexCategory\nx\u271d : IsIsomorphic X Y\nI : X \u2245 Y\n\u22a2 Fintype.card (Fin (len X + 1)) = Fintype.card (Fin (len Y + 1))", "state_after": "case f\nX Y : SimplexCategory\nx\u271d : IsIsomorphic X Y\nI : X \u2245 Y\n\u22a2 Fin (len X + 1) \u2243 Fin (len Y + 1)"}, {"tactic": "exact ((skeletalFunctor \u22d9 forget NonemptyFinLinOrd).mapIso I).toEquiv", "annotated_tactic": ["exact ((<a>skeletalFunctor</a> \u22d9 <a>forget</a> <a>NonemptyFinLinOrd</a>).<a>mapIso</a> I).<a>toEquiv</a>", [{"full_name": "SimplexCategory.skeletalFunctor", "def_path": "Mathlib/AlgebraicTopology/SimplexCategory.lean", "def_pos": [359, 5], "def_end_pos": [359, 20]}, {"full_name": "CategoryTheory.forget", "def_path": "Mathlib/CategoryTheory/ConcreteCategory/Basic.lean", "def_pos": [68, 5], "def_end_pos": [68, 11]}, {"full_name": "NonemptyFinLinOrd", "def_path": "Mathlib/Order/Category/NonemptyFinLinOrd.lean", "def_pos": [57, 5], "def_end_pos": [57, 22]}, {"full_name": "CategoryTheory.Functor.mapIso", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [589, 5], "def_end_pos": [589, 11]}, {"full_name": "CategoryTheory.Iso.toEquiv", "def_path": "Mathlib/CategoryTheory/Types.lean", "def_pos": [344, 5], "def_end_pos": [344, 12]}]], "state_before": "case f\nX Y : SimplexCategory\nx\u271d : IsIsomorphic X Y\nI : X \u2245 Y\n\u22a2 Fin (len X + 1) \u2243 Fin (len Y + 1)", "state_after": "no goals"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "X Y : SimplexCategory\nx\u271d : IsIsomorphic X Y\nI : X \u2245 Y\nthis : Fintype.card (Fin (len X + 1)) = Fintype.card (Fin (len Y + 1))\n\u22a2 X = Y", "state_after": "case a\nX Y : SimplexCategory\nx\u271d : IsIsomorphic X Y\nI : X \u2245 Y\nthis : Fintype.card (Fin (len X + 1)) = Fintype.card (Fin (len Y + 1))\n\u22a2 len X = len Y"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "case a\nX Y : SimplexCategory\nx\u271d : IsIsomorphic X Y\nI : X \u2245 Y\nthis : Fintype.card (Fin (len X + 1)) = Fintype.card (Fin (len Y + 1))\n\u22a2 len X = len Y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "full_name": "Polynomial.coeff_pow_mul_natDegree", "start": [1075, 1], "end": [1094, 44], "traced_tactics": [{"tactic": "induction' n with i hi", "annotated_tactic": ["induction' n with i hi", []], "state_before": "R : Type u\nS : Type v\na b c d : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\nn : \u2115\n\u22a2 coeff (p ^ n) (n * natDegree p) = leadingCoeff p ^ n", "state_after": "case zero\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\n\u22a2 coeff (p ^ Nat.zero) (Nat.zero * natDegree p) = leadingCoeff p ^ Nat.zero\n\ncase succ\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\n\u22a2 coeff (p ^ Nat.succ i) (Nat.succ i * natDegree p) = leadingCoeff p ^ Nat.succ i"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case zero\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\n\u22a2 coeff (p ^ Nat.zero) (Nat.zero * natDegree p) = leadingCoeff p ^ Nat.zero", "state_after": "no goals"}, {"tactic": "rw [pow_succ', pow_succ', Nat.succ_mul]", "annotated_tactic": ["rw [<a>pow_succ'</a>, <a>pow_succ'</a>, <a>Nat.succ_mul</a>]", [{"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [213, 9], "def_end_pos": [213, 25]}, {"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [213, 9], "def_end_pos": [213, 25]}, {"full_name": "Nat.succ_mul", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [169, 9], "def_end_pos": [169, 17]}]], "state_before": "case succ\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\n\u22a2 coeff (p ^ Nat.succ i) (Nat.succ i * natDegree p) = leadingCoeff p ^ Nat.succ i", "state_after": "case succ\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\n\u22a2 coeff (p ^ i * p) (i * natDegree p + natDegree p) = leadingCoeff p ^ i * leadingCoeff p"}, {"tactic": "by_cases hp1 : p.leadingCoeff ^ i = 0", "annotated_tactic": ["by_cases hp1 : p.leadingCoeff ^ i = 0", []], "state_before": "case succ\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\n\u22a2 coeff (p ^ i * p) (i * natDegree p + natDegree p) = leadingCoeff p ^ i * leadingCoeff p", "state_after": "case pos\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\n\u22a2 coeff (p ^ i * p) (i * natDegree p + natDegree p) = leadingCoeff p ^ i * leadingCoeff p\n\ncase neg\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : \u00acleadingCoeff p ^ i = 0\n\u22a2 coeff (p ^ i * p) (i * natDegree p + natDegree p) = leadingCoeff p ^ i * leadingCoeff p"}, {"tactic": "rw [hp1, zero_mul]", "annotated_tactic": ["rw [hp1, <a>zero_mul</a>]", [{"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}]], "state_before": "case pos\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\n\u22a2 coeff (p ^ i * p) (i * natDegree p + natDegree p) = leadingCoeff p ^ i * leadingCoeff p", "state_after": "case pos\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\n\u22a2 coeff (p ^ i * p) (i * natDegree p + natDegree p) = 0"}, {"tactic": "by_cases hp2 : p ^ i = 0", "annotated_tactic": ["by_cases hp2 : p ^ i = 0", []], "state_before": "case pos\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\n\u22a2 coeff (p ^ i * p) (i * natDegree p + natDegree p) = 0", "state_after": "case pos\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\nhp2 : p ^ i = 0\n\u22a2 coeff (p ^ i * p) (i * natDegree p + natDegree p) = 0\n\ncase neg\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\nhp2 : \u00acp ^ i = 0\n\u22a2 coeff (p ^ i * p) (i * natDegree p + natDegree p) = 0"}, {"tactic": "rw [hp2, zero_mul, coeff_zero]", "annotated_tactic": ["rw [hp2, <a>zero_mul</a>, <a>coeff_zero</a>]", [{"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "Polynomial.coeff_zero", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [695, 9], "def_end_pos": [695, 19]}]], "state_before": "case pos\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\nhp2 : p ^ i = 0\n\u22a2 coeff (p ^ i * p) (i * natDegree p + natDegree p) = 0", "state_after": "no goals"}, {"tactic": "apply coeff_eq_zero_of_natDegree_lt", "annotated_tactic": ["apply <a>coeff_eq_zero_of_natDegree_lt</a>", [{"full_name": "Polynomial.coeff_eq_zero_of_natDegree_lt", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [349, 9], "def_end_pos": [349, 38]}]], "state_before": "case neg\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\nhp2 : \u00acp ^ i = 0\n\u22a2 coeff (p ^ i * p) (i * natDegree p + natDegree p) = 0", "state_after": "case neg.h\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\nhp2 : \u00acp ^ i = 0\n\u22a2 natDegree (p ^ i * p) < i * natDegree p + natDegree p"}, {"tactic": "have h1 : (p ^ i).natDegree < i * p.natDegree := by\n  refine lt_of_le_of_ne natDegree_pow_le fun h => hp2 ?_\n  rw [\u2190 h, hp1] at hi\n  exact leadingCoeff_eq_zero.mp hi", "annotated_tactic": ["have h1 : (p ^ i).<a>natDegree</a> < i * p.natDegree := by\n          refine <a>lt_of_le_of_ne</a> <a>natDegree_pow_le</a> fun h => hp2 ?_\n          rw [\u2190 h, hp1] at hi\n          exact leadingCoeff_eq_zero.mp hi", [{"full_name": "Polynomial.natDegree", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [65, 5], "def_end_pos": [65, 14]}, {"full_name": "lt_of_le_of_ne", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [196, 9], "def_end_pos": [196, 23]}, {"full_name": "Polynomial.natDegree_pow_le", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [1062, 9], "def_end_pos": [1062, 25]}]], "state_before": "case neg.h\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\nhp2 : \u00acp ^ i = 0\n\u22a2 natDegree (p ^ i * p) < i * natDegree p + natDegree p", "state_after": "case neg.h\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\nhp2 : \u00acp ^ i = 0\nh1 : natDegree (p ^ i) < i * natDegree p\n\u22a2 natDegree (p ^ i * p) < i * natDegree p + natDegree p"}, {"tactic": "calc\n  (p ^ i * p).natDegree \u2264 (p ^ i).natDegree + p.natDegree := natDegree_mul_le\n  _ < i * p.natDegree + p.natDegree := add_lt_add_right h1 _", "annotated_tactic": ["calc\n          (p ^ i * p).<a>natDegree</a> \u2264 (p ^ i).<a>natDegree</a> + p.natDegree := <a>natDegree_mul_le</a>\n          _ < i * p.natDegree + p.natDegree := <a>add_lt_add_right</a> h1 _", [{"full_name": "Polynomial.natDegree", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [65, 5], "def_end_pos": [65, 14]}, {"full_name": "Polynomial.natDegree", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [65, 5], "def_end_pos": [65, 14]}, {"full_name": "Polynomial.natDegree_mul_le", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [1051, 9], "def_end_pos": [1051, 25]}, {"full_name": "add_lt_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [135, 15], "def_end_pos": [135, 31]}]], "state_before": "case neg.h\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\nhp2 : \u00acp ^ i = 0\nh1 : natDegree (p ^ i) < i * natDegree p\n\u22a2 natDegree (p ^ i * p) < i * natDegree p + natDegree p", "state_after": "no goals"}, {"tactic": "refine lt_of_le_of_ne natDegree_pow_le fun h => hp2 ?_", "annotated_tactic": ["refine <a>lt_of_le_of_ne</a> <a>natDegree_pow_le</a> fun h => hp2 ?_", [{"full_name": "lt_of_le_of_ne", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [196, 9], "def_end_pos": [196, 23]}, {"full_name": "Polynomial.natDegree_pow_le", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [1062, 9], "def_end_pos": [1062, 25]}]], "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\nhp2 : \u00acp ^ i = 0\n\u22a2 natDegree (p ^ i) < i * natDegree p", "state_after": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\nhp2 : \u00acp ^ i = 0\nh : natDegree (p ^ i) = i * natDegree p\n\u22a2 p ^ i = 0"}, {"tactic": "rw [\u2190 h, hp1] at hi", "annotated_tactic": ["rw [\u2190 h, hp1] at hi", []], "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\nhp2 : \u00acp ^ i = 0\nh : natDegree (p ^ i) = i * natDegree p\n\u22a2 p ^ i = 0", "state_after": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (natDegree (p ^ i)) = 0\nhp1 : leadingCoeff p ^ i = 0\nhp2 : \u00acp ^ i = 0\nh : natDegree (p ^ i) = i * natDegree p\n\u22a2 p ^ i = 0"}, {"tactic": "exact leadingCoeff_eq_zero.mp hi", "annotated_tactic": ["exact leadingCoeff_eq_zero.mp hi", []], "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (natDegree (p ^ i)) = 0\nhp1 : leadingCoeff p ^ i = 0\nhp2 : \u00acp ^ i = 0\nh : natDegree (p ^ i) = i * natDegree p\n\u22a2 p ^ i = 0", "state_after": "no goals"}, {"tactic": "rw [\u2190 natDegree_pow' hp1, \u2190 leadingCoeff_pow' hp1]", "annotated_tactic": ["rw [\u2190 <a>natDegree_pow'</a> hp1, \u2190 <a>leadingCoeff_pow'</a> hp1]", [{"full_name": "Polynomial.natDegree_pow'", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [1009, 9], "def_end_pos": [1009, 23]}, {"full_name": "Polynomial.leadingCoeff_pow'", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [993, 9], "def_end_pos": [993, 26]}]], "state_before": "case neg\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : \u00acleadingCoeff p ^ i = 0\n\u22a2 coeff (p ^ i * p) (i * natDegree p + natDegree p) = leadingCoeff p ^ i * leadingCoeff p", "state_after": "case neg\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : \u00acleadingCoeff p ^ i = 0\n\u22a2 coeff (p ^ i * p) (natDegree (p ^ i) + natDegree p) = leadingCoeff (p ^ i) * leadingCoeff p"}, {"tactic": "exact coeff_mul_degree_add_degree _ _", "annotated_tactic": ["exact <a>coeff_mul_degree_add_degree</a> _ _", [{"full_name": "Polynomial.coeff_mul_degree_add_degree", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [915, 9], "def_end_pos": [915, 36]}]], "state_before": "case neg\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np\u271d q : R[X]\n\u03b9 : Type u_1\np : R[X]\ni : \u2115\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : \u00acleadingCoeff p ^ i = 0\n\u22a2 coeff (p ^ i * p) (natDegree (p ^ i) + natDegree p) = leadingCoeff (p ^ i) * leadingCoeff p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/FilterBasis.lean", "full_name": "GroupFilterBasis.prod_subset_self", "start": [143, 1], "end": [144, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.bind_apply", "start": [284, 1], "end": [285, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/Basic.lean", "full_name": "Polynomial.toFinsupp_X_pow", "start": [934, 1], "end": [935, 45], "traced_tactics": [{"tactic": "rw [X_pow_eq_monomial, toFinsupp_monomial]", "annotated_tactic": ["rw [<a>X_pow_eq_monomial</a>, <a>toFinsupp_monomial</a>]", [{"full_name": "Polynomial.X_pow_eq_monomial", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [927, 9], "def_end_pos": [927, 26]}, {"full_name": "Polynomial.toFinsupp_monomial", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [438, 9], "def_end_pos": [438, 27]}]], "state_before": "R : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\n\u22a2 (X ^ n).toFinsupp = fun\u2080 | n => 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Instances/Rat.lean", "full_name": "Rat.uniformContinuous_abs", "start": [107, 1], "end": [110, 86], "traced_tactics": [{"tactic": "simpa [Rat.dist_eq] using abs_abs_sub_abs_le_abs_sub _ _", "annotated_tactic": ["simpa [<a>Rat.dist_eq</a>] using <a>abs_abs_sub_abs_le_abs_sub</a> _ _", [{"full_name": "Rat.dist_eq", "def_path": "Mathlib/Topology/Instances/Rat.lean", "def_pos": [28, 9], "def_end_pos": [28, 16]}, {"full_name": "abs_abs_sub_abs_le_abs_sub", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [312, 9], "def_end_pos": [312, 35]}]], "state_before": "\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\na\u271d b\u271d : \u211a\nh : dist a\u271d b\u271d < \u03b5\n\u22a2 dist |a\u271d| |b\u271d| \u2264 dist a\u271d b\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Hom/Bounded.lean", "full_name": "TopHom.ext", "start": [207, 1], "end": [208, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Moments.lean", "full_name": "ProbabilityTheory.centralMoment_one", "start": [81, 1], "end": [91, 29], "traced_tactics": [{"tactic": "by_cases h_int : Integrable X \u03bc", "annotated_tactic": ["by_cases h_int : <a>Integrable</a> X \u03bc", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\n\u22a2 centralMoment X 1 \u03bc = 0", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : Integrable X\n\u22a2 centralMoment X 1 \u03bc = 0\n\ncase neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\n\u22a2 centralMoment X 1 \u03bc = 0"}, {"tactic": "rw [centralMoment_one' h_int]", "annotated_tactic": ["rw [<a>centralMoment_one'</a> h_int]", [{"full_name": "ProbabilityTheory.centralMoment_one'", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [73, 9], "def_end_pos": [73, 27]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : Integrable X\n\u22a2 centralMoment X 1 \u03bc = 0", "state_after": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : Integrable X\n\u22a2 (1 - ENNReal.toReal (\u2191\u2191\u03bc Set.univ)) * \u222b (x : \u03a9), X x \u2202\u03bc = 0"}, {"tactic": "simp only [measure_univ, ENNReal.one_toReal, sub_self, zero_mul]", "annotated_tactic": ["simp only [<a>measure_univ</a>, <a>ENNReal.one_toReal</a>, <a>sub_self</a>, <a>zero_mul</a>]", [{"full_name": "MeasureTheory.IsProbabilityMeasure.measure_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [3027, 3], "def_end_pos": [3027, 15]}, {"full_name": "ENNReal.one_toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [230, 17], "def_end_pos": [230, 27]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}]], "state_before": "case pos\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : Integrable X\n\u22a2 (1 - ENNReal.toReal (\u2191\u2191\u03bc Set.univ)) * \u222b (x : \u03a9), X x \u2202\u03bc = 0", "state_after": "no goals"}, {"tactic": "simp only [centralMoment, Pi.sub_apply, pow_one]", "annotated_tactic": ["simp only [<a>centralMoment</a>, <a>Pi.sub_apply</a>, <a>pow_one</a>]", [{"full_name": "ProbabilityTheory.centralMoment", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [56, 5], "def_end_pos": [56, 18]}, {"full_name": "Pi.sub_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [200, 3], "def_end_pos": [200, 14]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\n\u22a2 centralMoment X 1 \u03bc = 0", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\n\u22a2 \u222b (x : \u03a9), X x - \u222b (x : \u03a9), X x \u2202\u03bc \u2202\u03bc = 0"}, {"tactic": "have : \u00acIntegrable (fun x => X x - integral \u03bc X) \u03bc := by\n  refine' fun h_sub => h_int _\n  have h_add : X = (fun x => X x - integral \u03bc X) + fun _ => integral \u03bc X := by ext1 x; simp\n  rw [h_add]\n  exact h_sub.add (integrable_const _)", "annotated_tactic": ["have : \u00ac<a>Integrable</a> (fun x => X x - <a>integral</a> \u03bc X) \u03bc := by\n      refine' fun h_sub => h_int _\n      have h_add : X = (fun x => X x - <a>integral</a> \u03bc X) + fun _ => <a>integral</a> \u03bc X := by ext1 x; simp\n      rw [h_add]\n      exact h_sub.add (<a>integrable_const</a> _)", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}, {"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}, {"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}, {"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}, {"full_name": "MeasureTheory.integrable_const", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [506, 9], "def_end_pos": [506, 25]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\n\u22a2 \u222b (x : \u03a9), X x - \u222b (x : \u03a9), X x \u2202\u03bc \u2202\u03bc = 0", "state_after": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\nthis : \u00acIntegrable fun x => X x - integral \u03bc X\n\u22a2 \u222b (x : \u03a9), X x - \u222b (x : \u03a9), X x \u2202\u03bc \u2202\u03bc = 0"}, {"tactic": "rw [integral_undef this]", "annotated_tactic": ["rw [<a>integral_undef</a> this]", [{"full_name": "MeasureTheory.integral_undef", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [836, 9], "def_end_pos": [836, 23]}]], "state_before": "case neg\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\nthis : \u00acIntegrable fun x => X x - integral \u03bc X\n\u22a2 \u222b (x : \u03a9), X x - \u222b (x : \u03a9), X x \u2202\u03bc \u2202\u03bc = 0", "state_after": "no goals"}, {"tactic": "refine' fun h_sub => h_int _", "annotated_tactic": ["refine' fun h_sub => h_int _", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\n\u22a2 \u00acIntegrable fun x => X x - integral \u03bc X", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\nh_sub : Integrable fun x => X x - integral \u03bc X\n\u22a2 Integrable X"}, {"tactic": "have h_add : X = (fun x => X x - integral \u03bc X) + fun _ => integral \u03bc X := by ext1 x; simp", "annotated_tactic": ["have h_add : X = (fun x => X x - <a>integral</a> \u03bc X) + fun _ => <a>integral</a> \u03bc X := by ext1 x; simp", [{"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}, {"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\nh_sub : Integrable fun x => X x - integral \u03bc X\n\u22a2 Integrable X", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\nh_sub : Integrable fun x => X x - integral \u03bc X\nh_add : X = (fun x => X x - integral \u03bc X) + fun x => integral \u03bc X\n\u22a2 Integrable X"}, {"tactic": "rw [h_add]", "annotated_tactic": ["rw [h_add]", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\nh_sub : Integrable fun x => X x - integral \u03bc X\nh_add : X = (fun x => X x - integral \u03bc X) + fun x => integral \u03bc X\n\u22a2 Integrable X", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\nh_sub : Integrable fun x => X x - integral \u03bc X\nh_add : X = (fun x => X x - integral \u03bc X) + fun x => integral \u03bc X\n\u22a2 Integrable ((fun x => X x - integral \u03bc X) + fun x => integral \u03bc X)"}, {"tactic": "exact h_sub.add (integrable_const _)", "annotated_tactic": ["exact h_sub.add (<a>integrable_const</a> _)", [{"full_name": "MeasureTheory.integrable_const", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [506, 9], "def_end_pos": [506, 25]}]], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\nh_sub : Integrable fun x => X x - integral \u03bc X\nh_add : X = (fun x => X x - integral \u03bc X) + fun x => integral \u03bc X\n\u22a2 Integrable ((fun x => X x - integral \u03bc X) + fun x => integral \u03bc X)", "state_after": "no goals"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\nh_sub : Integrable fun x => X x - integral \u03bc X\n\u22a2 X = (fun x => X x - integral \u03bc X) + fun x => integral \u03bc X", "state_after": "case h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\nh_sub : Integrable fun x => X x - integral \u03bc X\nx : \u03a9\n\u22a2 X x = ((fun x => X x - integral \u03bc X) + fun x => integral \u03bc X) x"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\n\u03a9 : Type u_1\n\u03b9 : Type u_2\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\np : \u2115\n\u03bc : Measure \u03a9\ninst\u271d : IsProbabilityMeasure \u03bc\nh_int : \u00acIntegrable X\nh_sub : Integrable fun x => X x - integral \u03bc X\nx : \u03a9\n\u22a2 X x = ((fun x => X x - integral \u03bc X) + fun x => integral \u03bc X) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Hom/Lattice.lean", "full_name": "Disjoint.map", "start": [266, 1], "end": [267, 50], "traced_tactics": [{"tactic": "rw [disjoint_iff, \u2190 map_inf, h.eq_bot, map_bot]", "annotated_tactic": ["rw [<a>disjoint_iff</a>, \u2190 <a>map_inf</a>, h.eq_bot, <a>map_bot</a>]", [{"full_name": "disjoint_iff", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [126, 9], "def_end_pos": [126, 21]}, {"full_name": "InfHomClass.map_inf", "def_path": "Mathlib/Order/Hom/Lattice.lean", "def_pos": [114, 3], "def_end_pos": [114, 10]}, {"full_name": "BotHomClass.map_bot", "def_path": "Mathlib/Order/Hom/Bounded.lean", "def_pos": [79, 3], "def_end_pos": [79, 10]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\n\u03b4 : Type u_6\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : BoundedOrder \u03b1\ninst\u271d\u00b2 : Lattice \u03b2\ninst\u271d\u00b9 : BoundedOrder \u03b2\ninst\u271d : BoundedLatticeHomClass F \u03b1 \u03b2\nf : F\na b : \u03b1\nh : Disjoint a b\n\u22a2 Disjoint (\u2191f a) (\u2191f b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Matrix/ZPow.lean", "full_name": "Matrix.zpow_neg_one", "start": [105, 1], "end": [107, 78], "traced_tactics": [{"tactic": "convert DivInvMonoid.zpow_neg' 0 A", "annotated_tactic": ["convert <a>DivInvMonoid.zpow_neg'</a> 0 A", [{"full_name": "DivInvMonoid.zpow_neg'", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [878, 13], "def_end_pos": [878, 22]}]], "state_before": "n' : Type u_1\ninst\u271d\u00b2 : DecidableEq n'\ninst\u271d\u00b9 : Fintype n'\nR : Type u_2\ninst\u271d : CommRing R\nA : M\n\u22a2 A ^ (-1) = A\u207b\u00b9", "state_after": "case h.e'_3.h.e'_3\nn' : Type u_1\ninst\u271d\u00b2 : DecidableEq n'\ninst\u271d\u00b9 : Fintype n'\nR : Type u_2\ninst\u271d : CommRing R\nA : M\n\u22a2 A = DivInvMonoid.zpow (\u2191(Nat.succ 0)) A"}, {"tactic": "simp only [zpow_one, Int.ofNat_zero, Int.ofNat_succ, zpow_eq_pow, zero_add]", "annotated_tactic": ["simp only [<a>zpow_one</a>, <a>Int.ofNat_zero</a>, <a>Int.ofNat_succ</a>, <a>zpow_eq_pow</a>, <a>zero_add</a>]", [{"full_name": "zpow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [285, 9], "def_end_pos": [285, 17]}, {"full_name": "Int.ofNat_zero", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [19, 17], "def_end_pos": [19, 27]}, {"full_name": "Int.ofNat_succ", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [37, 9], "def_end_pos": [37, 19]}, {"full_name": "zpow_eq_pow", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [936, 53], "def_end_pos": [936, 64]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "case h.e'_3.h.e'_3\nn' : Type u_1\ninst\u271d\u00b2 : DecidableEq n'\ninst\u271d\u00b9 : Fintype n'\nR : Type u_2\ninst\u271d : CommRing R\nA : M\n\u22a2 A = DivInvMonoid.zpow (\u2191(Nat.succ 0)) A", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/UniformSpace/Equicontinuity.lean", "full_name": "Filter.HasBasis.equicontinuousAt_iff_left", "start": [357, 1], "end": [363, 6], "traced_tactics": [{"tactic": "rw [equicontinuousAt_iff_continuousAt, ContinuousAt,\n  hX.tendsto_iff (UniformFun.hasBasis_nhds \u03b9 \u03b1 _)]", "annotated_tactic": ["rw [<a>equicontinuousAt_iff_continuousAt</a>, <a>ContinuousAt</a>,\n    hX.tendsto_iff (<a>UniformFun.hasBasis_nhds</a> \u03b9 \u03b1 _)]", [{"full_name": "equicontinuousAt_iff_continuousAt", "def_path": "Mathlib/Topology/UniformSpace/Equicontinuity.lean", "def_pos": [293, 9], "def_end_pos": [293, 42]}, {"full_name": "ContinuousAt", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1620, 5], "def_end_pos": [1620, 17]}, {"full_name": "UniformFun.hasBasis_nhds", "def_path": "Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean", "def_pos": [334, 19], "def_end_pos": [334, 32]}]], "state_before": "\u03b9 : Type u_1\n\u03ba\u271d : Type u_2\nX : Type u_3\nY : Type u_4\nZ : Type u_5\n\u03b1 : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\n\ud835\udcd5 : Type u_9\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : TopologicalSpace Z\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\n\u03ba : Type u_10\np : \u03ba \u2192 Prop\ns : \u03ba \u2192 Set X\nF : \u03b9 \u2192 X \u2192 \u03b1\nx\u2080 : X\nhX : HasBasis (\ud835\udcdd x\u2080) p s\n\u22a2 EquicontinuousAt F x\u2080 \u2194 \u2200 (U : Set (\u03b1 \u00d7 \u03b1)), U \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 k, p k \u2227 \u2200 (x : X), x \u2208 s k \u2192 \u2200 (i : \u03b9), (F i x\u2080, F i x) \u2208 U", "state_after": "\u03b9 : Type u_1\n\u03ba\u271d : Type u_2\nX : Type u_3\nY : Type u_4\nZ : Type u_5\n\u03b1 : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\n\ud835\udcd5 : Type u_9\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : TopologicalSpace Z\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\n\u03ba : Type u_10\np : \u03ba \u2192 Prop\ns : \u03ba \u2192 Set X\nF : \u03b9 \u2192 X \u2192 \u03b1\nx\u2080 : X\nhX : HasBasis (\ud835\udcdd x\u2080) p s\n\u22a2 (\u2200 (ib : Set (\u03b1 \u00d7 \u03b1)),\n      ib \u2208 \ud835\udce4 \u03b1 \u2192\n        \u2203 ia,\n          p ia \u2227\n            \u2200 (x : X),\n              x \u2208 s ia \u2192\n                (\u2191UniformFun.ofFun \u2218 Function.swap F) x \u2208\n                  {g | ((\u2191UniformFun.ofFun \u2218 Function.swap F) x\u2080, g) \u2208 UniformFun.gen \u03b9 \u03b1 ib}) \u2194\n    \u2200 (U : Set (\u03b1 \u00d7 \u03b1)), U \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 k, p k \u2227 \u2200 (x : X), x \u2208 s k \u2192 \u2200 (i : \u03b9), (F i x\u2080, F i x) \u2208 U"}, {"tactic": "simp only [Function.comp_apply, mem_setOf_eq, exists_prop]", "annotated_tactic": ["simp only [<a>Function.comp_apply</a>, <a>mem_setOf_eq</a>, <a>exists_prop</a>]", [{"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}]], "state_before": "\u03b9 : Type u_1\n\u03ba\u271d : Type u_2\nX : Type u_3\nY : Type u_4\nZ : Type u_5\n\u03b1 : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\n\ud835\udcd5 : Type u_9\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : TopologicalSpace Z\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\n\u03ba : Type u_10\np : \u03ba \u2192 Prop\ns : \u03ba \u2192 Set X\nF : \u03b9 \u2192 X \u2192 \u03b1\nx\u2080 : X\nhX : HasBasis (\ud835\udcdd x\u2080) p s\n\u22a2 (\u2200 (ib : Set (\u03b1 \u00d7 \u03b1)),\n      ib \u2208 \ud835\udce4 \u03b1 \u2192\n        \u2203 ia,\n          p ia \u2227\n            \u2200 (x : X),\n              x \u2208 s ia \u2192\n                (\u2191UniformFun.ofFun \u2218 Function.swap F) x \u2208\n                  {g | ((\u2191UniformFun.ofFun \u2218 Function.swap F) x\u2080, g) \u2208 UniformFun.gen \u03b9 \u03b1 ib}) \u2194\n    \u2200 (U : Set (\u03b1 \u00d7 \u03b1)), U \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 k, p k \u2227 \u2200 (x : X), x \u2208 s k \u2192 \u2200 (i : \u03b9), (F i x\u2080, F i x) \u2208 U", "state_after": "\u03b9 : Type u_1\n\u03ba\u271d : Type u_2\nX : Type u_3\nY : Type u_4\nZ : Type u_5\n\u03b1 : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\n\ud835\udcd5 : Type u_9\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : TopologicalSpace Z\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\n\u03ba : Type u_10\np : \u03ba \u2192 Prop\ns : \u03ba \u2192 Set X\nF : \u03b9 \u2192 X \u2192 \u03b1\nx\u2080 : X\nhX : HasBasis (\ud835\udcdd x\u2080) p s\n\u22a2 (\u2200 (ib : Set (\u03b1 \u00d7 \u03b1)),\n      ib \u2208 \ud835\udce4 \u03b1 \u2192\n        \u2203 ia,\n          p ia \u2227\n            \u2200 (x : X),\n              x \u2208 s ia \u2192\n                (\u2191UniformFun.ofFun (Function.swap F x\u2080), \u2191UniformFun.ofFun (Function.swap F x)) \u2208\n                  UniformFun.gen \u03b9 \u03b1 ib) \u2194\n    \u2200 (U : Set (\u03b1 \u00d7 \u03b1)), U \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 k, p k \u2227 \u2200 (x : X), x \u2208 s k \u2192 \u2200 (i : \u03b9), (F i x\u2080, F i x) \u2208 U"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b9 : Type u_1\n\u03ba\u271d : Type u_2\nX : Type u_3\nY : Type u_4\nZ : Type u_5\n\u03b1 : Type u_6\n\u03b2 : Type u_7\n\u03b3 : Type u_8\n\ud835\udcd5 : Type u_9\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : TopologicalSpace Z\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : UniformSpace \u03b3\n\u03ba : Type u_10\np : \u03ba \u2192 Prop\ns : \u03ba \u2192 Set X\nF : \u03b9 \u2192 X \u2192 \u03b1\nx\u2080 : X\nhX : HasBasis (\ud835\udcdd x\u2080) p s\n\u22a2 (\u2200 (ib : Set (\u03b1 \u00d7 \u03b1)),\n      ib \u2208 \ud835\udce4 \u03b1 \u2192\n        \u2203 ia,\n          p ia \u2227\n            \u2200 (x : X),\n              x \u2208 s ia \u2192\n                (\u2191UniformFun.ofFun (Function.swap F x\u2080), \u2191UniformFun.ofFun (Function.swap F x)) \u2208\n                  UniformFun.gen \u03b9 \u03b1 ib) \u2194\n    \u2200 (U : Set (\u03b1 \u00d7 \u03b1)), U \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 k, p k \u2227 \u2200 (x : X), x \u2208 s k \u2192 \u2200 (i : \u03b9), (F i x\u2080, F i x) \u2208 U", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Logic/Function/Basic.lean", "full_name": "Function.Surjective.of_comp_iff", "start": [174, 1], "end": [176, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Sites/Limits.lean", "full_name": "CategoryTheory.Sheaf.isSheaf_of_isLimit", "start": [139, 1], "end": [143, 47], "traced_tactics": [{"tactic": "rw [Presheaf.isSheaf_iff_multifork]", "annotated_tactic": ["rw [<a>Presheaf.isSheaf_iff_multifork</a>]", [{"full_name": "CategoryTheory.Presheaf.isSheaf_iff_multifork", "def_path": "Mathlib/CategoryTheory/Sites/Sheaf.lean", "def_pos": [510, 9], "def_end_pos": [510, 30]}]], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\n\u22a2 Presheaf.IsSheaf J E.pt", "state_after": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\n\u22a2 \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S E.pt))"}, {"tactic": "intro X S", "annotated_tactic": ["intro X S", []], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\n\u22a2 \u2200 (X : C) (S : GrothendieckTopology.Cover J X), Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S E.pt))", "state_after": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nS : GrothendieckTopology.Cover J X\n\u22a2 Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S E.pt))"}, {"tactic": "exact \u27e8isLimitMultiforkOfIsLimit _ _ hE _ _\u27e9", "annotated_tactic": ["exact \u27e8<a>isLimitMultiforkOfIsLimit</a> _ _ hE _ _\u27e9", [{"full_name": "CategoryTheory.Sheaf.isLimitMultiforkOfIsLimit", "def_path": "Mathlib/CategoryTheory/Sites/Limits.lean", "def_pos": [99, 5], "def_end_pos": [99, 30]}]], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst\u271d\u00b2 : Category.{max v u, w} D\nK : Type z\ninst\u271d\u00b9 : SmallCategory K\ninst\u271d : HasLimitsOfShape K D\nF : K \u2964 Sheaf J D\nE : Cone (F \u22d9 sheafToPresheaf J D)\nhE : IsLimit E\nX : C\nS : GrothendieckTopology.Cover J X\n\u22a2 Nonempty (IsLimit (GrothendieckTopology.Cover.multifork S E.pt))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Game/Impartial.lean", "full_name": "SetTheory.PGame.Impartial.equiv_zero_iff_le", "start": [191, 1], "end": [192, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Noetherian.lean", "full_name": "isNoetherian_top_iff", "start": [135, 1], "end": [138, 87], "traced_tactics": [{"tactic": "constructor <;> intro h", "annotated_tactic": ["constructor <;> intro h", []], "state_before": "R : Type u_1\nM : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid P\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R P\n\u22a2 IsNoetherian R { x // x \u2208 \u22a4 } \u2194 IsNoetherian R M", "state_after": "case mp\nR : Type u_1\nM : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid P\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R P\nh : IsNoetherian R { x // x \u2208 \u22a4 }\n\u22a2 IsNoetherian R M\n\ncase mpr\nR : Type u_1\nM : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid P\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R P\nh : IsNoetherian R M\n\u22a2 IsNoetherian R { x // x \u2208 \u22a4 }"}, {"tactic": "exact isNoetherian_of_linearEquiv (LinearEquiv.ofTop (\u22a4 : Submodule R M) rfl)", "annotated_tactic": ["exact <a>isNoetherian_of_linearEquiv</a> (<a>LinearEquiv.ofTop</a> (\u22a4 : <a>Submodule</a> R M) <a>rfl</a>)", [{"full_name": "isNoetherian_of_linearEquiv", "def_path": "Mathlib/RingTheory/Noetherian.lean", "def_pos": [131, 9], "def_end_pos": [131, 36]}, {"full_name": "LinearEquiv.ofTop", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [1990, 5], "def_end_pos": [1990, 10]}, {"full_name": "Submodule", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [41, 11], "def_end_pos": [41, 20]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case mp\nR : Type u_1\nM : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid P\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R P\nh : IsNoetherian R { x // x \u2208 \u22a4 }\n\u22a2 IsNoetherian R M", "state_after": "no goals"}, {"tactic": "exact isNoetherian_of_linearEquiv (LinearEquiv.ofTop (\u22a4 : Submodule R M) rfl).symm", "annotated_tactic": ["exact <a>isNoetherian_of_linearEquiv</a> (<a>LinearEquiv.ofTop</a> (\u22a4 : <a>Submodule</a> R M) <a>rfl</a>).<a>symm</a>", [{"full_name": "isNoetherian_of_linearEquiv", "def_path": "Mathlib/RingTheory/Noetherian.lean", "def_pos": [131, 9], "def_end_pos": [131, 36]}, {"full_name": "LinearEquiv.ofTop", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [1990, 5], "def_end_pos": [1990, 10]}, {"full_name": "Submodule", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [41, 11], "def_end_pos": [41, 20]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "LinearEquiv.symm", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [271, 5], "def_end_pos": [271, 9]}]], "state_before": "case mpr\nR : Type u_1\nM : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid P\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R P\nh : IsNoetherian R M\n\u22a2 IsNoetherian R { x // x \u2208 \u22a4 }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Init/Data/Nat/Bitwise.lean", "full_name": "Nat.div2_zero", "start": [104, 1], "end": [105, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/LeastGreatest.lean", "full_name": "Int.coe_leastOfBdd_eq", "start": [71, 1], "end": [76, 44], "traced_tactics": [{"tactic": "rcases leastOfBdd b Hb Hinh with \u27e8n, hn, h2n\u27e9", "annotated_tactic": ["rcases <a>leastOfBdd</a> b Hb Hinh with \u27e8n, hn, h2n\u27e9", [{"full_name": "Int.leastOfBdd", "def_path": "Mathlib/Data/Int/LeastGreatest.lean", "def_pos": [45, 5], "def_end_pos": [45, 15]}]], "state_before": "P : \u2124 \u2192 Prop\ninst\u271d : DecidablePred P\nb b' : \u2124\nHb : \u2200 (z : \u2124), P z \u2192 b \u2264 z\nHb' : \u2200 (z : \u2124), P z \u2192 b' \u2264 z\nHinh : \u2203 z, P z\n\u22a2 \u2191(leastOfBdd b Hb Hinh) = \u2191(leastOfBdd b' Hb' Hinh)", "state_after": "case mk.intro\nP : \u2124 \u2192 Prop\ninst\u271d : DecidablePred P\nb b' : \u2124\nHb : \u2200 (z : \u2124), P z \u2192 b \u2264 z\nHb' : \u2200 (z : \u2124), P z \u2192 b' \u2264 z\nHinh : \u2203 z, P z\nn : \u2124\nhn : P n\nh2n : \u2200 (z : \u2124), P z \u2192 n \u2264 z\n\u22a2 \u2191{ val := n, property := (_ : P n \u2227 \u2200 (z : \u2124), P z \u2192 n \u2264 z) } = \u2191(leastOfBdd b' Hb' Hinh)"}, {"tactic": "rcases leastOfBdd b' Hb' Hinh with \u27e8n', hn', h2n'\u27e9", "annotated_tactic": ["rcases <a>leastOfBdd</a> b' Hb' Hinh with \u27e8n', hn', h2n'\u27e9", [{"full_name": "Int.leastOfBdd", "def_path": "Mathlib/Data/Int/LeastGreatest.lean", "def_pos": [45, 5], "def_end_pos": [45, 15]}]], "state_before": "case mk.intro\nP : \u2124 \u2192 Prop\ninst\u271d : DecidablePred P\nb b' : \u2124\nHb : \u2200 (z : \u2124), P z \u2192 b \u2264 z\nHb' : \u2200 (z : \u2124), P z \u2192 b' \u2264 z\nHinh : \u2203 z, P z\nn : \u2124\nhn : P n\nh2n : \u2200 (z : \u2124), P z \u2192 n \u2264 z\n\u22a2 \u2191{ val := n, property := (_ : P n \u2227 \u2200 (z : \u2124), P z \u2192 n \u2264 z) } = \u2191(leastOfBdd b' Hb' Hinh)", "state_after": "case mk.intro.mk.intro\nP : \u2124 \u2192 Prop\ninst\u271d : DecidablePred P\nb b' : \u2124\nHb : \u2200 (z : \u2124), P z \u2192 b \u2264 z\nHb' : \u2200 (z : \u2124), P z \u2192 b' \u2264 z\nHinh : \u2203 z, P z\nn : \u2124\nhn : P n\nh2n : \u2200 (z : \u2124), P z \u2192 n \u2264 z\nn' : \u2124\nhn' : P n'\nh2n' : \u2200 (z : \u2124), P z \u2192 n' \u2264 z\n\u22a2 \u2191{ val := n, property := (_ : P n \u2227 \u2200 (z : \u2124), P z \u2192 n \u2264 z) } =\n    \u2191{ val := n', property := (_ : P n' \u2227 \u2200 (z : \u2124), P z \u2192 n' \u2264 z) }"}, {"tactic": "exact le_antisymm (h2n _ hn') (h2n' _ hn)", "annotated_tactic": ["exact <a>le_antisymm</a> (h2n _ hn') (h2n' _ hn)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "case mk.intro.mk.intro\nP : \u2124 \u2192 Prop\ninst\u271d : DecidablePred P\nb b' : \u2124\nHb : \u2200 (z : \u2124), P z \u2192 b \u2264 z\nHb' : \u2200 (z : \u2124), P z \u2192 b' \u2264 z\nHinh : \u2203 z, P z\nn : \u2124\nhn : P n\nh2n : \u2200 (z : \u2124), P z \u2192 n \u2264 z\nn' : \u2124\nhn' : P n'\nh2n' : \u2200 (z : \u2124), P z \u2192 n' \u2264 z\n\u22a2 \u2191{ val := n, property := (_ : P n \u2227 \u2200 (z : \u2124), P z \u2192 n \u2264 z) } =\n    \u2191{ val := n', property := (_ : P n' \u2227 \u2200 (z : \u2124), P z \u2192 n' \u2264 z) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Sum/Basic.lean", "full_name": "Sum.getLeft?_inl", "start": [86, 9], "end": [86, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Holor.lean", "full_name": "Holor.slice_eq", "start": [250, 1], "end": [259, 72], "traced_tactics": [{"tactic": "rw [\u2190 hiis]", "annotated_tactic": ["rw [\u2190 hiis]", []], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\nx y : Holor \u03b1 (d :: ds)\nh : slice x = slice y\nt : HolorIndex (d :: ds)\ni : \u2115\nis : List \u2115\nhiis : \u2191t = i :: is\n\u22a2 Forall\u2082 (fun x x_1 => x < x_1) (i :: is) (d :: ds)", "state_after": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\nx y : Holor \u03b1 (d :: ds)\nh : slice x = slice y\nt : HolorIndex (d :: ds)\ni : \u2115\nis : List \u2115\nhiis : \u2191t = i :: is\n\u22a2 Forall\u2082 (fun x x_1 => x < x_1) (\u2191t) (d :: ds)"}, {"tactic": "exact t.2", "annotated_tactic": ["exact t.2", []], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\nx y : Holor \u03b1 (d :: ds)\nh : slice x = slice y\nt : HolorIndex (d :: ds)\ni : \u2115\nis : List \u2115\nhiis : \u2191t = i :: is\n\u22a2 Forall\u2082 (fun x x_1 => x < x_1) (\u2191t) (d :: ds)", "state_after": "no goals"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\nx y : Holor \u03b1 (d :: ds)\nh : slice x = slice y\nt : HolorIndex (d :: ds)\ni : \u2115\nis : List \u2115\nhiis : \u2191t = i :: is\nhiisdds : Forall\u2082 (fun x x_1 => x < x_1) (i :: is) (d :: ds)\nhid : i < d\nhisds : Forall\u2082 (fun x x_1 => x < x_1) is ds\n\u22a2 slice x i hid { val := is, property := hisds } = slice y i hid { val := is, property := hisds }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Alternating/Basic.lean", "full_name": "AlternatingMap.domDomCongr_perm", "start": [826, 1], "end": [828, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "full_name": "Subgroup.mem_sup", "start": [3575, 1], "end": [3587, 57], "traced_tactics": [{"tactic": "rw [sup_eq_closure] at h", "annotated_tactic": ["rw [<a>sup_eq_closure</a>] at h", [{"full_name": "Subgroup.sup_eq_closure", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [1268, 9], "def_end_pos": [1268, 23]}]], "state_before": "G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\nh : x \u2208 s \u2294 t\n\u22a2 \u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = x", "state_after": "G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\nh : x \u2208 closure (\u2191s \u222a \u2191t)\n\u22a2 \u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = x"}, {"tactic": "refine Subgroup.closure_induction h ?_ ?_ ?_ ?_", "annotated_tactic": ["refine <a>Subgroup.closure_induction</a> h ?_ ?_ ?_ ?_", [{"full_name": "Subgroup.closure_induction", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [1149, 9], "def_end_pos": [1149, 26]}]], "state_before": "G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\nh : x \u2208 closure (\u2191s \u222a \u2191t)\n\u22a2 \u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = x", "state_after": "case refine_1\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\nh : x \u2208 closure (\u2191s \u222a \u2191t)\n\u22a2 \u2200 (x : C), x \u2208 \u2191s \u222a \u2191t \u2192 \u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = x\n\ncase refine_2\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\nh : x \u2208 closure (\u2191s \u222a \u2191t)\n\u22a2 \u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = 1\n\ncase refine_3\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\nh : x \u2208 closure (\u2191s \u222a \u2191t)\n\u22a2 \u2200 (x y : C),\n    (\u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = x) \u2192\n      (\u2203 y_1, y_1 \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y_1 * z = y) \u2192 \u2203 y_1, y_1 \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y_1 * z = x * y\n\ncase refine_4\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\nh : x \u2208 closure (\u2191s \u222a \u2191t)\n\u22a2 \u2200 (x : C), (\u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = x) \u2192 \u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = x\u207b\u00b9"}, {"tactic": "rintro y (h | h)", "annotated_tactic": ["rintro y (h | h)", []], "state_before": "case refine_1\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\nh : x \u2208 closure (\u2191s \u222a \u2191t)\n\u22a2 \u2200 (x : C), x \u2208 \u2191s \u222a \u2191t \u2192 \u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = x", "state_after": "case refine_1.inl\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\nh\u271d : x \u2208 closure (\u2191s \u222a \u2191t)\ny : C\nh : y \u2208 \u2191s\n\u22a2 \u2203 y_1, y_1 \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y_1 * z = y\n\ncase refine_1.inr\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\nh\u271d : x \u2208 closure (\u2191s \u222a \u2191t)\ny : C\nh : y \u2208 \u2191t\n\u22a2 \u2203 y_1, y_1 \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y_1 * z = y"}, {"tactic": "exact \u27e8y, h, 1, t.one_mem, by simp\u27e9", "annotated_tactic": ["exact \u27e8y, h, 1, t.one_mem, by simp\u27e9", []], "state_before": "case refine_1.inl\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\nh\u271d : x \u2208 closure (\u2191s \u222a \u2191t)\ny : C\nh : y \u2208 \u2191s\n\u22a2 \u2203 y_1, y_1 \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y_1 * z = y", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\nh\u271d : x \u2208 closure (\u2191s \u222a \u2191t)\ny : C\nh : y \u2208 \u2191s\n\u22a2 y * 1 = y", "state_after": "no goals"}, {"tactic": "exact \u27e81, s.one_mem, y, h, by simp\u27e9", "annotated_tactic": ["exact \u27e81, s.one_mem, y, h, by simp\u27e9", []], "state_before": "case refine_1.inr\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\nh\u271d : x \u2208 closure (\u2191s \u222a \u2191t)\ny : C\nh : y \u2208 \u2191t\n\u22a2 \u2203 y_1, y_1 \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y_1 * z = y", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\nh\u271d : x \u2208 closure (\u2191s \u222a \u2191t)\ny : C\nh : y \u2208 \u2191t\n\u22a2 1 * y = y", "state_after": "no goals"}, {"tactic": "exact \u27e81, s.one_mem, 1, \u27e8t.one_mem, mul_one 1\u27e9\u27e9", "annotated_tactic": ["exact \u27e81, s.one_mem, 1, \u27e8t.one_mem, <a>mul_one</a> 1\u27e9\u27e9", [{"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "case refine_2\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\nh : x \u2208 closure (\u2191s \u222a \u2191t)\n\u22a2 \u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = 1", "state_after": "no goals"}, {"tactic": "rintro _ _ \u27e8y\u2081, hy\u2081, z\u2081, hz\u2081, rfl\u27e9 \u27e8y\u2082, hy\u2082, z\u2082, hz\u2082, rfl\u27e9", "annotated_tactic": ["rintro _ _ \u27e8y\u2081, hy\u2081, z\u2081, hz\u2081, rfl\u27e9 \u27e8y\u2082, hy\u2082, z\u2082, hz\u2082, rfl\u27e9", []], "state_before": "case refine_3\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\nh : x \u2208 closure (\u2191s \u222a \u2191t)\n\u22a2 \u2200 (x y : C),\n    (\u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = x) \u2192\n      (\u2203 y_1, y_1 \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y_1 * z = y) \u2192 \u2203 y_1, y_1 \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y_1 * z = x * y", "state_after": "case refine_3.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\nh : x \u2208 closure (\u2191s \u222a \u2191t)\ny\u2081 : C\nhy\u2081 : y\u2081 \u2208 s\nz\u2081 : C\nhz\u2081 : z\u2081 \u2208 t\ny\u2082 : C\nhy\u2082 : y\u2082 \u2208 s\nz\u2082 : C\nhz\u2082 : z\u2082 \u2208 t\n\u22a2 \u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = y\u2081 * z\u2081 * (y\u2082 * z\u2082)"}, {"tactic": "exact \u27e8_, mul_mem hy\u2081 hy\u2082, _, mul_mem hz\u2081 hz\u2082, by simp [mul_assoc, mul_left_comm]\u27e9", "annotated_tactic": ["exact \u27e8_, <a>mul_mem</a> hy\u2081 hy\u2082, _, <a>mul_mem</a> hz\u2081 hz\u2082, by simp [<a>mul_assoc</a>, <a>mul_left_comm</a>]\u27e9", [{"full_name": "MulMemClass.mul_mem", "def_path": "Mathlib/GroupTheory/Subsemigroup/Basic.lean", "def_pos": [64, 3], "def_end_pos": [64, 10]}, {"full_name": "MulMemClass.mul_mem", "def_path": "Mathlib/GroupTheory/Subsemigroup/Basic.lean", "def_pos": [64, 3], "def_end_pos": [64, 10]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [96, 9], "def_end_pos": [96, 22]}]], "state_before": "case refine_3.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\nh : x \u2208 closure (\u2191s \u222a \u2191t)\ny\u2081 : C\nhy\u2081 : y\u2081 \u2208 s\nz\u2081 : C\nhz\u2081 : z\u2081 \u2208 t\ny\u2082 : C\nhy\u2082 : y\u2082 \u2208 s\nz\u2082 : C\nhz\u2082 : z\u2082 \u2208 t\n\u22a2 \u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = y\u2081 * z\u2081 * (y\u2082 * z\u2082)", "state_after": "no goals"}, {"tactic": "simp [mul_assoc, mul_left_comm]", "annotated_tactic": ["simp [<a>mul_assoc</a>, <a>mul_left_comm</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [96, 9], "def_end_pos": [96, 22]}]], "state_before": "G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\nh : x \u2208 closure (\u2191s \u222a \u2191t)\ny\u2081 : C\nhy\u2081 : y\u2081 \u2208 s\nz\u2081 : C\nhz\u2081 : z\u2081 \u2208 t\ny\u2082 : C\nhy\u2082 : y\u2082 \u2208 s\nz\u2082 : C\nhz\u2082 : z\u2082 \u2208 t\n\u22a2 y\u2081 * y\u2082 * (z\u2081 * z\u2082) = y\u2081 * z\u2081 * (y\u2082 * z\u2082)", "state_after": "no goals"}, {"tactic": "rintro _ \u27e8y, hy, z, hz, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8y, hy, z, hz, rfl\u27e9", []], "state_before": "case refine_4\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\nh : x \u2208 closure (\u2191s \u222a \u2191t)\n\u22a2 \u2200 (x : C), (\u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = x) \u2192 \u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = x\u207b\u00b9", "state_after": "case refine_4.intro.intro.intro.intro\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\nh : x \u2208 closure (\u2191s \u222a \u2191t)\ny : C\nhy : y \u2208 s\nz : C\nhz : z \u2208 t\n\u22a2 \u2203 y_1, y_1 \u2208 s \u2227 \u2203 z_1, z_1 \u2208 t \u2227 y_1 * z_1 = (y * z)\u207b\u00b9"}, {"tactic": "exact \u27e8_, inv_mem hy, _, inv_mem hz, mul_comm z y \u25b8 (mul_inv_rev z y).symm\u27e9", "annotated_tactic": ["exact \u27e8_, <a>inv_mem</a> hy, _, <a>inv_mem</a> hz, <a>mul_comm</a> z y \u25b8 (<a>mul_inv_rev</a> z y).<a>symm</a>\u27e9", [{"full_name": "InvMemClass.inv_mem", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [101, 3], "def_end_pos": [101, 10]}, {"full_name": "InvMemClass.inv_mem", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [101, 3], "def_end_pos": [101, 10]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}, {"full_name": "mul_inv_rev", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1050, 9], "def_end_pos": [1050, 20]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case refine_4.intro.intro.intro.intro\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\nh : x \u2208 closure (\u2191s \u222a \u2191t)\ny : C\nhy : y \u2208 s\nz : C\nhz : z \u2208 t\n\u22a2 \u2203 y_1, y_1 \u2208 s \u2227 \u2203 z_1, z_1 \u2208 t \u2227 y_1 * z_1 = (y * z)\u207b\u00b9", "state_after": "no goals"}, {"tactic": "rintro \u27e8y, hy, z, hz, rfl\u27e9", "annotated_tactic": ["rintro \u27e8y, hy, z, hz, rfl\u27e9", []], "state_before": "G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\nx : C\n\u22a2 (\u2203 y, y \u2208 s \u2227 \u2203 z, z \u2208 t \u2227 y * z = x) \u2192 x \u2208 s \u2294 t", "state_after": "case intro.intro.intro.intro\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\ny : C\nhy : y \u2208 s\nz : C\nhz : z \u2208 t\n\u22a2 y * z \u2208 s \u2294 t"}, {"tactic": "exact mul_mem_sup hy hz", "annotated_tactic": ["exact <a>mul_mem_sup</a> hy hz", [{"full_name": "Subgroup.mul_mem_sup", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [1046, 9], "def_end_pos": [1046, 20]}]], "state_before": "case intro.intro.intro.intro\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nN : Type u_5\ninst\u271d\u00b9 : Group N\nC : Type u_6\ninst\u271d : CommGroup C\ns t : Subgroup C\ny : C\nhy : y \u2208 s\nz : C\nhz : z \u2208 t\n\u22a2 y * z \u2208 s \u2294 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.coext", "start": [143, 11], "end": [144, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Matrix/Basic.lean", "full_name": "Matrix.conjTranspose_list_sum", "start": [2305, 1], "end": [2307, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Fourier/AddCircle.lean", "full_name": "coeFn_fourierLp", "start": [304, 1], "end": [306, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Prod.lean", "full_name": "Filter.prod_mono_right", "start": [237, 1], "end": [238, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Group/Abs.lean", "full_name": "abs_le_abs_of_nonpos", "start": [191, 1], "end": [193, 31], "traced_tactics": [{"tactic": "rw [abs_of_nonpos ha, abs_of_nonpos (hab.trans ha)]", "annotated_tactic": ["rw [<a>abs_of_nonpos</a> ha, <a>abs_of_nonpos</a> (hab.trans ha)]", [{"full_name": "abs_of_nonpos", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [115, 9], "def_end_pos": [115, 22]}, {"full_name": "abs_of_nonpos", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [115, 9], "def_end_pos": [115, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nha : a \u2264 0\nhab : b \u2264 a\n\u22a2 |a| \u2264 |b|", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nha : a \u2264 0\nhab : b \u2264 a\n\u22a2 -a \u2264 -b"}, {"tactic": "exact neg_le_neg_iff.mpr hab", "annotated_tactic": ["exact neg_le_neg_iff.mpr hab", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nha : a \u2264 0\nhab : b \u2264 a\n\u22a2 -a \u2264 -b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean", "full_name": "BoxIntegral.Prepartition.iUnion_biUnionTagged", "start": [147, 1], "end": [149, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Rat/NNRat.lean", "full_name": "NNRat.toNNRat_mono", "start": [181, 1], "end": [182, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/UniformSpace/Basic.lean", "full_name": "uniformContinuous_toMul", "start": [1453, 1], "end": [1454, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/Order/Lemmas.lean", "full_name": "Nat.dvd_left_iff_eq", "start": [242, 1], "end": [243, 87], "traced_tactics": [{"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "a b m\u271d n\u271d\u00b9 k m n\u271d : \u2115\nh : m = n\u271d\nn : \u2115\n\u22a2 n \u2223 m \u2194 n \u2223 n\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Real/EReal.lean", "full_name": "EReal.coe_coe_sign", "start": [1131, 1], "end": [1131, 82], "traced_tactics": [{"tactic": "cases x <;> rfl", "annotated_tactic": ["cases x <;> rfl", []], "state_before": "x : SignType\n\u22a2 \u2191\u2191x = \u2191x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/Padics/RingHoms.lean", "full_name": "PadicInt.lift_unique", "start": [647, 1], "end": [655, 90], "traced_tactics": [{"tactic": "ext1 r", "annotated_tactic": ["ext1 r", []], "state_before": "p : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\ng : R \u2192+* \u2124_[p]\nhg : \u2200 (n : \u2115), RingHom.comp (toZModPow n) g = f n\n\u22a2 lift f_compat = g", "state_after": "case a\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\ng : R \u2192+* \u2124_[p]\nhg : \u2200 (n : \u2115), RingHom.comp (toZModPow n) g = f n\nr : R\n\u22a2 \u2191(lift f_compat) r = \u2191g r"}, {"tactic": "apply eq_of_forall_dist_le", "annotated_tactic": ["apply <a>eq_of_forall_dist_le</a>", [{"full_name": "eq_of_forall_dist_le", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2944, 9], "def_end_pos": [2944, 29]}]], "state_before": "case a\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\ng : R \u2192+* \u2124_[p]\nhg : \u2200 (n : \u2115), RingHom.comp (toZModPow n) g = f n\nr : R\n\u22a2 \u2191(lift f_compat) r = \u2191g r", "state_after": "case a.h\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\ng : R \u2192+* \u2124_[p]\nhg : \u2200 (n : \u2115), RingHom.comp (toZModPow n) g = f n\nr : R\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 dist (\u2191(lift f_compat) r) (\u2191g r) \u2264 \u03b5"}, {"tactic": "intro \u03b5 h\u03b5", "annotated_tactic": ["intro \u03b5 h\u03b5", []], "state_before": "case a.h\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\ng : R \u2192+* \u2124_[p]\nhg : \u2200 (n : \u2115), RingHom.comp (toZModPow n) g = f n\nr : R\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 dist (\u2191(lift f_compat) r) (\u2191g r) \u2264 \u03b5", "state_after": "case a.h\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\ng : R \u2192+* \u2124_[p]\nhg : \u2200 (n : \u2115), RingHom.comp (toZModPow n) g = f n\nr : R\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 dist (\u2191(lift f_compat) r) (\u2191g r) \u2264 \u03b5"}, {"tactic": "obtain \u27e8n, hn\u27e9 := exists_pow_neg_lt p h\u03b5", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 := <a>exists_pow_neg_lt</a> p h\u03b5", [{"full_name": "PadicInt.exists_pow_neg_lt", "def_path": "Mathlib/NumberTheory/Padics/PadicIntegers.lean", "def_pos": [331, 9], "def_end_pos": [331, 26]}]], "state_before": "case a.h\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\ng : R \u2192+* \u2124_[p]\nhg : \u2200 (n : \u2115), RingHom.comp (toZModPow n) g = f n\nr : R\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 dist (\u2191(lift f_compat) r) (\u2191g r) \u2264 \u03b5", "state_after": "case a.h.intro\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\ng : R \u2192+* \u2124_[p]\nhg : \u2200 (n : \u2115), RingHom.comp (toZModPow n) g = f n\nr : R\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nn : \u2115\nhn : \u2191p ^ (-\u2191n) < \u03b5\n\u22a2 dist (\u2191(lift f_compat) r) (\u2191g r) \u2264 \u03b5"}, {"tactic": "apply le_trans _ (le_of_lt hn)", "annotated_tactic": ["apply <a>le_trans</a> _ (<a>le_of_lt</a> hn)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case a.h.intro\np : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\ng : R \u2192+* \u2124_[p]\nhg : \u2200 (n : \u2115), RingHom.comp (toZModPow n) g = f n\nr : R\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nn : \u2115\nhn : \u2191p ^ (-\u2191n) < \u03b5\n\u22a2 dist (\u2191(lift f_compat) r) (\u2191g r) \u2264 \u03b5", "state_after": "p : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\ng : R \u2192+* \u2124_[p]\nhg : \u2200 (n : \u2115), RingHom.comp (toZModPow n) g = f n\nr : R\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nn : \u2115\nhn : \u2191p ^ (-\u2191n) < \u03b5\n\u22a2 dist (\u2191(lift f_compat) r) (\u2191g r) \u2264 \u2191p ^ (-\u2191n)"}, {"tactic": "rw [dist_eq_norm, norm_le_pow_iff_mem_span_pow, \u2190 ker_toZModPow, RingHom.mem_ker,\n  RingHom.map_sub, \u2190 RingHom.comp_apply, \u2190 RingHom.comp_apply, lift_spec, hg, sub_self]", "annotated_tactic": ["rw [<a>dist_eq_norm</a>, <a>norm_le_pow_iff_mem_span_pow</a>, \u2190 <a>ker_toZModPow</a>, <a>RingHom.mem_ker</a>,\n    <a>RingHom.map_sub</a>, \u2190 <a>RingHom.comp_apply</a>, \u2190 <a>RingHom.comp_apply</a>, <a>lift_spec</a>, hg, <a>sub_self</a>]", [{"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [383, 7], "def_end_pos": [383, 19]}, {"full_name": "PadicInt.norm_le_pow_iff_mem_span_pow", "def_path": "Mathlib/NumberTheory/Padics/PadicIntegers.lean", "def_pos": [545, 9], "def_end_pos": [545, 37]}, {"full_name": "PadicInt.ker_toZModPow", "def_path": "Mathlib/NumberTheory/Padics/RingHoms.lean", "def_pos": [405, 9], "def_end_pos": [405, 22]}, {"full_name": "RingHom.mem_ker", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [2074, 9], "def_end_pos": [2074, 16]}, {"full_name": "RingHom.map_sub", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [618, 19], "def_end_pos": [618, 26]}, {"full_name": "RingHom.comp_apply", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [672, 9], "def_end_pos": [672, 19]}, {"full_name": "RingHom.comp_apply", "def_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "def_pos": [672, 9], "def_end_pos": [672, 19]}, {"full_name": "PadicInt.lift_spec", "def_path": "Mathlib/NumberTheory/Padics/RingHoms.lean", "def_pos": [637, 9], "def_end_pos": [637, 18]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}]], "state_before": "p : \u2115\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst\u271d : NonAssocSemiring R\nf : (k : \u2115) \u2192 R \u2192+* ZMod (p ^ k)\nf_compat : \u2200 (k1 k2 : \u2115) (hk : k1 \u2264 k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 \u2223 p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\ng : R \u2192+* \u2124_[p]\nhg : \u2200 (n : \u2115), RingHom.comp (toZModPow n) g = f n\nr : R\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nn : \u2115\nhn : \u2191p ^ (-\u2191n) < \u03b5\n\u22a2 dist (\u2191(lift f_compat) r) (\u2191g r) \u2264 \u2191p ^ (-\u2191n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/RegularExpressions.lean", "full_name": "RegularExpression.one_rmatch_iff", "start": [220, 1], "end": [221, 49], "traced_tactics": [{"tactic": "induction x <;> simp [rmatch, matchEpsilon, *]", "annotated_tactic": ["induction x <;> simp [<a>rmatch</a>, <a>matchEpsilon</a>, *]", [{"full_name": "RegularExpression.rmatch", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [210, 5], "def_end_pos": [210, 11]}, {"full_name": "RegularExpression.matchEpsilon", "def_path": "Mathlib/Computability/RegularExpressions.lean", "def_pos": [157, 5], "def_end_pos": [157, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ndec : DecidableEq \u03b1\na b : \u03b1\nx : List \u03b1\n\u22a2 rmatch 1 x = true \u2194 x = []", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Sites/CoverLifting.lean", "full_name": "CategoryTheory.RanIsSheafOfCoverLifting.gluedSection_isAmalgamation", "start": [255, 1], "end": [267, 49], "traced_tactics": [{"tactic": "intro V fV hV", "annotated_tactic": ["intro V fV hV", []], "state_before": "C D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\n\u22a2 IsAmalgamation x (gluedSection hu \u2131 hS hx)", "state_after": "C D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\n\u22a2 ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).map fV.op (gluedSection hu \u2131 hS hx) = x fV hV"}, {"tactic": "refine limit.hom_ext (\u03bb (W : StructuredArrow (op V) G.op) => ?_)", "annotated_tactic": ["refine <a>limit.hom_ext</a> (\u03bb (W : <a>StructuredArrow</a> (<a>op</a> V) G.op) => ?_)", [{"full_name": "CategoryTheory.Limits.limit.hom_ext", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [266, 9], "def_end_pos": [266, 22]}, {"full_name": "CategoryTheory.StructuredArrow", "def_path": "Mathlib/CategoryTheory/StructuredArrow.lean", "def_pos": [39, 5], "def_end_pos": [39, 20]}, {"full_name": "Opposite.op", "def_path": "Mathlib/Data/Opposite.lean", "def_pos": [53, 5], "def_end_pos": [53, 7]}]], "state_before": "C D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\n\u22a2 ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).map fV.op (gluedSection hu \u2131 hS hx) = x fV hV", "state_after": "C D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\n\u22a2 ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).map fV.op (gluedSection hu \u2131 hS hx) \u226b\n      limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W =\n    x fV hV \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W"}, {"tactic": "simp only [Functor.comp_map, limit.lift_pre, coyoneda_obj_map, ran_obj_map, gluedSection]", "annotated_tactic": ["simp only [<a>Functor.comp_map</a>, <a>limit.lift_pre</a>, <a>coyoneda_obj_map</a>, <a>ran_obj_map</a>, <a>gluedSection</a>]", [{"full_name": "CategoryTheory.Functor.comp_map", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 17]}, {"full_name": "CategoryTheory.Limits.limit.lift_pre", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [420, 9], "def_end_pos": [420, 23]}, {"full_name": "CategoryTheory.coyoneda_obj_map", "def_path": "Mathlib/CategoryTheory/Yoneda.lean", "def_pos": [51, 3], "def_end_pos": [51, 8]}, {"full_name": "CategoryTheory.ran_obj_map", "def_path": "Mathlib/CategoryTheory/Limits/KanExtension.lean", "def_pos": [161, 3], "def_end_pos": [161, 9]}, {"full_name": "CategoryTheory.RanIsSheafOfCoverLifting.gluedSection", "def_path": "Mathlib/CategoryTheory/Sites/CoverLifting.lean", "def_pos": [216, 5], "def_end_pos": [216, 17]}]], "state_before": "C D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\n\u22a2 ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).map fV.op (gluedSection hu \u2131 hS hx) \u226b\n      limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W =\n    x fV hV \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W", "state_after": "C D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\n\u22a2 limit.lift (StructuredArrow.map fV.op \u22d9 Ran.diagram G.op \u2131.val (op U))\n        (Cone.whisker (StructuredArrow.map fV.op) (gluedLimitCone hu \u2131 hS hx)) \u226b\n      limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W =\n    x fV hV \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W"}, {"tactic": "erw [limit.lift_\u03c0]", "annotated_tactic": ["erw [<a>limit.lift_\u03c0</a>]", [{"full_name": "CategoryTheory.Limits.limit.lift_\u03c0", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [192, 9], "def_end_pos": [192, 21]}]], "state_before": "C D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\n\u22a2 limit.lift (StructuredArrow.map fV.op \u22d9 Ran.diagram G.op \u2131.val (op U))\n        (Cone.whisker (StructuredArrow.map fV.op) (gluedLimitCone hu \u2131 hS hx)) \u226b\n      limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W =\n    x fV hV \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W", "state_after": "C D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\n\u22a2 (Cone.whisker (StructuredArrow.map fV.op) (gluedLimitCone hu \u2131 hS hx)).\u03c0.app W =\n    x fV hV \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "C D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\n\u22a2 (Cone.whisker (StructuredArrow.map fV.op) (gluedLimitCone hu \u2131 hS hx)).\u03c0.app W =\n    x fV hV \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W", "state_after": "C D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\n\u22a2 x fV hV \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W =\n    (Cone.whisker (StructuredArrow.map fV.op) (gluedLimitCone hu \u2131 hS hx)).\u03c0.app W"}, {"tactic": "convert helper hu \u2131 hS hx _ (x fV hV) _ _ using 1", "annotated_tactic": ["convert <a>helper</a> hu \u2131 hS hx _ (x fV hV) _ _ using 1", [{"full_name": "CategoryTheory.RanIsSheafOfCoverLifting.helper", "def_path": "Mathlib/CategoryTheory/Sites/CoverLifting.lean", "def_pos": [227, 9], "def_end_pos": [227, 15]}]], "state_before": "C D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\n\u22a2 x fV hV \u226b limit.\u03c0 (Ran.diagram G.op \u2131.val (op V)) W =\n    (Cone.whisker (StructuredArrow.map fV.op) (gluedLimitCone hu \u2131 hS hx)).\u03c0.app W", "state_after": "case convert_3\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\n\u22a2 \u2200 {V' : C} {fV_1 : G.obj V' \u27f6 V} (hV_1 : S.arrows (fV_1 \u226b fV)),\n    x fV hV \u226b ((ran G.op).obj \u2131.val).map fV_1.op = x (fV_1 \u226b fV) hV_1"}, {"tactic": "intro V' fV' hV'", "annotated_tactic": ["intro V' fV' hV'", []], "state_before": "case convert_3\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\n\u22a2 \u2200 {V' : C} {fV_1 : G.obj V' \u27f6 V} (hV_1 : S.arrows (fV_1 \u226b fV)),\n    x fV hV \u226b ((ran G.op).obj \u2131.val).map fV_1.op = x (fV_1 \u226b fV) hV_1", "state_after": "case convert_3\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\nV' : C\nfV' : G.obj V' \u27f6 V\nhV' : S.arrows (fV' \u226b fV)\n\u22a2 x fV hV \u226b ((ran G.op).obj \u2131.val).map fV'.op = x (fV' \u226b fV) hV'"}, {"tactic": "convert hx fV' (\ud835\udfd9 _) hV hV' (by rw [Category.id_comp])", "annotated_tactic": ["convert hx fV' (\ud835\udfd9 _) hV hV' (by rw [<a>Category.id_comp</a>])", [{"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [155, 3], "def_end_pos": [155, 10]}]], "state_before": "case convert_3\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\nV' : C\nfV' : G.obj V' \u27f6 V\nhV' : S.arrows (fV' \u226b fV)\n\u22a2 x fV hV \u226b ((ran G.op).obj \u2131.val).map fV'.op = x (fV' \u226b fV) hV'", "state_after": "case h.e'_3.h\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\nV' : C\nfV' : G.obj V' \u27f6 V\nhV' : S.arrows (fV' \u226b fV)\ne_1\u271d :\n  (X \u27f6 ((ran G.op).obj \u2131.val).obj (op (G.obj V'))) = ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).obj (op (G.obj V'))\n\u22a2 x (fV' \u226b fV) hV' = ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).map (\ud835\udfd9 (G.obj V')).op (x (fV' \u226b fV) hV')"}, {"tactic": "simp only [op_id, FunctorToTypes.map_id_apply]", "annotated_tactic": ["simp only [<a>op_id</a>, <a>FunctorToTypes.map_id_apply</a>]", [{"full_name": "CategoryTheory.op_id", "def_path": "Mathlib/CategoryTheory/Opposites.lean", "def_pos": [78, 9], "def_end_pos": [78, 14]}, {"full_name": "CategoryTheory.FunctorToTypes.map_id_apply", "def_path": "Mathlib/CategoryTheory/Types.lean", "def_pos": [148, 9], "def_end_pos": [148, 21]}]], "state_before": "case h.e'_3.h\nC D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\nV' : C\nfV' : G.obj V' \u27f6 V\nhV' : S.arrows (fV' \u226b fV)\ne_1\u271d :\n  (X \u27f6 ((ran G.op).obj \u2131.val).obj (op (G.obj V'))) = ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).obj (op (G.obj V'))\n\u22a2 x (fV' \u226b fV) hV' = ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)).map (\ud835\udfd9 (G.obj V')).op (x (fV' \u226b fV) hV')", "state_after": "no goals"}, {"tactic": "rw [Category.id_comp]", "annotated_tactic": ["rw [<a>Category.id_comp</a>]", [{"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [155, 3], "def_end_pos": [155, 10]}]], "state_before": "C D : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Category.{v, u} D\nA : Type w\ninst\u271d\u00b9 : Category.{max u v, w} A\ninst\u271d : HasLimits A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nG : C \u2964 D\nhu : CoverLifting J K G\n\u2131 : Sheaf J A\nX : A\nU : D\nS : Sieve U\nhS : S \u2208 GrothendieckTopology.sieves K U\nx : FamilyOfElements ((ran G.op).obj \u2131.val \u22d9 coyoneda.obj (op X)) S.arrows\nhx : Compatible x\nV : D\nfV : V \u27f6 U\nhV : S.arrows fV\nW : StructuredArrow (op V) G.op\nV' : C\nfV' : G.obj V' \u27f6 V\nhV' : S.arrows (fV' \u226b fV)\n\u22a2 fV' \u226b fV = \ud835\udfd9 (G.obj V') \u226b fV' \u226b fV", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/ContDiff.lean", "full_name": "contDiffOn_succ_of_fderiv_apply", "start": [1941, 1], "end": [1944, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/AlgebraMap.lean", "full_name": "Polynomial.coeff_zero_eq_aeval_zero", "start": [318, 1], "end": [319, 33], "traced_tactics": [{"tactic": "simp [coeff_zero_eq_eval_zero]", "annotated_tactic": ["simp [<a>coeff_zero_eq_eval_zero</a>]", [{"full_name": "Polynomial.coeff_zero_eq_eval_zero", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [509, 9], "def_end_pos": [509, 32]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type u_1\nB' : Type u_2\na b : R\nn : \u2115\ninst\u271d\u2076 : CommSemiring A'\ninst\u271d\u2075 : Semiring B'\ninst\u271d\u2074 : CommSemiring R\np\u271d q : R[X]\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\nB : Type u_3\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nx : A\np : R[X]\n\u22a2 coeff p 0 = \u2191(aeval 0) p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/CofilteredSystem.lean", "full_name": "nonempty_sections_of_finite_cofiltered_system.init", "start": [66, 1], "end": [74, 16], "traced_tactics": [{"tactic": "let F' : J \u2964 TopCat := F \u22d9 TopCat.discrete", "annotated_tactic": ["let F' : J \u2964 <a>TopCat</a> := F \u22d9 <a>TopCat.discrete</a>", [{"full_name": "TopCat", "def_path": "Mathlib/Topology/Category/TopCat/Basic.lean", "def_pos": [30, 5], "def_end_pos": [30, 11]}, {"full_name": "TopCat.discrete", "def_path": "Mathlib/Topology/Category/TopCat/Basic.lean", "def_pos": [104, 5], "def_end_pos": [104, 13]}]], "state_before": "J : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofilteredOrEmpty J\nF : J \u2964 Type u\nhf : \u2200 (j : J), Finite (F.obj j)\nhne : \u2200 (j : J), _root_.Nonempty (F.obj j)\n\u22a2 Set.Nonempty (Functor.sections F)", "state_after": "J : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofilteredOrEmpty J\nF : J \u2964 Type u\nhf : \u2200 (j : J), Finite (F.obj j)\nhne : \u2200 (j : J), _root_.Nonempty (F.obj j)\nF' : J \u2964 TopCat := F \u22d9 TopCat.discrete\n\u22a2 Set.Nonempty (Functor.sections F)"}, {"tactic": "haveI : \u2200 j, DiscreteTopology (F'.obj j) := fun _ => \u27e8rfl\u27e9", "annotated_tactic": ["haveI : \u2200 j, <a>DiscreteTopology</a> (F'.obj j) := fun _ => \u27e8<a>rfl</a>\u27e9", [{"full_name": "DiscreteTopology", "def_path": "Mathlib/Topology/Order.lean", "def_pos": [272, 7], "def_end_pos": [272, 23]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "J : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofilteredOrEmpty J\nF : J \u2964 Type u\nhf : \u2200 (j : J), Finite (F.obj j)\nhne : \u2200 (j : J), _root_.Nonempty (F.obj j)\nF' : J \u2964 TopCat := F \u22d9 TopCat.discrete\n\u22a2 Set.Nonempty (Functor.sections F)", "state_after": "J : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofilteredOrEmpty J\nF : J \u2964 Type u\nhf : \u2200 (j : J), Finite (F.obj j)\nhne : \u2200 (j : J), _root_.Nonempty (F.obj j)\nF' : J \u2964 TopCat := F \u22d9 TopCat.discrete\nthis : \u2200 (j : J), DiscreteTopology \u2191(F'.obj j)\n\u22a2 Set.Nonempty (Functor.sections F)"}, {"tactic": "haveI : \u2200 j, Finite (F'.obj j) := hf", "annotated_tactic": ["haveI : \u2200 j, <a>Finite</a> (F'.obj j) := hf", [{"full_name": "Finite", "def_path": "Mathlib/Data/Finite/Defs.lean", "def_pos": [58, 17], "def_end_pos": [58, 23]}]], "state_before": "J : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofilteredOrEmpty J\nF : J \u2964 Type u\nhf : \u2200 (j : J), Finite (F.obj j)\nhne : \u2200 (j : J), _root_.Nonempty (F.obj j)\nF' : J \u2964 TopCat := F \u22d9 TopCat.discrete\nthis : \u2200 (j : J), DiscreteTopology \u2191(F'.obj j)\n\u22a2 Set.Nonempty (Functor.sections F)", "state_after": "J : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofilteredOrEmpty J\nF : J \u2964 Type u\nhf : \u2200 (j : J), Finite (F.obj j)\nhne : \u2200 (j : J), _root_.Nonempty (F.obj j)\nF' : J \u2964 TopCat := F \u22d9 TopCat.discrete\nthis\u271d : \u2200 (j : J), DiscreteTopology \u2191(F'.obj j)\nthis : \u2200 (j : J), Finite \u2191(F'.obj j)\n\u22a2 Set.Nonempty (Functor.sections F)"}, {"tactic": "haveI : \u2200 j, Nonempty (F'.obj j) := hne", "annotated_tactic": ["haveI : \u2200 j, <a>Nonempty</a> (F'.obj j) := hne", [{"full_name": "Nonempty", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [686, 17], "def_end_pos": [686, 25]}]], "state_before": "J : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofilteredOrEmpty J\nF : J \u2964 Type u\nhf : \u2200 (j : J), Finite (F.obj j)\nhne : \u2200 (j : J), _root_.Nonempty (F.obj j)\nF' : J \u2964 TopCat := F \u22d9 TopCat.discrete\nthis\u271d : \u2200 (j : J), DiscreteTopology \u2191(F'.obj j)\nthis : \u2200 (j : J), Finite \u2191(F'.obj j)\n\u22a2 Set.Nonempty (Functor.sections F)", "state_after": "J : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofilteredOrEmpty J\nF : J \u2964 Type u\nhf : \u2200 (j : J), Finite (F.obj j)\nhne : \u2200 (j : J), _root_.Nonempty (F.obj j)\nF' : J \u2964 TopCat := F \u22d9 TopCat.discrete\nthis\u271d\u00b9 : \u2200 (j : J), DiscreteTopology \u2191(F'.obj j)\nthis\u271d : \u2200 (j : J), Finite \u2191(F'.obj j)\nthis : \u2200 (j : J), _root_.Nonempty \u2191(F'.obj j)\n\u22a2 Set.Nonempty (Functor.sections F)"}, {"tactic": "obtain \u27e8\u27e8u, hu\u27e9\u27e9 := TopCat.nonempty_limitCone_of_compact_t2_cofiltered_system.{u} F'", "annotated_tactic": ["obtain \u27e8\u27e8u, hu\u27e9\u27e9 := <a>TopCat.nonempty_limitCone_of_compact_t2_cofiltered_system</a>.{u} F'", [{"full_name": "TopCat.nonempty_limitCone_of_compact_t2_cofiltered_system", "def_path": "Mathlib/Topology/Category/TopCat/Limits/Konig.lean", "def_pos": [129, 9], "def_end_pos": [129, 59]}]], "state_before": "J : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofilteredOrEmpty J\nF : J \u2964 Type u\nhf : \u2200 (j : J), Finite (F.obj j)\nhne : \u2200 (j : J), _root_.Nonempty (F.obj j)\nF' : J \u2964 TopCat := F \u22d9 TopCat.discrete\nthis\u271d\u00b9 : \u2200 (j : J), DiscreteTopology \u2191(F'.obj j)\nthis\u271d : \u2200 (j : J), Finite \u2191(F'.obj j)\nthis : \u2200 (j : J), _root_.Nonempty \u2191(F'.obj j)\n\u22a2 Set.Nonempty (Functor.sections F)", "state_after": "case intro.mk\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofilteredOrEmpty J\nF : J \u2964 Type u\nhf : \u2200 (j : J), Finite (F.obj j)\nhne : \u2200 (j : J), _root_.Nonempty (F.obj j)\nF' : J \u2964 TopCat := F \u22d9 TopCat.discrete\nthis\u271d\u00b9 : \u2200 (j : J), DiscreteTopology \u2191(F'.obj j)\nthis\u271d : \u2200 (j : J), Finite \u2191(F'.obj j)\nthis : \u2200 (j : J), _root_.Nonempty \u2191(F'.obj j)\nu : (j : J) \u2192 \u2191(F'.obj j)\nhu : u \u2208 {u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(F'.map f) (u i) = u j}\n\u22a2 Set.Nonempty (Functor.sections F)"}, {"tactic": "exact \u27e8u, hu\u27e9", "annotated_tactic": ["exact \u27e8u, hu\u27e9", []], "state_before": "case intro.mk\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofilteredOrEmpty J\nF : J \u2964 Type u\nhf : \u2200 (j : J), Finite (F.obj j)\nhne : \u2200 (j : J), _root_.Nonempty (F.obj j)\nF' : J \u2964 TopCat := F \u22d9 TopCat.discrete\nthis\u271d\u00b9 : \u2200 (j : J), DiscreteTopology \u2191(F'.obj j)\nthis\u271d : \u2200 (j : J), Finite \u2191(F'.obj j)\nthis : \u2200 (j : J), _root_.Nonempty \u2191(F'.obj j)\nu : (j : J) \u2192 \u2191(F'.obj j)\nhu : u \u2208 {u | \u2200 {i j : J} (f : i \u27f6 j), \u2191(F'.map f) (u i) = u j}\n\u22a2 Set.Nonempty (Functor.sections F)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.range_eq_empty_of_isEmpty", "start": [535, 1], "end": [542, 13], "traced_tactics": [{"tactic": "rw [\u2190 Finset.not_nonempty_iff_eq_empty]", "annotated_tactic": ["rw [\u2190 <a>Finset.not_nonempty_iff_eq_empty</a>]", [{"full_name": "Finset.not_nonempty_iff_eq_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [609, 9], "def_end_pos": [609, 34]}]], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\n\u03b2 : Type u_5\nh\u03b1 : IsEmpty \u03b1\nf : \u03b1 \u2192\u209b \u03b2\n\u22a2 SimpleFunc.range f = \u2205", "state_after": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\n\u03b2 : Type u_5\nh\u03b1 : IsEmpty \u03b1\nf : \u03b1 \u2192\u209b \u03b2\n\u22a2 \u00acFinset.Nonempty (SimpleFunc.range f)"}, {"tactic": "by_contra h", "annotated_tactic": ["by_contra h", []], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\n\u03b2 : Type u_5\nh\u03b1 : IsEmpty \u03b1\nf : \u03b1 \u2192\u209b \u03b2\n\u22a2 \u00acFinset.Nonempty (SimpleFunc.range f)", "state_after": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\n\u03b2 : Type u_5\nh\u03b1 : IsEmpty \u03b1\nf : \u03b1 \u2192\u209b \u03b2\nh : Finset.Nonempty (SimpleFunc.range f)\n\u22a2 False"}, {"tactic": "obtain \u27e8y, hy_mem\u27e9 := h", "annotated_tactic": ["obtain \u27e8y, hy_mem\u27e9 := h", []], "state_before": "\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\n\u03b2 : Type u_5\nh\u03b1 : IsEmpty \u03b1\nf : \u03b1 \u2192\u209b \u03b2\nh : Finset.Nonempty (SimpleFunc.range f)\n\u22a2 False", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\n\u03b2 : Type u_5\nh\u03b1 : IsEmpty \u03b1\nf : \u03b1 \u2192\u209b \u03b2\ny : \u03b2\nhy_mem : y \u2208 SimpleFunc.range f\n\u22a2 False"}, {"tactic": "rw [SimpleFunc.mem_range, Set.mem_range] at hy_mem", "annotated_tactic": ["rw [<a>SimpleFunc.mem_range</a>, <a>Set.mem_range</a>] at hy_mem", [{"full_name": "MeasureTheory.SimpleFunc.mem_range", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [108, 9], "def_end_pos": [108, 18]}, {"full_name": "Set.mem_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [673, 9], "def_end_pos": [673, 18]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\n\u03b2 : Type u_5\nh\u03b1 : IsEmpty \u03b1\nf : \u03b1 \u2192\u209b \u03b2\ny : \u03b2\nhy_mem : y \u2208 SimpleFunc.range f\n\u22a2 False", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\n\u03b2 : Type u_5\nh\u03b1 : IsEmpty \u03b1\nf : \u03b1 \u2192\u209b \u03b2\ny : \u03b2\nhy_mem : \u2203 y_1, \u2191f y_1 = y\n\u22a2 False"}, {"tactic": "obtain \u27e8x, hxy\u27e9 := hy_mem", "annotated_tactic": ["obtain \u27e8x, hxy\u27e9 := hy_mem", []], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\n\u03b2 : Type u_5\nh\u03b1 : IsEmpty \u03b1\nf : \u03b1 \u2192\u209b \u03b2\ny : \u03b2\nhy_mem : \u2203 y_1, \u2191f y_1 = y\n\u22a2 False", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\n\u03b2 : Type u_5\nh\u03b1 : IsEmpty \u03b1\nf : \u03b1 \u2192\u209b \u03b2\ny : \u03b2\nx : \u03b1\nhxy : \u2191f x = y\n\u22a2 False"}, {"tactic": "rw [isEmpty_iff] at h\u03b1", "annotated_tactic": ["rw [<a>isEmpty_iff</a>] at h\u03b1", [{"full_name": "isEmpty_iff", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [94, 9], "def_end_pos": [94, 20]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\n\u03b2 : Type u_5\nh\u03b1 : IsEmpty \u03b1\nf : \u03b1 \u2192\u209b \u03b2\ny : \u03b2\nx : \u03b1\nhxy : \u2191f x = y\n\u22a2 False", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\n\u03b2 : Type u_5\nh\u03b1 : \u03b1 \u2192 False\nf : \u03b1 \u2192\u209b \u03b2\ny : \u03b2\nx : \u03b1\nhxy : \u2191f x = y\n\u22a2 False"}, {"tactic": "exact h\u03b1 x", "annotated_tactic": ["exact h\u03b1 x", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d : MeasurableSpace \u03b1\n\u03b2 : Type u_5\nh\u03b1 : \u03b1 \u2192 False\nf : \u03b1 \u2192\u209b \u03b2\ny : \u03b2\nx : \u03b1\nhxy : \u2191f x = y\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/Basic.lean", "full_name": "Nat.div_left_inj", "start": [712, 11], "end": [714, 63], "traced_tactics": [{"tactic": "refine \u27e8fun h => ?_, congr_arg fun n => n / d\u27e9", "annotated_tactic": ["refine \u27e8fun h => ?_, <a>congr_arg</a> fun n => n / d\u27e9", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}]], "state_before": "m n k a b d : \u2115\nhda : d \u2223 a\nhdb : d \u2223 b\n\u22a2 a / d = b / d \u2194 a = b", "state_after": "m n k a b d : \u2115\nhda : d \u2223 a\nhdb : d \u2223 b\nh : a / d = b / d\n\u22a2 a = b"}, {"tactic": "rw [\u2190 Nat.mul_div_cancel' hda, \u2190 Nat.mul_div_cancel' hdb, h]", "annotated_tactic": ["rw [\u2190 <a>Nat.mul_div_cancel'</a> hda, \u2190 <a>Nat.mul_div_cancel'</a> hdb, h]", [{"full_name": "Nat.mul_div_cancel'", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [943, 19], "def_end_pos": [943, 34]}, {"full_name": "Nat.mul_div_cancel'", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [943, 19], "def_end_pos": [943, 34]}]], "state_before": "m n k a b d : \u2115\nhda : d \u2223 a\nhdb : d \u2223 b\nh : a / d = b / d\n\u22a2 a = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.AEStronglyMeasurable.inv", "start": [1316, 11], "end": [1318, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Ring/BooleanRing.lean", "full_name": "RingHom.asBoolAlg_id", "start": [362, 1], "end": [363, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.nat_coe_zmod_eq_iff", "start": [532, 1], "end": [540, 16], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "p n : \u2115\nz : ZMod p\ninst\u271d : NeZero p\n\u22a2 \u2191n = z \u2194 \u2203 k, n = val z + p * k", "state_after": "case mp\np n : \u2115\nz : ZMod p\ninst\u271d : NeZero p\n\u22a2 \u2191n = z \u2192 \u2203 k, n = val z + p * k\n\ncase mpr\np n : \u2115\nz : ZMod p\ninst\u271d : NeZero p\n\u22a2 (\u2203 k, n = val z + p * k) \u2192 \u2191n = z"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "case mp\np n : \u2115\nz : ZMod p\ninst\u271d : NeZero p\n\u22a2 \u2191n = z \u2192 \u2203 k, n = val z + p * k", "state_after": "case mp\np n : \u2115\ninst\u271d : NeZero p\n\u22a2 \u2203 k, n = val \u2191n + p * k"}, {"tactic": "refine' \u27e8n / p, _\u27e9", "annotated_tactic": ["refine' \u27e8n / p, _\u27e9", []], "state_before": "case mp\np n : \u2115\ninst\u271d : NeZero p\n\u22a2 \u2203 k, n = val \u2191n + p * k", "state_after": "case mp\np n : \u2115\ninst\u271d : NeZero p\n\u22a2 n = val \u2191n + p * (n / p)"}, {"tactic": "rw [val_nat_cast, Nat.mod_add_div]", "annotated_tactic": ["rw [<a>val_nat_cast</a>, <a>Nat.mod_add_div</a>]", [{"full_name": "ZMod.val_nat_cast", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 21]}, {"full_name": "Nat.mod_add_div", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [550, 9], "def_end_pos": [550, 20]}]], "state_before": "case mp\np n : \u2115\ninst\u271d : NeZero p\n\u22a2 n = val \u2191n + p * (n / p)", "state_after": "no goals"}, {"tactic": "rintro \u27e8k, rfl\u27e9", "annotated_tactic": ["rintro \u27e8k, rfl\u27e9", []], "state_before": "case mpr\np n : \u2115\nz : ZMod p\ninst\u271d : NeZero p\n\u22a2 (\u2203 k, n = val z + p * k) \u2192 \u2191n = z", "state_after": "case mpr.intro\np : \u2115\nz : ZMod p\ninst\u271d : NeZero p\nk : \u2115\n\u22a2 \u2191(val z + p * k) = z"}, {"tactic": "rw [Nat.cast_add, nat_cast_zmod_val, Nat.cast_mul, nat_cast_self, zero_mul,\n  add_zero]", "annotated_tactic": ["rw [<a>Nat.cast_add</a>, <a>nat_cast_zmod_val</a>, <a>Nat.cast_mul</a>, <a>nat_cast_self</a>, <a>zero_mul</a>,\n      <a>add_zero</a>]", [{"full_name": "Nat.cast_add", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [146, 9], "def_end_pos": [146, 17]}, {"full_name": "ZMod.nat_cast_zmod_val", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [195, 9], "def_end_pos": [195, 26]}, {"full_name": "Nat.cast_mul", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [47, 9], "def_end_pos": [47, 17]}, {"full_name": "ZMod.nat_cast_self", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [132, 9], "def_end_pos": [132, 22]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "case mpr.intro\np : \u2115\nz : ZMod p\ninst\u271d : NeZero p\nk : \u2115\n\u22a2 \u2191(val z + p * k) = z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/Reverse.lean", "full_name": "Polynomial.reverse_natDegree_le", "start": [273, 1], "end": [278, 38], "traced_tactics": [{"tactic": "rw [natDegree_le_iff_degree_le, degree_le_iff_coeff_zero]", "annotated_tactic": ["rw [<a>natDegree_le_iff_degree_le</a>, <a>degree_le_iff_coeff_zero</a>]", [{"full_name": "Polynomial.natDegree_le_iff_degree_le", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [220, 9], "def_end_pos": [220, 35]}, {"full_name": "Polynomial.degree_le_iff_coeff_zero", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [1121, 9], "def_end_pos": [1121, 33]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\nf\u271d f : R[X]\n\u22a2 natDegree (reverse f) \u2264 natDegree f", "state_after": "R : Type u_1\ninst\u271d : Semiring R\nf\u271d f : R[X]\n\u22a2 \u2200 (m : \u2115), \u2191(natDegree f) < \u2191m \u2192 coeff (reverse f) m = 0"}, {"tactic": "intro n hn", "annotated_tactic": ["intro n hn", []], "state_before": "R : Type u_1\ninst\u271d : Semiring R\nf\u271d f : R[X]\n\u22a2 \u2200 (m : \u2115), \u2191(natDegree f) < \u2191m \u2192 coeff (reverse f) m = 0", "state_after": "R : Type u_1\ninst\u271d : Semiring R\nf\u271d f : R[X]\nn : \u2115\nhn : \u2191(natDegree f) < \u2191n\n\u22a2 coeff (reverse f) n = 0"}, {"tactic": "rw [Nat.cast_lt] at hn", "annotated_tactic": ["rw [<a>Nat.cast_lt</a>] at hn", [{"full_name": "Nat.cast_lt", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [96, 9], "def_end_pos": [96, 16]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\nf\u271d f : R[X]\nn : \u2115\nhn : \u2191(natDegree f) < \u2191n\n\u22a2 coeff (reverse f) n = 0", "state_after": "R : Type u_1\ninst\u271d : Semiring R\nf\u271d f : R[X]\nn : \u2115\nhn : natDegree f < n\n\u22a2 coeff (reverse f) n = 0"}, {"tactic": "rw [coeff_reverse, revAt, Function.Embedding.coeFn_mk, if_neg (not_le_of_gt hn),\n  coeff_eq_zero_of_natDegree_lt hn]", "annotated_tactic": ["rw [<a>coeff_reverse</a>, <a>revAt</a>, <a>Function.Embedding.coeFn_mk</a>, <a>if_neg</a> (<a>not_le_of_gt</a> hn),\n    <a>coeff_eq_zero_of_natDegree_lt</a> hn]", [{"full_name": "Polynomial.coeff_reverse", "def_path": "Mathlib/Data/Polynomial/Reverse.lean", "def_pos": [255, 9], "def_end_pos": [255, 22]}, {"full_name": "Polynomial.revAt", "def_path": "Mathlib/Data/Polynomial/Reverse.lean", "def_pos": [59, 5], "def_end_pos": [59, 10]}, {"full_name": "Function.Embedding.coeFn_mk", "def_path": "Mathlib/Logic/Embedding/Basic.lean", "def_pos": [115, 9], "def_end_pos": [115, 17]}, {"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}, {"full_name": "not_le_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [148, 9], "def_end_pos": [148, 21]}, {"full_name": "Polynomial.coeff_eq_zero_of_natDegree_lt", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [349, 9], "def_end_pos": [349, 38]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\nf\u271d f : R[X]\nn : \u2115\nhn : natDegree f < n\n\u22a2 coeff (reverse f) n = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/ContinuousFunction/Compact.lean", "full_name": "BoundedContinuousFunction.dist_toContinuousMap", "start": [118, 1], "end": [120, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/MeanValue.lean", "full_name": "StrictMonoOn.exists_slope_lt_deriv", "start": [1036, 1], "end": [1066, 13], "traced_tactics": [{"tactic": "by_cases h : \u2200 w \u2208 Ioo x y, deriv f w \u2260 0", "annotated_tactic": ["by_cases h : \u2200 w \u2208 <a>Ioo</a> x y, <a>deriv</a> f w \u2260 0", [{"full_name": "Set.Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "deriv", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [148, 5], "def_end_pos": [148, 10]}]], 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E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u00ac\u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a"}, {"tactic": "apply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h", "annotated_tactic": ["apply <a>StrictMonoOn.exists_slope_lt_deriv_aux</a> hf hxy hf'_mono h", [{"full_name": "StrictMonoOn.exists_slope_lt_deriv_aux", "def_path": "Mathlib/Analysis/Calculus/MeanValue.lean", "def_pos": [1023, 9], "def_end_pos": [1023, 47]}]], "state_before": "case pos\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a", "state_after": "no goals"}, {"tactic": "push_neg at h", "annotated_tactic": ["push_neg at h", []], "state_before": "case neg\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u00ac\u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a", "state_after": "case neg\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2203 w, w \u2208 Ioo x y \u2227 deriv f w = 0\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a"}, {"tactic": "rcases h with \u27e8w, \u27e8hxw, hwy\u27e9, hw\u27e9", "annotated_tactic": ["rcases h with \u27e8w, \u27e8hxw, hwy\u27e9, hw\u27e9", []], "state_before": "case neg\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2203 w, w \u2208 Ioo x y \u2227 deriv f w = 0\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a", "state_after": "case neg.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a"}, {"tactic": "refine' \u27e8b, \u27e8hxw.trans hwb, hby\u27e9, _\u27e9", "annotated_tactic": ["refine' \u27e8b, \u27e8hxw.trans hwb, hby\u27e9, _\u27e9", []], "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\nb : \u211d\nhb : (f y - f w) / (y - w) < deriv f b\nhwb : w < b\nhby : b < y\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\nb : \u211d\nhb : (f y - f w) / (y - w) < deriv f b\nhwb : w < b\nhby : b < y\n\u22a2 (f y - f x) / (y - x) < deriv f b"}, {"tactic": "simp only [div_lt_iff, hxy, hxw, hwy, sub_pos] at ha hb \u22a2", "annotated_tactic": ["simp only [<a>div_lt_iff</a>, hxy, hxw, hwy, <a>sub_pos</a>] at ha hb \u22a2", [{"full_name": "div_lt_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [169, 9], "def_end_pos": [169, 19]}, {"full_name": "sub_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [883, 30], "def_end_pos": [883, 37]}]], "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\nb : \u211d\nhb : (f y - f w) / (y - w) < deriv f b\nhwb : w < b\nhby : b < y\n\u22a2 (f y - f x) / (y - x) < deriv f b", "state_after": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nhxa : x < a\nhaw : a < w\nb : \u211d\nhwb : w < b\nhby : b < y\nha : f w - f x < deriv f a * (w - x)\nhb : f y - f w < deriv f b * (y - w)\n\u22a2 f y - f x < deriv f b * (y - x)"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nhxa : x < a\nhaw : a < w\nb : \u211d\nhwb : w < b\nhby : b < y\nha : f w - f x < deriv f a * (w - x)\nhb : f y - f w < deriv f b * (y - w)\nthis : deriv f a * (w - x) < deriv f b * (w - x)\n\u22a2 f y - f x < deriv f b * (y - x)", "state_after": "no goals"}, {"tactic": "apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _", "annotated_tactic": ["apply <a>StrictMonoOn.exists_slope_lt_deriv_aux</a> _ hxw _ _", [{"full_name": "StrictMonoOn.exists_slope_lt_deriv_aux", "def_path": "Mathlib/Analysis/Calculus/MeanValue.lean", "def_pos": [1023, 9], "def_end_pos": [1023, 47]}]], "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\n\u22a2 \u2203 a, a \u2208 Ioo x w \u2227 (f w - f x) / (w - x) < deriv f a", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\n\u22a2 ContinuousOn (fun w => f w) (Icc x w)\n\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\n\u22a2 StrictMonoOn (deriv fun w => f w) (Ioo x w)\n\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\n\u22a2 \u2200 (w_1 : \u211d), w_1 \u2208 Ioo x w \u2192 deriv (fun w => f w) w_1 \u2260 0"}, {"tactic": "exact hf.mono (Icc_subset_Icc le_rfl hwy.le)", "annotated_tactic": ["exact hf.mono (<a>Icc_subset_Icc</a> <a>le_rfl</a> hwy.le)", [{"full_name": "Set.Icc_subset_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [455, 9], "def_end_pos": [455, 23]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\n\u22a2 ContinuousOn (fun w => f w) (Icc x w)", "state_after": "no goals"}, {"tactic": "exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)", "annotated_tactic": ["exact hf'_mono.mono (<a>Ioo_subset_Ioo</a> <a>le_rfl</a> hwy.le)", [{"full_name": "Set.Ioo_subset_Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [431, 9], "def_end_pos": [431, 23]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\n\u22a2 StrictMonoOn (deriv fun w => f w) (Ioo x w)", "state_after": "no goals"}, {"tactic": "intro z hz", "annotated_tactic": ["intro z hz", []], "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\n\u22a2 \u2200 (w_1 : \u211d), w_1 \u2208 Ioo x w \u2192 deriv (fun w => f w) w_1 \u2260 0", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\nz : \u211d\nhz : z \u2208 Ioo x w\n\u22a2 deriv (fun w => f w) z \u2260 0"}, {"tactic": "rw [\u2190 hw]", "annotated_tactic": ["rw [\u2190 hw]", []], "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\nz : \u211d\nhz : z \u2208 Ioo x w\n\u22a2 deriv (fun w => f w) z \u2260 0", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\nz : \u211d\nhz : z \u2208 Ioo x w\n\u22a2 deriv (fun w => f w) z \u2260 deriv f w"}, {"tactic": "apply ne_of_lt", "annotated_tactic": ["apply <a>ne_of_lt</a>", [{"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}]], "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\nz : \u211d\nhz : z \u2208 Ioo x w\n\u22a2 deriv (fun w => f w) z \u2260 deriv f w", "state_after": "case h\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\nz : \u211d\nhz : z \u2208 Ioo x w\n\u22a2 deriv (fun w => f w) z < deriv f w"}, {"tactic": "exact hf'_mono \u27e8hz.1, hz.2.trans hwy\u27e9 \u27e8hxw, hwy\u27e9 hz.2", "annotated_tactic": ["exact hf'_mono \u27e8hz.1, hz.2.<a>trans</a> hwy\u27e9 \u27e8hxw, hwy\u27e9 hz.2", [{"full_name": "LT.lt.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [144, 7], "def_end_pos": [144, 18]}]], "state_before": "case h\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\nz : \u211d\nhz : z \u2208 Ioo x w\n\u22a2 deriv (fun w => f w) z < deriv f w", "state_after": "no goals"}, {"tactic": "apply StrictMonoOn.exists_slope_lt_deriv_aux _ hwy _ _", "annotated_tactic": ["apply <a>StrictMonoOn.exists_slope_lt_deriv_aux</a> _ hwy _ _", [{"full_name": "StrictMonoOn.exists_slope_lt_deriv_aux", "def_path": "Mathlib/Analysis/Calculus/MeanValue.lean", "def_pos": [1023, 9], "def_end_pos": [1023, 47]}]], "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\n\u22a2 \u2203 b, b \u2208 Ioo w y \u2227 (f y - f w) / (y - w) < deriv f b", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\n\u22a2 ContinuousOn (fun {y} => f y) (Icc w y)\n\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\n\u22a2 StrictMonoOn (deriv fun {y} => f y) (Ioo w y)\n\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\n\u22a2 \u2200 (w_1 : \u211d), w_1 \u2208 Ioo w y \u2192 deriv (fun {y} => f y) w_1 \u2260 0"}, {"tactic": "refine' hf.mono (Icc_subset_Icc hxw.le le_rfl)", "annotated_tactic": ["refine' hf.mono (<a>Icc_subset_Icc</a> hxw.le <a>le_rfl</a>)", [{"full_name": "Set.Icc_subset_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [455, 9], "def_end_pos": [455, 23]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\n\u22a2 ContinuousOn (fun {y} => f y) (Icc w y)", "state_after": "no goals"}, {"tactic": "exact hf'_mono.mono (Ioo_subset_Ioo hxw.le le_rfl)", "annotated_tactic": ["exact hf'_mono.mono (<a>Ioo_subset_Ioo</a> hxw.le <a>le_rfl</a>)", [{"full_name": "Set.Ioo_subset_Ioo", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [431, 9], "def_end_pos": [431, 23]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\n\u22a2 StrictMonoOn (deriv fun {y} => f y) (Ioo w y)", "state_after": "no goals"}, {"tactic": "intro z hz", "annotated_tactic": ["intro z hz", []], "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\n\u22a2 \u2200 (w_1 : \u211d), w_1 \u2208 Ioo w y \u2192 deriv (fun {y} => f y) w_1 \u2260 0", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\nz : \u211d\nhz : z \u2208 Ioo w y\n\u22a2 deriv (fun {y} => f y) z \u2260 0"}, {"tactic": "rw [\u2190 hw]", "annotated_tactic": ["rw [\u2190 hw]", []], "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\nz : \u211d\nhz : z \u2208 Ioo w y\n\u22a2 deriv (fun {y} => f y) z \u2260 0", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\nz : \u211d\nhz : z \u2208 Ioo w y\n\u22a2 deriv (fun {y} => f y) z \u2260 deriv f w"}, {"tactic": "apply ne_of_gt", "annotated_tactic": ["apply <a>ne_of_gt</a>", [{"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}]], "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\nz : \u211d\nhz : z \u2208 Ioo w y\n\u22a2 deriv (fun {y} => f y) z \u2260 deriv f w", "state_after": "case h\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\nz : \u211d\nhz : z \u2208 Ioo w y\n\u22a2 deriv f w < deriv (fun {y} => f y) z"}, {"tactic": "exact hf'_mono \u27e8hxw, hwy\u27e9 \u27e8hxw.trans hz.1, hz.2\u27e9 hz.1", "annotated_tactic": ["exact hf'_mono \u27e8hxw, hwy\u27e9 \u27e8hxw.trans hz.1, hz.2\u27e9 hz.1", []], "state_before": "case h\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\nz : \u211d\nhz : z \u2208 Ioo w y\n\u22a2 deriv f w < deriv (fun {y} => f y) z", "state_after": "no goals"}, {"tactic": "apply mul_lt_mul _ le_rfl (sub_pos.2 hxw) _", "annotated_tactic": ["apply <a>mul_lt_mul</a> _ <a>le_rfl</a> (<a>sub_pos</a>.2 hxw) _", [{"full_name": "mul_lt_mul", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [521, 9], "def_end_pos": [521, 19]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "sub_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [883, 30], "def_end_pos": [883, 37]}]], "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nhxa : x < a\nhaw : a < w\nb : \u211d\nhwb : w < b\nhby : b < y\nha : f w - f x < deriv f a * (w - x)\nhb : f y - f w < deriv f b * (y - w)\n\u22a2 deriv f a * (w - x) < deriv f b * (w - x)", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nhxa : x < a\nhaw : a < w\nb : \u211d\nhwb : w < b\nhby : b < y\nha : f w - f x < deriv f a * (w - x)\nhb : f y - f w < deriv f b * (y - w)\n\u22a2 deriv f a < deriv f b\n\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nhxa : x < a\nhaw : a < w\nb : \u211d\nhwb : w < b\nhby : b < y\nha : f w - f x < deriv f a * (w - x)\nhb : f y - f w < deriv f b * (y - w)\n\u22a2 0 \u2264 deriv f b"}, {"tactic": "exact hf'_mono \u27e8hxa, haw.trans hwy\u27e9 \u27e8hxw.trans hwb, hby\u27e9 (haw.trans hwb)", "annotated_tactic": ["exact hf'_mono \u27e8hxa, haw.trans hwy\u27e9 \u27e8hxw.trans hwb, hby\u27e9 (haw.trans hwb)", []], "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nhxa : x < a\nhaw : a < w\nb : \u211d\nhwb : w < b\nhby : b < y\nha : f w - f x < deriv f a * (w - x)\nhb : f y - f w < deriv f b * (y - w)\n\u22a2 deriv f a < deriv f b", "state_after": "no goals"}, {"tactic": "rw [\u2190 hw]", "annotated_tactic": ["rw [\u2190 hw]", []], "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nhxa : x < a\nhaw : a < w\nb : \u211d\nhwb : w < b\nhby : b < y\nha : f w - f x < deriv f a * (w - x)\nhb : f y - f w < deriv f b * (y - w)\n\u22a2 0 \u2264 deriv f b", "state_after": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nhxa : x < a\nhaw : a < w\nb : \u211d\nhwb : w < b\nhby : b < y\nha : f w - f x < deriv f a * (w - x)\nhb : f y - f w < deriv f b * (y - w)\n\u22a2 deriv f w \u2264 deriv f b"}, {"tactic": "exact (hf'_mono \u27e8hxw, hwy\u27e9 \u27e8hxw.trans hwb, hby\u27e9 hwb).le", "annotated_tactic": ["exact (hf'_mono \u27e8hxw, hwy\u27e9 \u27e8hxw.trans hwb, hby\u27e9 hwb).<a>le</a>", [{"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "E : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nhxa : x < a\nhaw : a < w\nb : \u211d\nhwb : w < b\nhby : b < y\nha : f w - f x < deriv f a * (w - x)\nhb : f y - f w < deriv f b * (y - w)\n\u22a2 deriv f w \u2264 deriv f b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/InnerProductSpace/l2Space.lean", "full_name": "HilbertBasis.tsum_inner_mul_inner", "start": [496, 11], "end": [498, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Order/Floor.lean", "full_name": "tendsto_ceil_atTop", "start": [42, 1], "end": [43, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "full_name": "SimpleGraph.Walk.length_concat", "start": [405, 1], "end": [406, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.mem_range'", "start": [1990, 1], "end": [1995, 71], "traced_tactics": [{"tactic": "simp [range', Nat.not_lt_zero]", "annotated_tactic": ["simp [<a>range'</a>, <a>Nat.not_lt_zero</a>]", [{"full_name": "List.range'", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1195, 5], "def_end_pos": [1195, 11]}, {"full_name": "Nat.not_lt_zero", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1575, 9], "def_end_pos": [1575, 24]}]], "state_before": "m s step : Nat\n\u22a2 m \u2208 range' s 0 step \u2194 \u2203 i, i < 0 \u2227 m = s + step * i", "state_after": "no goals"}, {"tactic": "have h (i) : i \u2264 n \u2194 i = 0 \u2228 \u2203 j, i = succ j \u2227 j < n := by cases i <;> simp [Nat.succ_le]", "annotated_tactic": ["have h (i) : i \u2264 n \u2194 i = 0 \u2228 \u2203 j, i = <a>succ</a> j \u2227 j < n := by cases i <;> simp [<a>Nat.succ_le</a>]", [{"full_name": "Nat.succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1044, 5], "def_end_pos": [1044, 9]}, {"full_name": "Nat.succ_le", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [221, 9], "def_end_pos": [221, 16]}]], "state_before": "m s step n : Nat\n\u22a2 m \u2208 range' s (n + 1) step \u2194 \u2203 i, i < n + 1 \u2227 m = s + step * i", "state_after": "m s step n : Nat\nh : \u2200 (i : Nat), i \u2264 n \u2194 i = 0 \u2228 \u2203 j, i = succ j \u2227 j < n\n\u22a2 m \u2208 range' s (n + 1) step \u2194 \u2203 i, i < n + 1 \u2227 m = s + step * i"}, {"tactic": "simp [range', mem_range', Nat.lt_succ, h]", "annotated_tactic": ["simp [<a>range'</a>, mem_range', <a>Nat.lt_succ</a>, h]", [{"full_name": "List.range'", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1195, 5], "def_end_pos": [1195, 11]}, {"full_name": "Nat.lt_succ", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [223, 9], "def_end_pos": [223, 16]}]], "state_before": "m s step n : Nat\nh : \u2200 (i : Nat), i \u2264 n \u2194 i = 0 \u2228 \u2203 j, i = succ j \u2227 j < n\n\u22a2 m \u2208 range' s (n + 1) step \u2194 \u2203 i, i < n + 1 \u2227 m = s + step * i", "state_after": "m s step n : Nat\nh : \u2200 (i : Nat), i \u2264 n \u2194 i = 0 \u2228 \u2203 j, i = succ j \u2227 j < n\n\u22a2 (m = s \u2228 \u2203 i, i < n \u2227 m = s + step + step * i) \u2194 m = s \u2228 \u2203 a, (\u2203 j, a = succ j \u2227 j < n) \u2227 m = s + step * a"}, {"tactic": "simp only [\u2190 exists_and_right, and_assoc]", "annotated_tactic": ["simp only [\u2190 <a>exists_and_right</a>, <a>and_assoc</a>]", [{"full_name": "exists_and_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [468, 17], "def_end_pos": [468, 33]}, {"full_name": "and_assoc", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [177, 9], "def_end_pos": [177, 18]}]], "state_before": "m s step n : Nat\nh : \u2200 (i : Nat), i \u2264 n \u2194 i = 0 \u2228 \u2203 j, i = succ j \u2227 j < n\n\u22a2 (m = s \u2228 \u2203 i, i < n \u2227 m = s + step + step * i) \u2194 m = s \u2228 \u2203 a, (\u2203 j, a = succ j \u2227 j < n) \u2227 m = s + step * a", "state_after": "m s step n : Nat\nh : \u2200 (i : Nat), i \u2264 n \u2194 i = 0 \u2228 \u2203 j, i = succ j \u2227 j < n\n\u22a2 (m = s \u2228 \u2203 i, i < n \u2227 m = s + step + step * i) \u2194 m = s \u2228 \u2203 a x, a = succ x \u2227 x < n \u2227 m = s + step * a"}, {"tactic": "rw [exists_comm]", "annotated_tactic": ["rw [<a>exists_comm</a>]", [{"full_name": "exists_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [496, 9], "def_end_pos": [496, 20]}]], "state_before": "m s step n : Nat\nh : \u2200 (i : Nat), i \u2264 n \u2194 i = 0 \u2228 \u2203 j, i = succ j \u2227 j < n\n\u22a2 (m = s \u2228 \u2203 i, i < n \u2227 m = s + step + step * i) \u2194 m = s \u2228 \u2203 a x, a = succ x \u2227 x < n \u2227 m = s + step * a", "state_after": "m s step n : Nat\nh : \u2200 (i : Nat), i \u2264 n \u2194 i = 0 \u2228 \u2203 j, i = succ j \u2227 j < n\n\u22a2 (m = s \u2228 \u2203 i, i < n \u2227 m = s + step + step * i) \u2194 m = s \u2228 \u2203 b a, a = succ b \u2227 b < n \u2227 m = s + step * a"}, {"tactic": "simp [Nat.mul_succ, Nat.add_assoc, Nat.add_comm]", "annotated_tactic": ["simp [<a>Nat.mul_succ</a>, <a>Nat.add_assoc</a>, <a>Nat.add_comm</a>]", [{"full_name": "Nat.mul_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [162, 9], "def_end_pos": [162, 17]}, {"full_name": "Nat.add_assoc", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [138, 19], "def_end_pos": [138, 28]}, {"full_name": "Nat.add_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [131, 19], "def_end_pos": [131, 27]}]], "state_before": "m s step n : Nat\nh : \u2200 (i : Nat), i \u2264 n \u2194 i = 0 \u2228 \u2203 j, i = succ j \u2227 j < n\n\u22a2 (m = s \u2228 \u2203 i, i < n \u2227 m = s + step + step * i) \u2194 m = s \u2228 \u2203 b a, a = succ b \u2227 b < n \u2227 m = s + step * a", "state_after": "no goals"}, {"tactic": "cases i <;> simp [Nat.succ_le]", "annotated_tactic": ["cases i <;> simp [<a>Nat.succ_le</a>]", [{"full_name": "Nat.succ_le", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [221, 9], "def_end_pos": [221, 16]}]], "state_before": "m s step n i : Nat\n\u22a2 i \u2264 n \u2194 i = 0 \u2228 \u2203 j, i = succ j \u2227 j < n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Arrow.lean", "full_name": "CategoryTheory.Arrow.iso_w", "start": [171, 1], "end": [174, 44], "traced_tactics": [{"tactic": "have eq := Arrow.hom.congr_right e.inv_hom_id", "annotated_tactic": ["have eq := <a>Arrow.hom.congr_right</a> e.inv_hom_id", [{"full_name": "CategoryTheory.Arrow.hom.congr_right", "def_path": "Mathlib/CategoryTheory/Arrow.lean", "def_pos": [167, 9], "def_end_pos": [167, 24]}]], "state_before": "T : Type u\ninst\u271d : Category.{v, u} T\nf g : Arrow T\ne : f \u2245 g\n\u22a2 g.hom = e.inv.left \u226b f.hom \u226b e.hom.right", "state_after": "T : Type u\ninst\u271d : Category.{v, u} T\nf g : Arrow T\ne : f \u2245 g\neq : (e.inv \u226b e.hom).right = (\ud835\udfd9 g).right\n\u22a2 g.hom = e.inv.left \u226b f.hom \u226b e.hom.right"}, {"tactic": "rw [Arrow.comp_right, Arrow.id_right] at eq", "annotated_tactic": ["rw [<a>Arrow.comp_right</a>, <a>Arrow.id_right</a>] at eq", [{"full_name": "CategoryTheory.Arrow.comp_right", "def_path": "Mathlib/CategoryTheory/Arrow.lean", "def_pos": [74, 9], "def_end_pos": [74, 19]}, {"full_name": "CategoryTheory.Arrow.id_right", "def_path": "Mathlib/CategoryTheory/Arrow.lean", "def_pos": [63, 9], "def_end_pos": [63, 17]}]], "state_before": "T : Type u\ninst\u271d : Category.{v, u} T\nf g : Arrow T\ne : f \u2245 g\neq : (e.inv \u226b e.hom).right = (\ud835\udfd9 g).right\n\u22a2 g.hom = e.inv.left \u226b f.hom \u226b e.hom.right", "state_after": "T : Type u\ninst\u271d : Category.{v, u} T\nf g : Arrow T\ne : f \u2245 g\neq : e.inv.right \u226b e.hom.right = \ud835\udfd9 g.right\n\u22a2 g.hom = e.inv.left \u226b f.hom \u226b e.hom.right"}, {"tactic": "erw [Arrow.w_assoc, eq, Category.comp_id]", "annotated_tactic": ["erw [<a>Arrow.w_assoc</a>, eq, <a>Category.comp_id</a>]", [{"full_name": "CategoryTheory.Arrow.w_assoc", "def_path": "Mathlib/CategoryTheory/Arrow.lean", "def_pos": [126, 3], "def_end_pos": [126, 40]}, {"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [157, 3], "def_end_pos": [157, 10]}]], "state_before": "T : Type u\ninst\u271d : Category.{v, u} T\nf g : Arrow T\ne : f \u2245 g\neq : e.inv.right \u226b e.hom.right = \ud835\udfd9 g.right\n\u22a2 g.hom = e.inv.left \u226b f.hom \u226b e.hom.right", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.condexpL1_congr_ae", "start": [546, 1], "end": [548, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Convex/Combination.lean", "full_name": "Finset.centerMass_eq_of_sum_1", "start": [81, 1], "end": [83, 55], "traced_tactics": [{"tactic": "simp only [Finset.centerMass, hw, inv_one, one_smul]", "annotated_tactic": ["simp only [<a>Finset.centerMass</a>, hw, <a>inv_one</a>, <a>one_smul</a>]", [{"full_name": "Finset.centerMass", "def_path": "Mathlib/Analysis/Convex/Combination.lean", "def_pos": [41, 5], "def_end_pos": [41, 22]}, {"full_name": "inv_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1015, 9], "def_end_pos": [1015, 16]}, {"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}]], "state_before": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\n\u03b9 : Type u_5\n\u03b9' : Type u_6\n\u03b1 : Type u_7\ninst\u271d\u2078 : LinearOrderedField R\ninst\u271d\u2077 : LinearOrderedField R'\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1\ninst\u271d\u00b3 : Module R E\ninst\u271d\u00b2 : Module R F\ninst\u271d\u00b9 : Module R \u03b1\ninst\u271d : OrderedSMul R \u03b1\ns : Set E\ni j : \u03b9\nc : R\nt : Finset \u03b9\nw : \u03b9 \u2192 R\nz : \u03b9 \u2192 E\nhw : \u2211 i in t, w i = 1\n\u22a2 centerMass t w z = \u2211 i in t, w i \u2022 z i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/IsTensorProduct.lean", "full_name": "IsBaseChange.inductionOn", "start": [201, 8], "end": [203, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/ModelTheory/FinitelyGenerated.lean", "full_name": "FirstOrder.Language.Substructure.cg_closure", "start": [142, 1], "end": [143, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Bounded.lean", "full_name": "Set.bounded_le_inter_lt", "start": [328, 1], "end": [330, 46], "traced_tactics": [{"tactic": "simp_rw [\u2190 not_le, bounded_le_inter_not_le]", "annotated_tactic": ["simp_rw [\u2190 <a>not_le</a>, <a>bounded_le_inter_not_le</a>]", [{"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "Set.bounded_le_inter_not_le", "def_path": "Mathlib/Order/Bounded.lean", "def_pos": [317, 9], "def_end_pos": [317, 32]}]], "state_before": "\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\n\u22a2 Bounded (fun x x_1 => x \u2264 x_1) (s \u2229 {b | a < b}) \u2194 Bounded (fun x x_1 => x \u2264 x_1) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "full_name": "ModelWithCorners.right_inv", "start": [285, 11], "end": [286, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.toNat_lt_of_lt_of_lt_aleph0", "start": [1743, 1], "end": [1745, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "isOpen_gt'", "start": [893, 1], "end": [894, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Disjoint.lean", "full_name": "Disjoint.inf_right", "start": [160, 1], "end": [161, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Pointwise.lean", "full_name": "sInf_inv", "start": [68, 1], "end": [70, 43], "traced_tactics": [{"tactic": "rw [\u2190 image_inv, sInf_image]", "annotated_tactic": ["rw [\u2190 <a>image_inv</a>, <a>sInf_image</a>]", [{"full_name": "Set.image_inv", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 18]}, {"full_name": "sInf_image", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1456, 9], "def_end_pos": [1456, 19]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : CompleteLattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ns\u271d t s : Set \u03b1\n\u22a2 sInf s\u207b\u00b9 = (sSup s)\u207b\u00b9", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b3 : CompleteLattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ns\u271d t s : Set \u03b1\n\u22a2 \u2a05 a \u2208 s, a\u207b\u00b9 = (sSup s)\u207b\u00b9"}, {"tactic": "exact ((OrderIso.inv \u03b1).map_sSup _).symm", "annotated_tactic": ["exact ((<a>OrderIso.inv</a> \u03b1).<a>map_sSup</a> _).<a>symm</a>", [{"full_name": "OrderIso.inv", "def_path": "Mathlib/Algebra/Order/Group/OrderIso.lean", "def_pos": [37, 5], "def_end_pos": [37, 17]}, {"full_name": "OrderIso.map_sSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1037, 9], "def_end_pos": [1037, 26]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : CompleteLattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ns\u271d t s : Set \u03b1\n\u22a2 \u2a05 a \u2208 s, a\u207b\u00b9 = (sSup s)\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/BoxIntegral/Partition/Filter.lean", "full_name": "BoxIntegral.IntegrationParams.toFilteriUnion_congr", "start": [462, 1], "end": [464, 58], "traced_tactics": [{"tactic": "simp only [toFilteriUnion, toFilterDistortioniUnion, h]", "annotated_tactic": ["simp only [<a>toFilteriUnion</a>, <a>toFilterDistortioniUnion</a>, h]", [{"full_name": "BoxIntegral.IntegrationParams.toFilteriUnion", "def_path": "Mathlib/Analysis/BoxIntegral/Partition/Filter.lean", "def_pos": [342, 5], "def_end_pos": [342, 19]}, {"full_name": "BoxIntegral.IntegrationParams.toFilterDistortioniUnion", "def_path": "Mathlib/Analysis/BoxIntegral/Partition/Filter.lean", "def_pos": [335, 5], "def_end_pos": [335, 29]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nI\u271d J : Box \u03b9\nc c\u2081 c\u2082 : \u211d\u22650\nr r\u2081 r\u2082 : (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u03c0 \u03c0\u2081\u271d \u03c0\u2082\u271d : TaggedPrepartition I\u271d\nl\u271d l\u2081 l\u2082 : IntegrationParams\nI : Box \u03b9\nl : IntegrationParams\n\u03c0\u2081 \u03c0\u2082 : Prepartition I\nh : Prepartition.iUnion \u03c0\u2081 = Prepartition.iUnion \u03c0\u2082\n\u22a2 toFilteriUnion l I \u03c0\u2081 = toFilteriUnion l I \u03c0\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.mem_inf_iff", "start": [427, 1], "end": [428, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean", "full_name": "Orientation.norm_div_cos_oangle_sub_right_of_oangle_eq_pi_div_two", "start": [437, 1], "end": [444, 60], "traced_tactics": [{"tactic": "have hs : (o.oangle y (y - x)).sign = 1 := by\n  rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two]", "annotated_tactic": ["have hs : (o.oangle y (y - x)).<a>sign</a> = 1 := by\n    rw [<a>oangle_sign_sub_right_swap</a>, h, <a>Real.Angle.sign_coe_pi_div_two</a>]", [{"full_name": "Real.Angle.sign", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [849, 5], "def_end_pos": [849, 9]}, {"full_name": "Orientation.oangle_sign_sub_right_swap", "def_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean", "def_pos": [1019, 9], "def_end_pos": [1019, 35]}, {"full_name": "Real.Angle.sign_coe_pi_div_two", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [967, 9], "def_end_pos": [967, 28]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nhd2 : Fact (finrank \u211d V = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nh : oangle o x y = \u2191(\u03c0 / 2)\n\u22a2 \u2016y\u2016 / Real.Angle.cos (oangle o y (y - x)) = \u2016y - x\u2016", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nhd2 : Fact (finrank \u211d V = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nh : oangle o x y = \u2191(\u03c0 / 2)\nhs : Real.Angle.sign (oangle o y (y - x)) = 1\n\u22a2 \u2016y\u2016 / Real.Angle.cos (oangle o y (y - x)) = \u2016y - x\u2016"}, {"tactic": "rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.cos_coe,\n  InnerProductGeometry.norm_div_cos_angle_sub_of_inner_eq_zero\n    (o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h)\n    (Or.inl (o.right_ne_zero_of_oangle_eq_pi_div_two h))]", "annotated_tactic": ["rw [o.oangle_eq_angle_of_sign_eq_one hs, <a>Real.Angle.cos_coe</a>,\n    <a>InnerProductGeometry.norm_div_cos_angle_sub_of_inner_eq_zero</a>\n      (o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h)\n      (<a>Or.inl</a> (o.right_ne_zero_of_oangle_eq_pi_div_two h))]", [{"full_name": "Real.Angle.cos_coe", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}, {"full_name": "InnerProductGeometry.norm_div_cos_angle_sub_of_inner_eq_zero", "def_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean", "def_pos": [329, 9], "def_end_pos": [329, 48]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nhd2 : Fact (finrank \u211d V = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nh : oangle o x y = \u2191(\u03c0 / 2)\nhs : Real.Angle.sign (oangle o y (y - x)) = 1\n\u22a2 \u2016y\u2016 / Real.Angle.cos (oangle o y (y - x)) = \u2016y - x\u2016", "state_after": "no goals"}, {"tactic": "rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two]", "annotated_tactic": ["rw [<a>oangle_sign_sub_right_swap</a>, h, <a>Real.Angle.sign_coe_pi_div_two</a>]", [{"full_name": "Orientation.oangle_sign_sub_right_swap", "def_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean", "def_pos": [1019, 9], "def_end_pos": [1019, 35]}, {"full_name": "Real.Angle.sign_coe_pi_div_two", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [967, 9], "def_end_pos": [967, 28]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nhd2 : Fact (finrank \u211d V = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nh : oangle o x y = \u2191(\u03c0 / 2)\n\u22a2 Real.Angle.sign (oangle o y (y - x)) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/LocalHomeomorph.lean", "full_name": "LocalHomeomorph.refl_prod_refl", "start": [1034, 1], "end": [1036, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Irreducible.lean", "full_name": "Subtype.preirreducibleSpace", "start": [228, 1], "end": [233, 51], "traced_tactics": [{"tactic": "rintro _ _ \u27e8u, hu, rfl\u27e9 \u27e8v, hv, rfl\u27e9 \u27e8\u27e8x, hxs\u27e9, -, hxu\u27e9 \u27e8\u27e8y, hys\u27e9, -, hyv\u27e9", "annotated_tactic": ["rintro _ _ \u27e8u, hu, rfl\u27e9 \u27e8v, hv, rfl\u27e9 \u27e8\u27e8x, hxs\u27e9, -, hxu\u27e9 \u27e8\u27e8y, hys\u27e9, -, hyv\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nh : IsPreirreducible s\n\u22a2 IsPreirreducible univ", "state_after": "case intro.intro.intro.intro.intro.mk.intro.intro.mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nh : IsPreirreducible s\nu : Set \u03b1\nhu : IsOpen u\nv : Set \u03b1\nhv : IsOpen v\nx : \u03b1\nhxs : x \u2208 s\nhxu : { val := x, property := hxs } \u2208 val \u207b\u00b9' u\ny : \u03b1\nhys : y \u2208 s\nhyv : { val := y, property := hys } \u2208 val \u207b\u00b9' v\n\u22a2 Set.Nonempty (univ \u2229 (val \u207b\u00b9' u \u2229 val \u207b\u00b9' v))"}, {"tactic": "rcases h u v hu hv \u27e8x, hxs, hxu\u27e9 \u27e8y, hys, hyv\u27e9 with \u27e8z, hzs, \u27e8hzu, hzv\u27e9\u27e9", "annotated_tactic": ["rcases h u v hu hv \u27e8x, hxs, hxu\u27e9 \u27e8y, hys, hyv\u27e9 with \u27e8z, hzs, \u27e8hzu, hzv\u27e9\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.mk.intro.intro.mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nh : IsPreirreducible s\nu : Set \u03b1\nhu : IsOpen u\nv : Set \u03b1\nhv : IsOpen v\nx : \u03b1\nhxs : x \u2208 s\nhxu : { val := x, property := hxs } \u2208 val \u207b\u00b9' u\ny : \u03b1\nhys : y \u2208 s\nhyv : { val := y, property := hys } \u2208 val \u207b\u00b9' v\n\u22a2 Set.Nonempty (univ \u2229 (val \u207b\u00b9' u \u2229 val \u207b\u00b9' v))", "state_after": "case intro.intro.intro.intro.intro.mk.intro.intro.mk.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nh : IsPreirreducible s\nu : Set \u03b1\nhu : IsOpen u\nv : Set \u03b1\nhv : IsOpen v\nx : \u03b1\nhxs : x \u2208 s\nhxu : { val := x, property := hxs } \u2208 val \u207b\u00b9' u\ny : \u03b1\nhys : y \u2208 s\nhyv : { val := y, property := hys } \u2208 val \u207b\u00b9' v\nz : \u03b1\nhzs : z \u2208 s\nhzu : z \u2208 u\nhzv : z \u2208 v\n\u22a2 Set.Nonempty (univ \u2229 (val \u207b\u00b9' u \u2229 val \u207b\u00b9' v))"}, {"tactic": "exact \u27e8\u27e8z, hzs\u27e9, \u27e8Set.mem_univ _, \u27e8hzu, hzv\u27e9\u27e9\u27e9", "annotated_tactic": ["exact \u27e8\u27e8z, hzs\u27e9, \u27e8<a>Set.mem_univ</a> _, \u27e8hzu, hzv\u27e9\u27e9\u27e9", [{"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [676, 9], "def_end_pos": [676, 17]}]], "state_before": "case intro.intro.intro.intro.intro.mk.intro.intro.mk.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s : Set \u03b1\nh : IsPreirreducible s\nu : Set \u03b1\nhu : IsOpen u\nv : Set \u03b1\nhv : IsOpen v\nx : \u03b1\nhxs : x \u2208 s\nhxu : { val := x, property := hxs } \u2208 val \u207b\u00b9' u\ny : \u03b1\nhys : y \u2208 s\nhyv : { val := y, property := hys } \u2208 val \u207b\u00b9' v\nz : \u03b1\nhzs : z \u2208 s\nhzu : z \u2208 u\nhzv : z \u2208 v\n\u22a2 Set.Nonempty (univ \u2229 (val \u207b\u00b9' u \u2229 val \u207b\u00b9' v))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.mk_preimage_of_injective_of_subset_range_lift", "start": [2320, 1], "end": [2322, 94], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/FieldTheory/PerfectClosure.lean", "full_name": "PerfectClosure.liftOn_mk", "start": [66, 1], "end": [68, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.iInter\u2082_mono'", "start": [529, 1], "end": [533, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "LipschitzWith.continuous_compLp", "start": [1074, 1], "end": [1076, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Subobject/Basic.lean", "full_name": "CategoryTheory.Subobject.mk_eq_of_comm", "start": [299, 1], "end": [301, 80], "traced_tactics": [{"tactic": "rw [Iso.symm_hom, Iso.inv_comp_eq, w]", "annotated_tactic": ["rw [<a>Iso.symm_hom</a>, <a>Iso.inv_comp_eq</a>, w]", [{"full_name": "CategoryTheory.Iso.symm_hom", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [103, 9], "def_end_pos": [103, 17]}, {"full_name": "CategoryTheory.Iso.inv_comp_eq", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [214, 9], "def_end_pos": [214, 20]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nB A : C\nX : Subobject B\nf : A \u27f6 B\ninst\u271d : Mono f\ni : A \u2245 underlying.obj X\nw : i.hom \u226b arrow X = f\n\u22a2 i.symm.hom \u226b f = arrow X", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/PNat/Prime.lean", "full_name": "PNat.Coprime.coprime_dvd_left", "start": [255, 1], "end": [258, 37], "traced_tactics": [{"tactic": "rw [dvd_iff]", "annotated_tactic": ["rw [<a>dvd_iff</a>]", [{"full_name": "PNat.dvd_iff", "def_path": "Mathlib/Data/PNat/Basic.lean", "def_pos": [394, 9], "def_end_pos": [394, 16]}]], "state_before": "m k n : \u2115+\n\u22a2 m \u2223 k \u2192 Coprime k n \u2192 Coprime m n", "state_after": "m k n : \u2115+\n\u22a2 \u2191m \u2223 \u2191k \u2192 Coprime k n \u2192 Coprime m n"}, {"tactic": "repeat' rw [\u2190 coprime_coe]", "annotated_tactic": ["repeat' rw [\u2190 <a>coprime_coe</a>]", [{"full_name": "PNat.coprime_coe", "def_path": "Mathlib/Data/PNat/Prime.lean", "def_pos": [168, 9], "def_end_pos": [168, 20]}]], "state_before": "m k n : \u2115+\n\u22a2 \u2191m \u2223 \u2191k \u2192 Coprime k n \u2192 Coprime m n", "state_after": "m k n : \u2115+\n\u22a2 \u2191m \u2223 \u2191k \u2192 Nat.Coprime \u2191k \u2191n \u2192 Nat.Coprime \u2191m \u2191n"}, {"tactic": "apply Nat.Coprime.coprime_dvd_left", "annotated_tactic": ["apply <a>Nat.Coprime.coprime_dvd_left</a>", [{"full_name": "Nat.Coprime.coprime_dvd_left", "def_path": "lake-packages/std/Std/Data/Nat/Gcd.lean", "def_pos": [312, 9], "def_end_pos": [312, 33]}]], "state_before": "m k n : \u2115+\n\u22a2 \u2191m \u2223 \u2191k \u2192 Nat.Coprime \u2191k \u2191n \u2192 Nat.Coprime \u2191m \u2191n", "state_after": "no goals"}, {"tactic": "rw [\u2190 coprime_coe]", "annotated_tactic": ["rw [\u2190 <a>coprime_coe</a>]", [{"full_name": "PNat.coprime_coe", "def_path": "Mathlib/Data/PNat/Prime.lean", "def_pos": [168, 9], "def_end_pos": [168, 20]}]], "state_before": "m k n : \u2115+\n\u22a2 \u2191m \u2223 \u2191k \u2192 Nat.Coprime \u2191k \u2191n \u2192 Coprime m n", "state_after": "m k n : \u2115+\n\u22a2 \u2191m \u2223 \u2191k \u2192 Nat.Coprime \u2191k \u2191n \u2192 Nat.Coprime \u2191m \u2191n"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Basic.lean", "full_name": "LinearEquivClass.range", "start": [2035, 11], "end": [2037, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Normed/Order/Lattice.lean", "full_name": "norm_sup_sub_sup_le_add_norm", "start": [132, 1], "end": [143, 43], "traced_tactics": [{"tactic": "rw [\u2190 norm_abs_eq_norm (a - c), \u2190 norm_abs_eq_norm (b - d)]", "annotated_tactic": ["rw [\u2190 <a>norm_abs_eq_norm</a> (a - c), \u2190 <a>norm_abs_eq_norm</a> (b - d)]", [{"full_name": "norm_abs_eq_norm", "def_path": "Mathlib/Analysis/Normed/Order/Lattice.lean", "def_pos": [114, 9], "def_end_pos": [114, 25]}, {"full_name": "norm_abs_eq_norm", "def_path": "Mathlib/Analysis/Normed/Order/Lattice.lean", "def_pos": [114, 9], "def_end_pos": [114, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 \u2016a \u2294 b - c \u2294 d\u2016 \u2264 \u2016a - c\u2016 + \u2016b - d\u2016", "state_after": "\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 \u2016a \u2294 b - c \u2294 d\u2016 \u2264 \u2016|a - c|\u2016 + \u2016|b - d|\u2016"}, {"tactic": "refine' le_trans (solid _) (norm_add_le |a - c| |b - d|)", "annotated_tactic": ["refine' <a>le_trans</a> (<a>solid</a> _) (<a>norm_add_le</a> |a - c| |b - d|)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "HasSolidNorm.solid", "def_path": "Mathlib/Analysis/Normed/Order/Lattice.lean", "def_pos": [49, 3], "def_end_pos": [49, 8]}, {"full_name": "norm_add_le", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [482, 15], "def_end_pos": [482, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 \u2016a \u2294 b - c \u2294 d\u2016 \u2264 \u2016|a - c|\u2016 + \u2016|b - d|\u2016", "state_after": "\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |a \u2294 b - c \u2294 d| \u2264 ||a - c| + |b - d||"}, {"tactic": "rw [abs_of_nonneg (|a - c| + |b - d|) (add_nonneg (abs_nonneg (a - c)) (abs_nonneg (b - d)))]", "annotated_tactic": ["rw [<a>abs_of_nonneg</a> (|a - c| + |b - d|) (<a>add_nonneg</a> (<a>abs_nonneg</a> (a - c)) (<a>abs_nonneg</a> (b - d)))]", [{"full_name": "LatticeOrderedGroup.abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [353, 15], "def_end_pos": [353, 28]}, {"full_name": "add_nonneg", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [1092, 24], "def_end_pos": [1092, 34]}, {"full_name": "LatticeOrderedGroup.abs_nonneg", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [380, 15], "def_end_pos": [380, 25]}, {"full_name": "LatticeOrderedGroup.abs_nonneg", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [380, 15], "def_end_pos": [380, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |a \u2294 b - c \u2294 d| \u2264 ||a - c| + |b - d||", "state_after": "\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |a \u2294 b - c \u2294 d| \u2264 |a - c| + |b - d|"}, {"tactic": "rw [sub_add_sub_cancel]", "annotated_tactic": ["rw [<a>sub_add_sub_cancel</a>]", [{"full_name": "sub_add_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [789, 30], "def_end_pos": [789, 48]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |a \u2294 b - c \u2294 d| = |a \u2294 b - c \u2294 b + (c \u2294 b - c \u2294 d)|", "state_after": "no goals"}, {"tactic": "apply add_le_add", "annotated_tactic": ["apply <a>add_le_add</a>", [{"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |a \u2294 b - c \u2294 b| + |c \u2294 b - c \u2294 d| \u2264 |a - c| + |b - d|", "state_after": "case h\u2081\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |a \u2294 b - c \u2294 b| \u2264 |a - c|\n\ncase h\u2082\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |c \u2294 b - c \u2294 d| \u2264 |b - d|"}, {"tactic": "exact abs_sup_sub_sup_le_abs _ _ _", "annotated_tactic": ["exact <a>abs_sup_sub_sup_le_abs</a> _ _ _", [{"full_name": "LatticeOrderedCommGroup.abs_sup_sub_sup_le_abs", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [593, 15], "def_end_pos": [593, 37]}]], "state_before": "case h\u2081\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |a \u2294 b - c \u2294 b| \u2264 |a - c|", "state_after": "no goals"}, {"tactic": "rw [@sup_comm _ _ c, @sup_comm _ _ c]", "annotated_tactic": ["rw [@<a>sup_comm</a> _ _ c, @<a>sup_comm</a> _ _ c]", [{"full_name": "sup_comm", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [250, 9], "def_end_pos": [250, 17]}, {"full_name": "sup_comm", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [250, 9], "def_end_pos": [250, 17]}]], "state_before": "case h\u2082\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |c \u2294 b - c \u2294 d| \u2264 |b - d|", "state_after": "case h\u2082\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |b \u2294 c - d \u2294 c| \u2264 |b - d|"}, {"tactic": "exact abs_sup_sub_sup_le_abs _ _ _", "annotated_tactic": ["exact <a>abs_sup_sub_sup_le_abs</a> _ _ _", [{"full_name": "LatticeOrderedCommGroup.abs_sup_sub_sup_le_abs", "def_path": "Mathlib/Algebra/Order/LatticeGroup.lean", "def_pos": [593, 15], "def_end_pos": [593, 37]}]], "state_before": "case h\u2082\n\u03b1 : Type u_1\ninst\u271d : NormedLatticeAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |b \u2294 c - d \u2294 c| \u2264 |b - d|", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "continuous_const", "start": [1723, 1], "end": [1724, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/DFinsupp/NeLocus.lean", "full_name": "DFinsupp.neLocus_add_left", "start": [122, 1], "end": [124, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Ideal.lean", "full_name": "Order.Ideal.carrier_eq_coe", "start": [116, 1], "end": [117, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Control/Fold.lean", "full_name": "Traversable.foldlm_map", "start": [420, 1], "end": [422, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Basis/Flag.lean", "full_name": "Basis.flag_strictMono", "start": [63, 1], "end": [64, 59], "traced_tactics": [{"tactic": "simp [flag_succ]", "annotated_tactic": ["simp [<a>flag_succ</a>]", [{"full_name": "Basis.flag_succ", "def_path": "Mathlib/LinearAlgebra/Basis/Flag.lean", "def_pos": [41, 9], "def_end_pos": [41, 18]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\nn : \u2115\ninst\u271d : Nontrivial R\nb : Basis (Fin n) R M\nx\u271d : Fin n\n\u22a2 flag b (Fin.castSucc x\u271d) < flag b (Fin.succ x\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Ordinal/FixedPoint.lean", "full_name": "Ordinal.derivFamily_isNormal", "start": [178, 1], "end": [180, 45], "traced_tactics": [{"tactic": "rw [derivFamily_succ, \u2190 succ_le_iff]", "annotated_tactic": ["rw [<a>derivFamily_succ</a>, \u2190 <a>succ_le_iff</a>]", [{"full_name": "Ordinal.derivFamily_succ", "def_path": "Mathlib/SetTheory/Ordinal/FixedPoint.lean", "def_pos": [168, 9], "def_end_pos": [168, 25]}, {"full_name": "Order.succ_le_iff", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [343, 9], "def_end_pos": [343, 20]}]], "state_before": "\u03b9 : Type u\nf\u271d : \u03b9 \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v}\nf : \u03b9 \u2192 Ordinal.{max u u_1} \u2192 Ordinal.{max u u_1}\no : Ordinal.{max u_1 u}\n\u22a2 derivFamily f o < derivFamily f (succ o)", "state_after": "\u03b9 : Type u\nf\u271d : \u03b9 \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v}\nf : \u03b9 \u2192 Ordinal.{max u u_1} \u2192 Ordinal.{max u u_1}\no : Ordinal.{max u_1 u}\n\u22a2 succ (derivFamily f o) \u2264 nfpFamily f (succ (derivFamily f o))"}, {"tactic": "apply le_nfpFamily", "annotated_tactic": ["apply <a>le_nfpFamily</a>", [{"full_name": "Ordinal.le_nfpFamily", "def_path": "Mathlib/SetTheory/Ordinal/FixedPoint.lean", "def_pos": [67, 9], "def_end_pos": [67, 21]}]], "state_before": "\u03b9 : Type u\nf\u271d : \u03b9 \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v}\nf : \u03b9 \u2192 Ordinal.{max u u_1} \u2192 Ordinal.{max u u_1}\no : Ordinal.{max u_1 u}\n\u22a2 succ (derivFamily f o) \u2264 nfpFamily f (succ (derivFamily f o))", "state_after": "no goals"}, {"tactic": "rw [derivFamily_limit _ l, bsup_le_iff]", "annotated_tactic": ["rw [<a>derivFamily_limit</a> _ l, <a>bsup_le_iff</a>]", [{"full_name": "Ordinal.derivFamily_limit", "def_path": "Mathlib/SetTheory/Ordinal/FixedPoint.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}, {"full_name": "Ordinal.bsup_le_iff", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [1485, 9], "def_end_pos": [1485, 20]}]], "state_before": "\u03b9 : Type u\nf\u271d : \u03b9 \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v}\nf : \u03b9 \u2192 Ordinal.{max u u_1} \u2192 Ordinal.{max u u_1}\no : Ordinal.{max u_1 u}\nl : IsLimit o\na : Ordinal.{max u_1 u}\n\u22a2 derivFamily f o \u2264 a \u2194 \u2200 (b : Ordinal.{max u_1 u}), b < o \u2192 derivFamily f b \u2264 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Subsemiring/Basic.lean", "full_name": "Subsemiring.range_fst", "start": [1267, 1], "end": [1268, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.VectorMeasure.of_diff", "start": [196, 1], "end": [199, 49], "traced_tactics": [{"tactic": "rw [\u2190 of_add_of_diff hA hB h, add_sub_cancel']", "annotated_tactic": ["rw [\u2190 <a>of_add_of_diff</a> hA hB h, <a>add_sub_cancel'</a>]", [{"full_name": "MeasureTheory.VectorMeasure.of_add_of_diff", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [191, 9], "def_end_pos": [191, 23]}, {"full_name": "add_sub_cancel'", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [948, 30], "def_end_pos": [948, 45]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nM\u271d : Type u_3\ninst\u271d\u2075 : AddCommMonoid M\u271d\ninst\u271d\u2074 : TopologicalSpace M\u271d\ninst\u271d\u00b3 : T2Space M\u271d\nv\u271d : VectorMeasure \u03b1 M\u271d\nf : \u2115 \u2192 Set \u03b1\nM : Type u_4\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : T2Space M\nv : VectorMeasure \u03b1 M\nA B : Set \u03b1\nhA : MeasurableSet A\nhB : MeasurableSet B\nh : A \u2286 B\n\u22a2 \u2191v (B \\ A) = \u2191v B - \u2191v A", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Group.lean", "full_name": "Set.add_mem_Ioc_iff_left", "start": [69, 1], "end": [70, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "full_name": "MeasureTheory.ae_eq_univ", "start": [462, 1], "end": [463, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Valuation/Basic.lean", "full_name": "Valuation.map_sub_le", "start": [303, 1], "end": [305, 63], "traced_tactics": [{"tactic": "rw [sub_eq_add_neg]", "annotated_tactic": ["rw [<a>sub_eq_add_neg</a>]", [{"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}]], "state_before": "K : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx\u271d y\u271d z x y : R\ng : (fun x => \u0393\u2080) x\nhx : \u2191v x \u2264 g\nhy : \u2191v y \u2264 g\n\u22a2 \u2191v (x - y) \u2264 g", "state_after": "K : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx\u271d y\u271d z x y : R\ng : (fun x => \u0393\u2080) x\nhx : \u2191v x \u2264 g\nhy : \u2191v y \u2264 g\n\u22a2 \u2191v (x + -y) \u2264 g"}, {"tactic": "exact v.map_add_le hx (le_trans (le_of_eq (v.map_neg y)) hy)", "annotated_tactic": ["exact v.map_add_le hx (<a>le_trans</a> (<a>le_of_eq</a> (v.map_neg y)) hy)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "K : Type u_1\nF : Type u_2\nR : Type u_3\ninst\u271d\u00b3 : DivisionRing K\n\u0393\u2080 : Type u_4\n\u0393'\u2080 : Type u_5\n\u0393''\u2080 : Type u_6\ninst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx\u271d y\u271d z x y : R\ng : (fun x => \u0393\u2080) x\nhx : \u2191v x \u2264 g\nhy : \u2191v y \u2264 g\n\u22a2 \u2191v (x + -y) \u2264 g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Subgroup/Pointwise.lean", "full_name": "AddSubgroup.pointwise_smul_le_iff\u2080", "start": [547, 1], "end": [549, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Init/Algebra/Classes.lean", "full_name": "total_of", "start": [368, 1], "end": [369, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/LocalProperties.lean", "full_name": "IsLocalization.exists_smul_mem_of_mem_adjoin", "start": [574, 1], "end": [592, 14], "traced_tactics": [{"tactic": "let g : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'", "annotated_tactic": ["let g : S \u2192\u2090[R] S' := <a>IsScalarTower.toAlgHom</a> R S S'", [{"full_name": "IsScalarTower.toAlgHom", "def_path": "Mathlib/Algebra/Algebra/Tower.lean", "def_pos": [140, 5], "def_end_pos": [140, 13]}]], "state_before": "R S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\n\u22a2 \u2203 m, m \u2022 x \u2208 A", "state_after": "R S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\n\u22a2 \u2203 m, m \u2022 x \u2208 A"}, {"tactic": "let y := IsLocalization.commonDenomOfFinset M s", "annotated_tactic": ["let y := <a>IsLocalization.commonDenomOfFinset</a> M s", [{"full_name": "IsLocalization.commonDenomOfFinset", "def_path": "Mathlib/RingTheory/Localization/Integer.lean", "def_pos": [141, 19], "def_end_pos": [141, 38]}]], "state_before": "R S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\n\u22a2 \u2203 m, m \u2022 x \u2208 A", "state_after": "R S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\n\u22a2 \u2203 m, m \u2022 x \u2208 A"}, {"tactic": "have hx\u2081 : (y : S) \u2022 (s : Set S') = g '' _ :=\n  (IsLocalization.finsetIntegerMultiple_image _ s).symm", "annotated_tactic": ["have hx\u2081 : (y : S) \u2022 (s : <a>Set</a> S') = g '' _ :=\n    (<a>IsLocalization.finsetIntegerMultiple_image</a> _ s).<a>symm</a>", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "IsLocalization.finsetIntegerMultiple_image", "def_path": "Mathlib/RingTheory/Localization/Integer.lean", "def_pos": [152, 9], "def_end_pos": [152, 36]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "R S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\n\u22a2 \u2203 m, m \u2022 x \u2208 A", "state_after": "R S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\n\u22a2 \u2203 m, m \u2022 x \u2208 A"}, {"tactic": "obtain \u27e8n, hn\u27e9 :=\n  Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin (y : S) (s : Set S') (A.map g)\n    (by rw [hx\u2081]; exact Set.image_subset _ hA\u2081) hx (Set.mem_image_of_mem _ (hA\u2082 y.2))", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 :=\n    <a>Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin</a> (y : S) (s : <a>Set</a> S') (A.map g)\n      (by rw [hx\u2081]; exact <a>Set.image_subset</a> _ hA\u2081) hx (<a>Set.mem_image_of_mem</a> _ (hA\u2082 y.2))", [{"full_name": "Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin", "def_path": "Mathlib/RingTheory/Adjoin/Basic.lean", "def_pos": [364, 9], "def_end_pos": [364, 50]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [321, 9], "def_end_pos": [321, 21]}, {"full_name": "Set.mem_image_of_mem", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [240, 9], "def_end_pos": [240, 25]}]], "state_before": "R S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\n\u22a2 \u2203 m, m \u2022 x \u2208 A", "state_after": "case intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\nn : \u2115\nhn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A\n\u22a2 \u2203 m, m \u2022 x \u2208 A"}, {"tactic": "obtain \u27e8x', hx', hx''\u27e9 := hn n (le_of_eq rfl)", "annotated_tactic": ["obtain \u27e8x', hx', hx''\u27e9 := hn n (<a>le_of_eq</a> <a>rfl</a>)", [{"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\nn : \u2115\nhn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A\n\u22a2 \u2203 m, m \u2022 x \u2208 A", "state_after": "case intro.intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\nn : \u2115\nhn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A\nx' : S\nhx' : x' \u2208 \u2191A.toSubsemiring\nhx'' : \u2191\u2191g x' = \u2191y ^ n \u2022 \u2191(algebraMap S S') x\n\u22a2 \u2203 m, m \u2022 x \u2208 A"}, {"tactic": "rw [Algebra.smul_def, \u2190 _root_.map_mul] at hx''", "annotated_tactic": ["rw [<a>Algebra.smul_def</a>, \u2190 <a>_root_.map_mul</a>] at hx''", [{"full_name": "Algebra.smul_def", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [347, 9], "def_end_pos": [347, 17]}, {"full_name": "map_mul", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [299, 9], "def_end_pos": [299, 16]}]], "state_before": "case intro.intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\nn : \u2115\nhn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A\nx' : S\nhx' : x' \u2208 \u2191A.toSubsemiring\nhx'' : \u2191\u2191g x' = \u2191y ^ n \u2022 \u2191(algebraMap S S') x\n\u22a2 \u2203 m, m \u2022 x \u2208 A", "state_after": "case intro.intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\nn : \u2115\nhn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A\nx' : S\nhx' : x' \u2208 \u2191A.toSubsemiring\nhx'' : \u2191\u2191g x' = \u2191(algebraMap S S') (\u2191y ^ n * x)\n\u22a2 \u2203 m, m \u2022 x \u2208 A"}, {"tactic": "obtain \u27e8a, ha\u2082\u27e9 := (IsLocalization.eq_iff_exists M S').mp hx''", "annotated_tactic": ["obtain \u27e8a, ha\u2082\u27e9 := (<a>IsLocalization.eq_iff_exists</a> M S').<a>mp</a> hx''", [{"full_name": "IsLocalization.eq_iff_exists", "def_path": "Mathlib/RingTheory/Localization/Basic.lean", "def_pos": [130, 9], "def_end_pos": [130, 22]}, {"full_name": "Iff.mp", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [90, 3], "def_end_pos": [90, 5]}]], "state_before": "case intro.intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\nn : \u2115\nhn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A\nx' : S\nhx' : x' \u2208 \u2191A.toSubsemiring\nhx'' : \u2191\u2191g x' = \u2191(algebraMap S S') (\u2191y ^ n * x)\n\u22a2 \u2203 m, m \u2022 x \u2208 A", "state_after": "case intro.intro.intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\nn : \u2115\nhn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A\nx' : S\nhx' : x' \u2208 \u2191A.toSubsemiring\nhx'' : \u2191\u2191g x' = \u2191(algebraMap S S') (\u2191y ^ n * x)\na : { x // x \u2208 M }\nha\u2082 : \u2191a * x' = \u2191a * (\u2191y ^ n * x)\n\u22a2 \u2203 m, m \u2022 x \u2208 A"}, {"tactic": "use a * y ^ n", "annotated_tactic": ["use a * y ^ n", []], "state_before": "case intro.intro.intro.intro\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\nn : \u2115\nhn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A\nx' : S\nhx' : x' \u2208 \u2191A.toSubsemiring\nhx'' : \u2191\u2191g x' = \u2191(algebraMap S S') (\u2191y ^ n * x)\na : { x // x \u2208 M }\nha\u2082 : \u2191a * x' = \u2191a * (\u2191y ^ n * x)\n\u22a2 \u2203 m, m \u2022 x \u2208 A", "state_after": "case h\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\nn : \u2115\nhn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A\nx' : S\nhx' : x' \u2208 \u2191A.toSubsemiring\nhx'' : \u2191\u2191g x' = \u2191(algebraMap S S') (\u2191y ^ n * x)\na : { x // x \u2208 M }\nha\u2082 : \u2191a * x' = \u2191a * (\u2191y ^ n * x)\n\u22a2 (a * y ^ n) \u2022 x \u2208 A"}, {"tactic": "convert A.mul_mem hx' (hA\u2082 a.prop) using 1", "annotated_tactic": ["convert A.mul_mem hx' (hA\u2082 a.prop) using 1", []], "state_before": "case h\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\nn : \u2115\nhn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A\nx' : S\nhx' : x' \u2208 \u2191A.toSubsemiring\nhx'' : \u2191\u2191g x' = \u2191(algebraMap S S') (\u2191y ^ n * x)\na : { x // x \u2208 M }\nha\u2082 : \u2191a * x' = \u2191a * (\u2191y ^ n * x)\n\u22a2 (a * y ^ n) \u2022 x \u2208 A", "state_after": "case h.e'_4\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\nn : \u2115\nhn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A\nx' : S\nhx' : x' \u2208 \u2191A.toSubsemiring\nhx'' : \u2191\u2191g x' = \u2191(algebraMap S S') (\u2191y ^ n * x)\na : { x // x \u2208 M }\nha\u2082 : \u2191a * x' = \u2191a * (\u2191y ^ n * x)\n\u22a2 (a * y ^ n) \u2022 x = x' * \u2191a"}, {"tactic": "rw [Submonoid.smul_def, smul_eq_mul, Submonoid.coe_mul, SubmonoidClass.coe_pow, mul_assoc, \u2190 ha\u2082,\n  mul_comm]", "annotated_tactic": ["rw [<a>Submonoid.smul_def</a>, <a>smul_eq_mul</a>, <a>Submonoid.coe_mul</a>, <a>SubmonoidClass.coe_pow</a>, <a>mul_assoc</a>, \u2190 ha\u2082,\n    <a>mul_comm</a>]", [{"full_name": "Submonoid.smul_def", "def_path": "Mathlib/GroupTheory/Submonoid/Operations.lean", "def_pos": [1529, 9], "def_end_pos": [1529, 17]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}, {"full_name": "Submonoid.coe_mul", "def_path": "Mathlib/GroupTheory/Submonoid/Operations.lean", "def_pos": [678, 9], "def_end_pos": [678, 16]}, {"full_name": "SubmonoidClass.coe_pow", "def_path": "Mathlib/GroupTheory/Submonoid/Operations.lean", "def_pos": [562, 9], "def_end_pos": [562, 16]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [302, 9], "def_end_pos": [302, 17]}]], "state_before": "case h.e'_4\nR S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\nn : \u2115\nhn : \u2200 (n_1 : \u2115), n_1 \u2265 n \u2192 \u2191y ^ n_1 \u2022 \u2191(algebraMap S S') x \u2208 Subalgebra.map g A\nx' : S\nhx' : x' \u2208 \u2191A.toSubsemiring\nhx'' : \u2191\u2191g x' = \u2191(algebraMap S S') (\u2191y ^ n * x)\na : { x // x \u2208 M }\nha\u2082 : \u2191a * x' = \u2191a * (\u2191y ^ n * x)\n\u22a2 (a * y ^ n) \u2022 x = x' * \u2191a", "state_after": "no goals"}, {"tactic": "rw [hx\u2081]", "annotated_tactic": ["rw [hx\u2081]", []], "state_before": "R S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\n\u22a2 \u2191y \u2022 \u2191s \u2286 \u2191(Subalgebra.map g A)", "state_after": "R S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\n\u22a2 \u2191g '' \u2191(finsetIntegerMultiple M s) \u2286 \u2191(Subalgebra.map g A)"}, {"tactic": "exact Set.image_subset _ hA\u2081", "annotated_tactic": ["exact <a>Set.image_subset</a> _ hA\u2081", [{"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [321, 9], "def_end_pos": [321, 21]}]], "state_before": "R S : Type u\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\nM\u271d : Submonoid R\nN : Submonoid S\nR' S' : Type u\ninst\u271d\u2077 : CommRing R'\ninst\u271d\u2076 : CommRing S'\nf : R \u2192+* S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra S S'\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra R S'\ninst\u271d\u00b9 : IsScalarTower R S S'\nM : Submonoid S\ninst\u271d : IsLocalization M S'\nx : S\ns : Finset S'\nA : Subalgebra R S\nhA\u2081 : \u2191(finsetIntegerMultiple M s) \u2286 \u2191A\nhA\u2082 : M \u2264 A.toSubmonoid\nhx : \u2191(algebraMap S S') x \u2208 Algebra.adjoin R \u2191s\ng : S \u2192\u2090[R] S' := IsScalarTower.toAlgHom R S S'\ny : { x // x \u2208 M } := commonDenomOfFinset M s\nhx\u2081 : \u2191y \u2022 \u2191s = \u2191g '' \u2191(finsetIntegerMultiple M s)\n\u22a2 \u2191g '' \u2191(finsetIntegerMultiple M s) \u2286 \u2191(Subalgebra.map g A)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/NumberTheory/Modular.lean", "full_name": "ModularGroup.lcRow0_apply", "start": [174, 1], "end": [176, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "Monotone.map_sSup_of_continuousAt'", "start": [2705, 1], "end": [2710, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Sylow.lean", "full_name": "Sylow.subsingleton_of_normal", "start": [713, 1], "end": [720, 18], "traced_tactics": [{"tactic": "apply Subsingleton.intro", "annotated_tactic": ["apply <a>Subsingleton.intro</a>", [{"full_name": "Subsingleton.intro", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [869, 3], "def_end_pos": [869, 8]}]], "state_before": "G : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b2 : Group G\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nh : Normal \u2191P\n\u22a2 Subsingleton (Sylow p G)", "state_after": "case allEq\nG : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b2 : Group G\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nh : Normal \u2191P\n\u22a2 \u2200 (a b : Sylow p G), a = b"}, {"tactic": "intro Q R", "annotated_tactic": ["intro Q R", []], "state_before": "case allEq\nG : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b2 : Group G\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nh : Normal \u2191P\n\u22a2 \u2200 (a b : Sylow p G), a = b", "state_after": "case allEq\nG : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b2 : Group G\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nh : Normal \u2191P\nQ R : Sylow p G\n\u22a2 Q = R"}, {"tactic": "obtain \u27e8x, h1\u27e9 := exists_smul_eq G P Q", "annotated_tactic": ["obtain \u27e8x, h1\u27e9 := <a>exists_smul_eq</a> G P Q", [{"full_name": "MulAction.exists_smul_eq", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [152, 9], "def_end_pos": [152, 23]}]], "state_before": "case allEq\nG : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b2 : Group G\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nh : Normal \u2191P\nQ R : Sylow p G\n\u22a2 Q = R", "state_after": "case allEq.intro\nG : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b2 : Group G\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nh : Normal \u2191P\nQ R : Sylow p G\nx : G\nh1 : x \u2022 P = Q\n\u22a2 Q = R"}, {"tactic": "obtain \u27e8x, h2\u27e9 := exists_smul_eq G P R", "annotated_tactic": ["obtain \u27e8x, h2\u27e9 := <a>exists_smul_eq</a> G P R", [{"full_name": "MulAction.exists_smul_eq", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [152, 9], "def_end_pos": [152, 23]}]], "state_before": "case allEq.intro\nG : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b2 : Group G\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nh : Normal \u2191P\nQ R : Sylow p G\nx : G\nh1 : x \u2022 P = Q\n\u22a2 Q = R", "state_after": "case allEq.intro.intro\nG : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b2 : Group G\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nh : Normal \u2191P\nQ R : Sylow p G\nx\u271d : G\nh1 : x\u271d \u2022 P = Q\nx : G\nh2 : x \u2022 P = R\n\u22a2 Q = R"}, {"tactic": "rw [Sylow.smul_eq_of_normal] at h1 h2", "annotated_tactic": ["rw [<a>Sylow.smul_eq_of_normal</a>] at h1 h2", [{"full_name": "Sylow.smul_eq_of_normal", "def_path": "Mathlib/GroupTheory/Sylow.lean", "def_pos": [270, 9], "def_end_pos": [270, 32]}]], "state_before": "case allEq.intro.intro\nG : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b2 : Group G\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nh : Normal \u2191P\nQ R : Sylow p G\nx\u271d : G\nh1 : x\u271d \u2022 P = Q\nx : G\nh2 : x \u2022 P = R\n\u22a2 Q = R", "state_after": "case allEq.intro.intro\nG : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b2 : Group G\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nh : Normal \u2191P\nQ R : Sylow p G\nx\u271d : G\nh1 : P = Q\nx : G\nh2 : P = R\n\u22a2 Q = R"}, {"tactic": "rw [\u2190 h1, \u2190 h2]", "annotated_tactic": ["rw [\u2190 h1, \u2190 h2]", []], "state_before": "case allEq.intro.intro\nG : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b2 : Group G\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nh : Normal \u2191P\nQ R : Sylow p G\nx\u271d : G\nh1 : P = Q\nx : G\nh2 : P = R\n\u22a2 Q = R", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Monotone/Basic.lean", "full_name": "StrictMono.minimal_of_minimal_image", "start": [893, 1], "end": [895, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/ModelTheory/Fraisse.lean", "full_name": "FirstOrder.Language.age_directLimit", "start": [187, 1], "end": [208, 63], "traced_tactics": [{"tactic": "ext M", "annotated_tactic": ["ext M", []], "state_before": "L : Language\nK : Set (Bundled (Structure L))\nM : Type w\ninst\u271d\u2076 : Structure L M\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\n\u22a2 age L (DirectLimit G f) = \u22c3 i, age L (G i)", "state_after": "case h\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\n\u22a2 M \u2208 age L (DirectLimit G f) \u2194 M \u2208 \u22c3 i, age L (G i)"}, {"tactic": "simp only [mem_iUnion]", "annotated_tactic": ["simp only [<a>mem_iUnion</a>]", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}]], "state_before": "case h\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\n\u22a2 M \u2208 age L (DirectLimit G f) \u2194 M \u2208 \u22c3 i, age L (G i)", "state_after": "case h\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\n\u22a2 M \u2208 age L (DirectLimit G f) \u2194 \u2203 i, M \u2208 age L (G i)"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\n\u22a2 M \u2208 age L (DirectLimit G f) \u2194 \u2203 i, M \u2208 age L (G i)", "state_after": "case h.mp\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\n\u22a2 M \u2208 age L (DirectLimit G f) \u2192 \u2203 i, M \u2208 age L (G i)\n\ncase h.mpr\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\n\u22a2 (\u2203 i, M \u2208 age L (G i)) \u2192 M \u2208 age L (DirectLimit G f)"}, {"tactic": "rintro \u27e8Mfg, \u27e8e\u27e9\u27e9", "annotated_tactic": ["rintro \u27e8Mfg, \u27e8e\u27e9\u27e9", []], "state_before": "case h.mp\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\n\u22a2 M \u2208 age L (DirectLimit G f) \u2192 \u2203 i, M \u2208 age L (G i)", "state_after": "case h.mp.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\n\u22a2 \u2203 i, M \u2208 age L (G i)"}, {"tactic": "obtain \u27e8s, hs\u27e9 := Mfg.range e.toHom", "annotated_tactic": ["obtain \u27e8s, hs\u27e9 := Mfg.range e.toHom", []], "state_before": "case h.mp.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\n\u22a2 \u2203 i, M \u2208 age L (G i)", "state_after": "case h.mp.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\n\u22a2 \u2203 i, M \u2208 age L (G i)"}, {"tactic": "let out := @Quotient.out _ (DirectLimit.setoid G f)", "annotated_tactic": ["let out := @<a>Quotient.out</a> _ (<a>DirectLimit.setoid</a> G f)", [{"full_name": "Quotient.out", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [373, 19], "def_end_pos": [373, 31]}, {"full_name": "FirstOrder.Language.DirectLimit.setoid", "def_path": "Mathlib/ModelTheory/DirectLimit.lean", "def_pos": [132, 5], "def_end_pos": [132, 11]}]], "state_before": "case h.mp.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\n\u22a2 \u2203 i, M \u2208 age L (G i)", "state_after": "case h.mp.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\n\u22a2 \u2203 i, M \u2208 age L (G i)"}, {"tactic": "obtain \u27e8i, hi\u27e9 := Finset.exists_le (s.image (Sigma.fst \u2218 out))", "annotated_tactic": ["obtain \u27e8i, hi\u27e9 := <a>Finset.exists_le</a> (s.image (<a>Sigma.fst</a> \u2218 out))", [{"full_name": "Finset.exists_le", "def_path": "Mathlib/Data/Finset/Order.lean", "def_pos": [29, 9], "def_end_pos": [29, 25]}, {"full_name": "Sigma.fst", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [142, 3], "def_end_pos": [142, 6]}]], "state_before": "case h.mp.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\n\u22a2 \u2203 i, M \u2208 age L (G i)", "state_after": "case h.mp.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\n\u22a2 \u2203 i, M \u2208 age L (G i)"}, {"tactic": "have e' := (DirectLimit.of L \u03b9 G f i).equivRange.symm.toEmbedding", "annotated_tactic": ["have e' := (<a>DirectLimit.of</a> L \u03b9 G f i).equivRange.symm.toEmbedding", [{"full_name": "FirstOrder.Language.DirectLimit.of", "def_path": "Mathlib/ModelTheory/DirectLimit.lean", "def_pos": [293, 5], "def_end_pos": [293, 7]}]], "state_before": "case h.mp.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\n\u22a2 \u2203 i, M \u2208 age L (G i)", "state_after": "case h.mp.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\ne' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i\n\u22a2 \u2203 i, M \u2208 age L (G i)"}, {"tactic": "refine' \u27e8i, Mfg, \u27e8e'.comp ((Substructure.inclusion _).comp e.equivRange.toEmbedding)\u27e9\u27e9", "annotated_tactic": ["refine' \u27e8i, Mfg, \u27e8e'.comp ((<a>Substructure.inclusion</a> _).<a>comp</a> e.equivRange.toEmbedding)\u27e9\u27e9", [{"full_name": "FirstOrder.Language.Substructure.inclusion", "def_path": "Mathlib/ModelTheory/Substructures.lean", "def_pos": [998, 5], "def_end_pos": [998, 14]}, {"full_name": "FirstOrder.Language.Embedding.comp", "def_path": "Mathlib/ModelTheory/Basic.lean", "def_pos": [706, 5], "def_end_pos": [706, 9]}]], "state_before": "case h.mp.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\ne' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i\n\u22a2 \u2203 i, M \u2208 age L (G i)", "state_after": "case h.mp.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\ne' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i\n\u22a2 Hom.range (Embedding.toHom e) \u2264 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i))"}, {"tactic": "rw [\u2190 hs, closure_le]", "annotated_tactic": ["rw [\u2190 hs, <a>closure_le</a>]", [{"full_name": "FirstOrder.Language.Substructure.closure_le", "def_path": "Mathlib/ModelTheory/Substructures.lean", "def_pos": [283, 9], "def_end_pos": [283, 19]}]], "state_before": "case h.mp.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\ne' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i\n\u22a2 Hom.range (Embedding.toHom e) \u2264 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i))", "state_after": "case h.mp.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\ne' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i\n\u22a2 \u2191s \u2286 \u2191(Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)))"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case h.mp.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\ne' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i\n\u22a2 \u2191s \u2286 \u2191(Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)))", "state_after": "case h.mp.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\ne' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i\nx : DirectLimit G f\nhx : x \u2208 \u2191s\n\u22a2 x \u2208 \u2191(Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)))"}, {"tactic": "refine' \u27e8f (out x).1 i (hi (out x).1 (Finset.mem_image_of_mem _ hx)) (out x).2, _\u27e9", "annotated_tactic": ["refine' \u27e8f (out x).1 i (hi (out x).1 (<a>Finset.mem_image_of_mem</a> _ hx)) (out x).2, _\u27e9", [{"full_name": "Finset.mem_image_of_mem", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [334, 9], "def_end_pos": [334, 25]}]], "state_before": "case h.mp.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\ne' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i\nx : DirectLimit G f\nhx : x \u2208 \u2191s\n\u22a2 x \u2208 \u2191(Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)))", "state_after": "case h.mp.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\ne' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i\nx : DirectLimit G f\nhx : x \u2208 \u2191s\n\u22a2 \u2191(Embedding.toHom (DirectLimit.of L \u03b9 G f i)) (\u2191(f (out x).fst i (_ : (out x).fst \u2264 i)) (out x).snd) = x"}, {"tactic": "rw [Embedding.coe_toHom, DirectLimit.of_apply, @Quotient.mk_eq_iff_out _ (_),\n  DirectLimit.equiv_iff G f _ (hi (out x).1 (Finset.mem_image_of_mem _ hx)),\n  DirectedSystem.map_self]", "annotated_tactic": ["rw [<a>Embedding.coe_toHom</a>, <a>DirectLimit.of_apply</a>, @<a>Quotient.mk_eq_iff_out</a> _ (_),\n      <a>DirectLimit.equiv_iff</a> G f _ (hi (out x).1 (<a>Finset.mem_image_of_mem</a> _ hx)),\n      <a>DirectedSystem.map_self</a>]", [{"full_name": "FirstOrder.Language.Embedding.coe_toHom", "def_path": "Mathlib/ModelTheory/Basic.lean", "def_pos": [641, 9], "def_end_pos": [641, 18]}, {"full_name": "FirstOrder.Language.DirectLimit.of_apply", "def_path": "Mathlib/ModelTheory/DirectLimit.lean", "def_pos": [314, 9], "def_end_pos": [314, 17]}, {"full_name": "Quotient.mk_eq_iff_out", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [386, 9], "def_end_pos": [386, 31]}, {"full_name": "FirstOrder.Language.DirectLimit.equiv_iff", "def_path": "Mathlib/ModelTheory/DirectLimit.lean", "def_pos": [180, 9], "def_end_pos": [180, 18]}, {"full_name": "Finset.mem_image_of_mem", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [334, 9], "def_end_pos": [334, 25]}, {"full_name": "FirstOrder.Language.DirectedSystem.map_self", "def_path": "Mathlib/ModelTheory/DirectLimit.lean", "def_pos": [43, 16], "def_end_pos": [43, 24]}]], "state_before": "case h.mp.intro.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\ne' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i\nx : DirectLimit G f\nhx : x \u2208 \u2191s\n\u22a2 \u2191(Embedding.toHom (DirectLimit.of L \u03b9 G f i)) (\u2191(f (out x).fst i (_ : (out x).fst \u2264 i)) (out x).snd) = x", "state_after": "L : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\ne' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i\nx : DirectLimit G f\nhx : x \u2208 \u2191s\n\u22a2 (Structure.Sigma.mk f i (\u2191(f (out x).fst i (_ : (out x).fst \u2264 i)) (out x).snd)).fst \u2264 i\n\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\ne' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i\nx : DirectLimit G f\nhx : x \u2208 \u2191s\n\u22a2 (Structure.Sigma.mk f i (\u2191(f (out x).fst i (_ : (out x).fst \u2264 i)) (out x).snd)).fst \u2264 i"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "L : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\ne' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i\nx : DirectLimit G f\nhx : x \u2208 \u2191s\n\u22a2 (Structure.Sigma.mk f i (\u2191(f (out x).fst i (_ : (out x).fst \u2264 i)) (out x).snd)).fst \u2264 i\n\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] DirectLimit G f\ns : Finset (DirectLimit G f)\nhs : LowerAdjoint.toFun (closure L) \u2191s = Hom.range (Embedding.toHom e)\nout : Quotient (DirectLimit.setoid G f) \u2192 Structure.Sigma f := Quotient.out\ni : \u03b9\nhi : \u2200 (i_1 : \u03b9), i_1 \u2208 Finset.image (Sigma.fst \u2218 out) s \u2192 i_1 \u2264 i\ne' : { x // x \u2208 Hom.range (Embedding.toHom (DirectLimit.of L \u03b9 G f i)) } \u21aa[L] G i\nx : DirectLimit G f\nhx : x \u2208 \u2191s\n\u22a2 (Structure.Sigma.mk f i (\u2191(f (out x).fst i (_ : (out x).fst \u2264 i)) (out x).snd)).fst \u2264 i", "state_after": "no goals"}, {"tactic": "rintro \u27e8i, Mfg, \u27e8e\u27e9\u27e9", "annotated_tactic": ["rintro \u27e8i, Mfg, \u27e8e\u27e9\u27e9", []], "state_before": "case h.mpr\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\n\u22a2 (\u2203 i, M \u2208 age L (G i)) \u2192 M \u2208 age L (DirectLimit G f)", "state_after": "case h.mpr.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\ni : \u03b9\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] G i\n\u22a2 M \u2208 age L (DirectLimit G f)"}, {"tactic": "exact \u27e8Mfg, \u27e8Embedding.comp (DirectLimit.of L \u03b9 G f i) e\u27e9\u27e9", "annotated_tactic": ["exact \u27e8Mfg, \u27e8<a>Embedding.comp</a> (<a>DirectLimit.of</a> L \u03b9 G f i) e\u27e9\u27e9", [{"full_name": "FirstOrder.Language.Embedding.comp", "def_path": "Mathlib/ModelTheory/Basic.lean", "def_pos": [706, 5], "def_end_pos": [706, 9]}, {"full_name": "FirstOrder.Language.DirectLimit.of", "def_path": "Mathlib/ModelTheory/DirectLimit.lean", "def_pos": [293, 5], "def_end_pos": [293, 7]}]], "state_before": "case h.mpr.intro.intro.intro\nL : Language\nK : Set (Bundled (Structure L))\nM\u271d : Type w\ninst\u271d\u2076 : Structure L M\u271d\nN : Type w\ninst\u271d\u2075 : Structure L N\n\u03b9 : Type w\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type (max w w')\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nM : Bundled (Structure L)\ni : \u03b9\nMfg : Structure.FG L \u2191M\ne : \u2191M \u21aa[L] G i\n\u22a2 M \u2208 age L (DirectLimit G f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Bitvec/Lemmas.lean", "full_name": "Bitvec.toNat_ofNat", "start": [70, 1], "end": [74, 85], "traced_tactics": [{"tactic": "induction' k with k ih generalizing n", "annotated_tactic": ["induction' k with k ih generalizing n", []], "state_before": "k n : \u2115\n\u22a2 Bitvec.toNat (Bitvec.ofNat k n) = n % 2 ^ k", "state_after": "case zero\nn\u271d n : \u2115\n\u22a2 Bitvec.toNat (Bitvec.ofNat zero n) = n % 2 ^ zero\n\ncase succ\nn\u271d k : \u2115\nih : \u2200 {n : \u2115}, Bitvec.toNat (Bitvec.ofNat k n) = n % 2 ^ k\nn : \u2115\n\u22a2 Bitvec.toNat (Bitvec.ofNat (succ k) n) = n % 2 ^ succ k"}, {"tactic": "simp [Nat.mod_one]", "annotated_tactic": ["simp [<a>Nat.mod_one</a>]", [{"full_name": "Nat.mod_one", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [157, 9], "def_end_pos": [157, 16]}]], "state_before": "case zero\nn\u271d n : \u2115\n\u22a2 Bitvec.toNat (Bitvec.ofNat zero n) = n % 2 ^ zero", "state_after": "case zero\nn\u271d n : \u2115\n\u22a2 Bitvec.toNat (Bitvec.ofNat 0 n) = 0"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case zero\nn\u271d n : \u2115\n\u22a2 Bitvec.toNat (Bitvec.ofNat 0 n) = 0", "state_after": "no goals"}, {"tactic": "rw [ofNat_succ, toNat_append, ih, bits_toNat_decide, mod_pow_succ, Nat.mul_comm]", "annotated_tactic": ["rw [<a>ofNat_succ</a>, <a>toNat_append</a>, ih, <a>bits_toNat_decide</a>, <a>mod_pow_succ</a>, <a>Nat.mul_comm</a>]", [{"full_name": "Bitvec.ofNat_succ", "def_path": "Mathlib/Data/Bitvec/Lemmas.lean", "def_pos": [65, 9], "def_end_pos": [65, 19]}, {"full_name": "Bitvec.toNat_append", "def_path": "Mathlib/Data/Bitvec/Lemmas.lean", "def_pos": [33, 9], "def_end_pos": [33, 21]}, {"full_name": "Bitvec.bits_toNat_decide", "def_path": "Mathlib/Data/Bitvec/Lemmas.lean", "def_pos": [59, 9], "def_end_pos": [59, 26]}, {"full_name": "Nat.mod_pow_succ", "def_path": "Mathlib/Data/Nat/Pow.lean", "def_pos": [163, 9], "def_end_pos": [163, 21]}, {"full_name": "Nat.mul_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [174, 19], "def_end_pos": [174, 27]}]], "state_before": "case succ\nn\u271d k : \u2115\nih : \u2200 {n : \u2115}, Bitvec.toNat (Bitvec.ofNat k n) = n % 2 ^ k\nn : \u2115\n\u22a2 Bitvec.toNat (Bitvec.ofNat (succ k) n) = n % 2 ^ succ k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Homotopy/Equiv.lean", "full_name": "ContinuousMap.HomotopyEquiv.toFun_eq_coe", "start": [63, 1], "end": [64, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "full_name": "MeasureTheory.IsFundamentalDomain.set_integral_eq", "start": [462, 11], "end": [471, 43], "traced_tactics": [{"tactic": "by_cases hfs : IntegrableOn f s \u03bc", "annotated_tactic": ["by_cases hfs : <a>IntegrableOn</a> f s \u03bc", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 E\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\n\u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc", "state_after": "case pos\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 E\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\nhfs : IntegrableOn f s\n\u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc\n\ncase neg\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 E\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\nhfs : \u00acIntegrableOn f s\n\u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc"}, {"tactic": "have hft : IntegrableOn f t \u03bc := by rwa [ht.integrableOn_iff hs hf]", "annotated_tactic": ["have hft : <a>IntegrableOn</a> f t \u03bc := by rwa [ht.integrableOn_iff hs hf]", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}]], "state_before": "case pos\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 E\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\nhfs : IntegrableOn f s\n\u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc", "state_after": "case pos\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 E\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\n\u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc"}, {"tactic": "calc\n  \u222b x in s, f x \u2202\u03bc = \u2211' g : G, \u222b x in s \u2229 g \u2022 t, f x \u2202\u03bc := ht.set_integral_eq_tsum hfs\n  _ = \u2211' g : G, \u222b x in g \u2022 t \u2229 s, f (g\u207b\u00b9 \u2022 x) \u2202\u03bc := by simp only [hf, inter_comm]\n  _ = \u222b x in t, f x \u2202\u03bc := (hs.set_integral_eq_tsum' hft).symm", "annotated_tactic": ["calc\n      \u222b x in s, f x \u2202\u03bc = \u2211' g : G, \u222b x in s \u2229 g \u2022 t, f x \u2202\u03bc := ht.set_integral_eq_tsum hfs\n      _ = \u2211' g : G, \u222b x in g \u2022 t \u2229 s, f (g\u207b\u00b9 \u2022 x) \u2202\u03bc := by simp only [hf, <a>inter_comm</a>]\n      _ = \u222b x in t, f x \u2202\u03bc := (hs.set_integral_eq_tsum' hft).<a>symm</a>", [{"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case pos\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 E\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\n\u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rwa [ht.integrableOn_iff hs hf]", "annotated_tactic": ["rwa [ht.integrableOn_iff hs hf]", []], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 E\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\nhfs : IntegrableOn f s\n\u22a2 IntegrableOn f t", "state_after": "no goals"}, {"tactic": "simp only [hf, inter_comm]", "annotated_tactic": ["simp only [hf, <a>inter_comm</a>]", [{"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [940, 9], "def_end_pos": [940, 19]}]], "state_before": "G : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 E\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\nhfs : IntegrableOn f s\nhft : IntegrableOn f t\n\u22a2 \u2211' (g : G), \u222b (x : \u03b1) in s \u2229 g \u2022 t, f x \u2202\u03bc = \u2211' (g : G), \u222b (x : \u03b1) in g \u2022 t \u2229 s, f (g\u207b\u00b9 \u2022 x) \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [integral_undef hfs, integral_undef]", "annotated_tactic": ["rw [<a>integral_undef</a> hfs, <a>integral_undef</a>]", [{"full_name": "MeasureTheory.integral_undef", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [836, 9], "def_end_pos": [836, 23]}, {"full_name": "MeasureTheory.integral_undef", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [836, 9], "def_end_pos": [836, 23]}]], "state_before": "case neg\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 E\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\nhfs : \u00acIntegrableOn f s\n\u22a2 \u222b (x : \u03b1) in s, f x \u2202\u03bc = \u222b (x : \u03b1) in t, f x \u2202\u03bc", "state_after": "case neg\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 E\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\nhfs : \u00acIntegrableOn f s\n\u22a2 \u00acIntegrable fun x => f x"}, {"tactic": "rwa [hs.integrableOn_iff ht hf] at hfs", "annotated_tactic": ["rwa [hs.integrableOn_iff ht hf] at hfs", []], "state_before": "case neg\nG : Type u_1\nH : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nE : Type u_5\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : Group H\ninst\u271d\u00b9\u2070 : MulAction G \u03b1\ninst\u271d\u2079 : MeasurableSpace \u03b1\ninst\u271d\u2078 : MulAction H \u03b2\ninst\u271d\u2077 : MeasurableSpace \u03b2\ninst\u271d\u2076 : NormedAddCommGroup E\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : MeasurableSMul G \u03b1\ninst\u271d\u00b3 : SMulInvariantMeasure G \u03b1 \u03bc\ninst\u271d\u00b2 : Countable G\n\u03bd : Measure \u03b1\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nhs : IsFundamentalDomain G s\nht : IsFundamentalDomain G t\nf : \u03b1 \u2192 E\nhf : \u2200 (g : G) (x : \u03b1), f (g \u2022 x) = f x\nhfs : \u00acIntegrableOn f s\n\u22a2 \u00acIntegrable fun x => f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Ring/Units.lean", "full_name": "Units.neg_divp", "start": [49, 1], "end": [49, 88], "traced_tactics": [{"tactic": "simp only [divp, neg_mul]", "annotated_tactic": ["simp only [<a>divp</a>, <a>neg_mul</a>]", [{"full_name": "divp", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [434, 5], "def_end_pos": [434, 9]}, {"full_name": "neg_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 16]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : HasDistribNeg \u03b1\na\u271d b a : \u03b1\nu : \u03b1\u02e3\n\u22a2 -(a /\u209a u) = -a /\u209a u", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/ODE/PicardLindelof.lean", "full_name": "PicardLindelof.continuousOn", "start": [107, 11], "end": [112, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean", "full_name": "AffineMap.linear_eq_zero_iff_exists_const", "start": [189, 1], "end": [197, 30], "traced_tactics": [{"tactic": "refine' \u27e8fun h => _, fun h => _\u27e9", "annotated_tactic": ["refine' \u27e8fun h => _, fun h => _\u27e9", []], "state_before": "k : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\n\u22a2 f.linear = 0 \u2194 \u2203 q, f = const k P1 q", "state_after": "case refine'_1\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\nh : f.linear = 0\n\u22a2 \u2203 q, f = const k P1 q\n\ncase refine'_2\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\nh : \u2203 q, f = const k P1 q\n\u22a2 f.linear = 0"}, {"tactic": "use f (Classical.arbitrary P1)", "annotated_tactic": ["use f (<a>Classical.arbitrary</a> P1)", [{"full_name": "Classical.arbitrary", "def_path": "Mathlib/Logic/Nonempty.lean", "def_pos": [134, 29], "def_end_pos": [134, 48]}]], "state_before": "case refine'_1\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\nh : f.linear = 0\n\u22a2 \u2203 q, f = const k P1 q", "state_after": "case h\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\nh : f.linear = 0\n\u22a2 f = const k P1 (\u2191f (Classical.arbitrary P1))"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "case h\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\nh : f.linear = 0\n\u22a2 f = const k P1 (\u2191f (Classical.arbitrary P1))", "state_after": "case h.h\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\nh : f.linear = 0\np\u271d : P1\n\u22a2 \u2191f p\u271d = \u2191(const k P1 (\u2191f (Classical.arbitrary P1))) p\u271d"}, {"tactic": "rw [coe_const, Function.const_apply, \u2190 @vsub_eq_zero_iff_eq V2, \u2190 f.linearMap_vsub, h,\n  LinearMap.zero_apply]", "annotated_tactic": ["rw [<a>coe_const</a>, <a>Function.const_apply</a>, \u2190 @<a>vsub_eq_zero_iff_eq</a> V2, \u2190 f.linearMap_vsub, h,\n      <a>LinearMap.zero_apply</a>]", [{"full_name": "AffineMap.coe_const", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean", "def_pos": [174, 9], "def_end_pos": [174, 18]}, {"full_name": "Function.const_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [31, 17], "def_end_pos": [31, 37]}, {"full_name": "vsub_eq_zero_iff_eq", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [138, 9], "def_end_pos": [138, 28]}, {"full_name": "LinearMap.zero_apply", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [882, 9], "def_end_pos": [882, 19]}]], "state_before": "case h.h\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\nh : f.linear = 0\np\u271d : P1\n\u22a2 \u2191f p\u271d = \u2191(const k P1 (\u2191f (Classical.arbitrary P1))) p\u271d", "state_after": "no goals"}, {"tactic": "rcases h with \u27e8q, rfl\u27e9", "annotated_tactic": ["rcases h with \u27e8q, rfl\u27e9", []], "state_before": "case refine'_2\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\nh : \u2203 q, f = const k P1 q\n\u22a2 f.linear = 0", "state_after": "case refine'_2.intro\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nq : P2\n\u22a2 (const k P1 q).linear = 0"}, {"tactic": "exact const_linear k P1 q", "annotated_tactic": ["exact <a>const_linear</a> k P1 q", [{"full_name": "AffineMap.const_linear", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean", "def_pos": [183, 9], "def_end_pos": [183, 21]}]], "state_before": "case refine'_2.intro\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nq : P2\n\u22a2 (const k P1 q).linear = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Coset.lean", "full_name": "rightCoset_eq_iff", "start": [285, 1], "end": [296, 35], "traced_tactics": [{"tactic": "rw [Set.ext_iff]", "annotated_tactic": ["rw [<a>Set.ext_iff</a>]", [{"full_name": "Set.ext_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [241, 9], "def_end_pos": [241, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : Group \u03b1\ns : Subgroup \u03b1\nx y : \u03b1\n\u22a2 \u2191s *r x = \u2191s *r y \u2194 y * x\u207b\u00b9 \u2208 s", "state_after": "\u03b1 : Type u_1\ninst\u271d : Group \u03b1\ns : Subgroup \u03b1\nx y : \u03b1\n\u22a2 (\u2200 (x_1 : \u03b1), x_1 \u2208 \u2191s *r x \u2194 x_1 \u2208 \u2191s *r y) \u2194 y * x\u207b\u00b9 \u2208 s"}, {"tactic": "simp_rw [mem_rightCoset_iff, SetLike.mem_coe]", "annotated_tactic": ["simp_rw [<a>mem_rightCoset_iff</a>, <a>SetLike.mem_coe</a>]", [{"full_name": "mem_rightCoset_iff", "def_path": "Mathlib/GroupTheory/Coset.lean", "def_pos": [206, 9], "def_end_pos": [206, 27]}, {"full_name": "SetLike.mem_coe", "def_path": "Mathlib/Data/SetLike/Basic.lean", "def_pos": [163, 9], "def_end_pos": [163, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : Group \u03b1\ns : Subgroup \u03b1\nx y : \u03b1\n\u22a2 (\u2200 (x_1 : \u03b1), x_1 \u2208 \u2191s *r x \u2194 x_1 \u2208 \u2191s *r y) \u2194 y * x\u207b\u00b9 \u2208 s", "state_after": "\u03b1 : Type u_1\ninst\u271d : Group \u03b1\ns : Subgroup \u03b1\nx y : \u03b1\n\u22a2 (\u2200 (x_1 : \u03b1), x_1 * x\u207b\u00b9 \u2208 s \u2194 x_1 * y\u207b\u00b9 \u2208 s) \u2194 y * x\u207b\u00b9 \u2208 s"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : Group \u03b1\ns : Subgroup \u03b1\nx y : \u03b1\n\u22a2 (\u2200 (x_1 : \u03b1), x_1 * x\u207b\u00b9 \u2208 s \u2194 x_1 * y\u207b\u00b9 \u2208 s) \u2194 y * x\u207b\u00b9 \u2208 s", "state_after": "case mp\n\u03b1 : Type u_1\ninst\u271d : Group \u03b1\ns : Subgroup \u03b1\nx y : \u03b1\n\u22a2 (\u2200 (x_1 : \u03b1), x_1 * x\u207b\u00b9 \u2208 s \u2194 x_1 * y\u207b\u00b9 \u2208 s) \u2192 y * x\u207b\u00b9 \u2208 s\n\ncase mpr\n\u03b1 : Type u_1\ninst\u271d : Group \u03b1\ns : Subgroup \u03b1\nx y : \u03b1\n\u22a2 y * x\u207b\u00b9 \u2208 s \u2192 \u2200 (x_1 : \u03b1), x_1 * x\u207b\u00b9 \u2208 s \u2194 x_1 * y\u207b\u00b9 \u2208 s"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mp\n\u03b1 : Type u_1\ninst\u271d : Group \u03b1\ns : Subgroup \u03b1\nx y : \u03b1\n\u22a2 (\u2200 (x_1 : \u03b1), x_1 * x\u207b\u00b9 \u2208 s \u2194 x_1 * y\u207b\u00b9 \u2208 s) \u2192 y * x\u207b\u00b9 \u2208 s", "state_after": "case mp\n\u03b1 : Type u_1\ninst\u271d : Group \u03b1\ns : Subgroup \u03b1\nx y : \u03b1\nh : \u2200 (x_1 : \u03b1), x_1 * x\u207b\u00b9 \u2208 s \u2194 x_1 * y\u207b\u00b9 \u2208 s\n\u22a2 y * x\u207b\u00b9 \u2208 s"}, {"tactic": "apply (h y).mpr", "annotated_tactic": ["apply (h y).<a>mpr</a>", [{"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "case mp\n\u03b1 : Type u_1\ninst\u271d : Group \u03b1\ns : Subgroup \u03b1\nx y : \u03b1\nh : \u2200 (x_1 : \u03b1), x_1 * x\u207b\u00b9 \u2208 s \u2194 x_1 * y\u207b\u00b9 \u2208 s\n\u22a2 y * x\u207b\u00b9 \u2208 s", "state_after": "case mp\n\u03b1 : Type u_1\ninst\u271d : Group \u03b1\ns : Subgroup \u03b1\nx y : \u03b1\nh : \u2200 (x_1 : \u03b1), x_1 * x\u207b\u00b9 \u2208 s \u2194 x_1 * y\u207b\u00b9 \u2208 s\n\u22a2 y * y\u207b\u00b9 \u2208 s"}, {"tactic": "rw [mul_right_inv]", "annotated_tactic": ["rw [<a>mul_right_inv</a>]", [{"full_name": "mul_right_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1135, 9], "def_end_pos": [1135, 22]}]], "state_before": "case mp\n\u03b1 : Type u_1\ninst\u271d : Group \u03b1\ns : Subgroup \u03b1\nx y : \u03b1\nh : \u2200 (x_1 : \u03b1), x_1 * x\u207b\u00b9 \u2208 s \u2194 x_1 * y\u207b\u00b9 \u2208 s\n\u22a2 y * y\u207b\u00b9 \u2208 s", "state_after": "case mp\n\u03b1 : Type u_1\ninst\u271d : Group \u03b1\ns : Subgroup \u03b1\nx y : \u03b1\nh : \u2200 (x_1 : \u03b1), x_1 * x\u207b\u00b9 \u2208 s \u2194 x_1 * y\u207b\u00b9 \u2208 s\n\u22a2 1 \u2208 s"}, {"tactic": "exact s.one_mem", "annotated_tactic": ["exact s.one_mem", []], "state_before": "case mp\n\u03b1 : Type u_1\ninst\u271d : Group \u03b1\ns : Subgroup \u03b1\nx y : \u03b1\nh : \u2200 (x_1 : \u03b1), x_1 * x\u207b\u00b9 \u2208 s \u2194 x_1 * y\u207b\u00b9 \u2208 s\n\u22a2 1 \u2208 s", "state_after": "no goals"}, {"tactic": "intro h z", "annotated_tactic": ["intro h z", []], "state_before": "case mpr\n\u03b1 : Type u_1\ninst\u271d : Group \u03b1\ns : Subgroup \u03b1\nx y : \u03b1\n\u22a2 y * x\u207b\u00b9 \u2208 s \u2192 \u2200 (x_1 : \u03b1), x_1 * x\u207b\u00b9 \u2208 s \u2194 x_1 * y\u207b\u00b9 \u2208 s", "state_after": "case mpr\n\u03b1 : Type u_1\ninst\u271d : Group \u03b1\ns : Subgroup \u03b1\nx y : \u03b1\nh : y * x\u207b\u00b9 \u2208 s\nz : \u03b1\n\u22a2 z * x\u207b\u00b9 \u2208 s \u2194 z * y\u207b\u00b9 \u2208 s"}, {"tactic": "rw [\u2190 inv_mul_cancel_left y x\u207b\u00b9]", "annotated_tactic": ["rw [\u2190 <a>inv_mul_cancel_left</a> y x\u207b\u00b9]", [{"full_name": "inv_mul_cancel_left", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1147, 9], "def_end_pos": [1147, 28]}]], "state_before": "case mpr\n\u03b1 : Type u_1\ninst\u271d : Group \u03b1\ns : Subgroup \u03b1\nx y : \u03b1\nh : y * x\u207b\u00b9 \u2208 s\nz : \u03b1\n\u22a2 z * x\u207b\u00b9 \u2208 s \u2194 z * y\u207b\u00b9 \u2208 s", "state_after": "case mpr\n\u03b1 : Type u_1\ninst\u271d : Group \u03b1\ns : Subgroup \u03b1\nx y : \u03b1\nh : y * x\u207b\u00b9 \u2208 s\nz : \u03b1\n\u22a2 z * (y\u207b\u00b9 * (y * x\u207b\u00b9)) \u2208 s \u2194 z * y\u207b\u00b9 \u2208 s"}, {"tactic": "rw [\u2190 mul_assoc]", "annotated_tactic": ["rw [\u2190 <a>mul_assoc</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "case mpr\n\u03b1 : Type u_1\ninst\u271d : Group \u03b1\ns : Subgroup \u03b1\nx y : \u03b1\nh : y * x\u207b\u00b9 \u2208 s\nz : \u03b1\n\u22a2 z * (y\u207b\u00b9 * (y * x\u207b\u00b9)) \u2208 s \u2194 z * y\u207b\u00b9 \u2208 s", "state_after": "case mpr\n\u03b1 : Type u_1\ninst\u271d : Group \u03b1\ns : Subgroup \u03b1\nx y : \u03b1\nh : y * x\u207b\u00b9 \u2208 s\nz : \u03b1\n\u22a2 z * y\u207b\u00b9 * (y * x\u207b\u00b9) \u2208 s \u2194 z * y\u207b\u00b9 \u2208 s"}, {"tactic": "exact s.mul_mem_cancel_right h", "annotated_tactic": ["exact s.mul_mem_cancel_right h", []], "state_before": "case mpr\n\u03b1 : Type u_1\ninst\u271d : Group \u03b1\ns : Subgroup \u03b1\nx y : \u03b1\nh : y * x\u207b\u00b9 \u2208 s\nz : \u03b1\n\u22a2 z * y\u207b\u00b9 * (y * x\u207b\u00b9) \u2208 s \u2194 z * y\u207b\u00b9 \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean", "full_name": "CategoryTheory.Limits.pullbackZeroZeroIso_hom_fst", "start": [184, 1], "end": [185, 91], "traced_tactics": [{"tactic": "simp [\u2190 Iso.eq_inv_comp]", "annotated_tactic": ["simp [\u2190 <a>Iso.eq_inv_comp</a>]", [{"full_name": "CategoryTheory.Iso.eq_inv_comp", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [218, 9], "def_end_pos": [218, 20]}]], "state_before": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_2, u_1} C\ninst\u271d\u00b2 : HasZeroObject C\ninst\u271d\u00b9 : HasZeroMorphisms C\nX Y : C\ninst\u271d : HasBinaryProduct X Y\n\u22a2 (pullbackZeroZeroIso X Y).hom \u226b prod.fst = pullback.fst", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "full_name": "Rat.neg_divInt", "start": [220, 1], "end": [221, 83], "traced_tactics": [{"tactic": "rcases Int.eq_nat_or_neg d with \u27e8_, rfl | rfl\u27e9 <;> simp [divInt_neg', neg_mkRat]", "annotated_tactic": ["rcases <a>Int.eq_nat_or_neg</a> d with \u27e8_, rfl | rfl\u27e9 <;> simp [<a>divInt_neg'</a>, <a>neg_mkRat</a>]", [{"full_name": "Int.eq_nat_or_neg", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [166, 9], "def_end_pos": [166, 22]}, {"full_name": "Rat.divInt_neg'", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [139, 9], "def_end_pos": [139, 20]}, {"full_name": "Rat.neg_mkRat", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [217, 9], "def_end_pos": [217, 18]}]], "state_before": "n d : Int\n\u22a2 -(n /. d) = -n /. d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean", "full_name": "AlgebraicGeometry.isOpenImmersionCat_comp_of_sourceAffineLocally", "start": [312, 1], "end": [320, 28], "traced_tactics": [{"tactic": "rw [\u2190 h\u2081.cancel_right_isIso _\n  (Scheme.\u0393.map (IsOpenImmersion.isoOfRangeEq (Y.ofRestrict _) f _).hom.op),\n  \u2190 Functor.map_comp, \u2190 op_comp]", "annotated_tactic": ["rw [\u2190 h\u2081.cancel_right_isIso _\n    (Scheme.\u0393.map (<a>IsOpenImmersion.isoOfRangeEq</a> (Y.ofRestrict _) f _).hom.op),\n    \u2190 <a>Functor.map_comp</a>, \u2190 <a>op_comp</a>]", [{"full_name": "AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq", "def_path": "Mathlib/AlgebraicGeometry/OpenImmersion.lean", "def_pos": [628, 5], "def_end_pos": [628, 17]}, {"full_name": "CategoryTheory.Functor.map_comp", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [43, 3], "def_end_pos": [43, 11]}, {"full_name": "CategoryTheory.op_comp", "def_path": "Mathlib/CategoryTheory/Opposites.lean", "def_pos": [73, 9], "def_end_pos": [73, 16]}]], "state_before": "P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 P (Scheme.\u0393.map (f \u226b g).op)", "state_after": "P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 P (Scheme.\u0393.map ((IsOpenImmersion.isoOfRangeEq (Scheme.ofRestrict Y ?m.145001) f ?m.145008).hom \u226b f \u226b g).op)\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 TopCat\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 ?m.144999 \u27f6 TopCat.of \u2191\u2191Y.toPresheafedSpace\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 OpenEmbedding \u2191?m.145000\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 Set.range \u2191(Scheme.ofRestrict Y ?m.145001).val.base = Set.range \u2191f.val.base\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 TopCat\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 ?m.144999 \u27f6 TopCat.of \u2191\u2191Y.toPresheafedSpace\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 OpenEmbedding \u2191?m.145000\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 Set.range \u2191(Scheme.ofRestrict Y ?m.145001).val.base = Set.range \u2191f.val.base\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 TopCat\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 ?m.144999 \u27f6 TopCat.of \u2191\u2191Y.toPresheafedSpace\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 OpenEmbedding \u2191?m.145000\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 Set.range \u2191(Scheme.ofRestrict Y ?m.145001).val.base = Set.range \u2191f.val.base"}, {"tactic": "convert h\u2082 \u27e8_, rangeIsAffineOpenOfOpenImmersion f\u27e9 using 3", "annotated_tactic": ["convert h\u2082 \u27e8_, <a>rangeIsAffineOpenOfOpenImmersion</a> f\u27e9 using 3", [{"full_name": "AlgebraicGeometry.rangeIsAffineOpenOfOpenImmersion", "def_path": "Mathlib/AlgebraicGeometry/AffineScheme.lean", "def_pos": [179, 9], "def_end_pos": [179, 41]}]], "state_before": "P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 P (Scheme.\u0393.map ((IsOpenImmersion.isoOfRangeEq (Scheme.ofRestrict Y ?m.145001) f ?m.145008).hom \u226b f \u226b g).op)\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 TopCat\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 ?m.144999 \u27f6 TopCat.of \u2191\u2191Y.toPresheafedSpace\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 OpenEmbedding \u2191?m.145000\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 Set.range \u2191(Scheme.ofRestrict Y ?m.145001).val.base = Set.range \u2191f.val.base\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 TopCat\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 ?m.144999 \u27f6 TopCat.of \u2191\u2191Y.toPresheafedSpace\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 OpenEmbedding \u2191?m.145000\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 Set.range \u2191(Scheme.ofRestrict Y ?m.145001).val.base = Set.range \u2191f.val.base\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 TopCat\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 ?m.144999 \u27f6 TopCat.of \u2191\u2191Y.toPresheafedSpace\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 OpenEmbedding \u2191?m.145000\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 Set.range \u2191(Scheme.ofRestrict Y ?m.145001).val.base = Set.range \u2191f.val.base", "state_after": "case h.e'_5.h.h.e'_8.h.e'_5\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 (IsOpenImmersion.isoOfRangeEq\n          (Scheme.ofRestrict Y\n            (_ :\n              OpenEmbedding\n                \u2191(Opens.inclusion\n                    \u2191{ val := Scheme.Hom.opensRange f, property := (_ : IsAffineOpen (Scheme.Hom.opensRange f)) })))\n          f ?m.145008).hom \u226b\n      f \u226b g =\n    Scheme.ofRestrict Y\n        (_ :\n          OpenEmbedding\n            \u2191(Opens.inclusion\n                \u2191{ val := Scheme.Hom.opensRange f, property := (_ : IsAffineOpen (Scheme.Hom.opensRange f)) })) \u226b\n      g\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 Set.range\n      \u2191(Scheme.ofRestrict Y\n              (_ :\n                OpenEmbedding\n                  \u2191(Opens.inclusion\n                      \u2191{ val := Scheme.Hom.opensRange f,\n                          property := (_ : IsAffineOpen (Scheme.Hom.opensRange f)) }))).val.base =\n    Set.range \u2191f.val.base\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 Set.range\n      \u2191(Scheme.ofRestrict Y\n              (_ :\n                OpenEmbedding\n                  \u2191(Opens.inclusion\n                      \u2191{ val := Scheme.Hom.opensRange f,\n                          property := (_ : IsAffineOpen (Scheme.Hom.opensRange f)) }))).val.base =\n    Set.range \u2191f.val.base\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 Set.range\n      \u2191(Scheme.ofRestrict Y\n              (_ :\n                OpenEmbedding\n                  \u2191(Opens.inclusion\n                      \u2191{ val := Scheme.Hom.opensRange f,\n                          property := (_ : IsAffineOpen (Scheme.Hom.opensRange f)) }))).val.base =\n    Set.range \u2191f.val.base"}, {"tactic": "rw [IsOpenImmersion.isoOfRangeEq_hom_fac_assoc]", "annotated_tactic": ["rw [<a>IsOpenImmersion.isoOfRangeEq_hom_fac_assoc</a>]", [{"full_name": "AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq_hom_fac_assoc", "def_path": "Mathlib/AlgebraicGeometry/OpenImmersion.lean", "def_pos": [635, 9], "def_end_pos": [635, 16]}]], "state_before": "case h.e'_5.h.h.e'_8.h.e'_5\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 (IsOpenImmersion.isoOfRangeEq\n          (Scheme.ofRestrict Y\n            (_ :\n              OpenEmbedding\n                \u2191(Opens.inclusion\n                    \u2191{ val := Scheme.Hom.opensRange f, property := (_ : IsAffineOpen (Scheme.Hom.opensRange f)) })))\n          f ?m.145008).hom \u226b\n      f \u226b g =\n    Scheme.ofRestrict Y\n        (_ :\n          OpenEmbedding\n            \u2191(Opens.inclusion\n                \u2191{ val := Scheme.Hom.opensRange f, property := (_ : IsAffineOpen (Scheme.Hom.opensRange f)) })) \u226b\n      g", "state_after": "case h.e'_5.h.h.e'_8.h.e'_5.e\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 Set.range\n      \u2191(Scheme.ofRestrict Y\n              (_ :\n                OpenEmbedding\n                  \u2191(Opens.inclusion\n                      \u2191{ val := Scheme.Hom.opensRange f,\n                          property := (_ : IsAffineOpen (Scheme.Hom.opensRange f)) }))).val.base =\n    Set.range \u2191f.val.base"}, {"tactic": "exact Subtype.range_coe", "annotated_tactic": ["exact <a>Subtype.range_coe</a>", [{"full_name": "Subtype.range_coe", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 18]}]], "state_before": "case h.e'_5.h.h.e'_8.h.e'_5.e\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RingHom.PropertyIsLocal P\nh\u2081 : RingHom.RespectsIso P\nX Y Z : Scheme\ninst\u271d\u00b2 : IsAffine X\ninst\u271d\u00b9 : IsAffine Z\nf : X \u27f6 Y\ninst\u271d : IsOpenImmersion f\ng : Y \u27f6 Z\nh\u2082 : sourceAffineLocally P g\n\u22a2 Set.range\n      \u2191(Scheme.ofRestrict Y\n              (_ :\n                OpenEmbedding\n                  \u2191(Opens.inclusion\n                      \u2191{ val := Scheme.Hom.opensRange f,\n                          property := (_ : IsAffineOpen (Scheme.Hom.opensRange f)) }))).val.base =\n    Set.range \u2191f.val.base", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Nat.pos_of_floor_pos", "start": [215, 1], "end": [216, 88], "traced_tactics": [{"tactic": "rwa [floor_of_nonpos ha] at h", "annotated_tactic": ["rwa [<a>floor_of_nonpos</a> ha] at h", [{"full_name": "Nat.floor_of_nonpos", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [179, 9], "def_end_pos": [179, 24]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\nh : 0 < \u230aa\u230b\u208a\nha : a \u2264 0\n\u22a2 0 < 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/Additive/Behrend.lean", "full_name": "Behrend.le_sqrt_log", "start": [307, 1], "end": [329, 43], "traced_tactics": [{"tactic": "refine' (mul_le_mul_of_nonneg_right ((log_le_log _ <| by norm_num1).2\n    two_div_one_sub_two_div_e_le_eight) <| by norm_num1).trans _", "annotated_tactic": ["refine' (<a>mul_le_mul_of_nonneg_right</a> ((<a>log_le_log</a> _ <| by norm_num1).2\n      <a>two_div_one_sub_two_div_e_le_eight</a>) <| by norm_num1).<a>trans</a> _", [{"full_name": "mul_le_mul_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [156, 9], "def_end_pos": [156, 35]}, {"full_name": "Real.log_le_log", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [144, 9], "def_end_pos": [144, 19]}, {"full_name": "Behrend.two_div_one_sub_two_div_e_le_eight", "def_path": "Mathlib/Combinatorics/Additive/Behrend.lean", "def_pos": [300, 9], "def_end_pos": [300, 43]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 log (2 / (1 - 2 / rexp 1)) * (69 / 50) \u2264 Real.sqrt (log \u2191N)", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 0 < 2 / (1 - 2 / rexp 1)\n\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 log 8 * (69 / 50) \u2264 Real.sqrt (log \u2191N)"}, {"tactic": "have l8 : log 8 = (3 : \u2115) * log 2 := by\n  rw [\u2190 log_rpow zero_lt_two, rpow_nat_cast]\n  norm_num", "annotated_tactic": ["have l8 : <a>log</a> 8 = (3 : \u2115) * <a>log</a> 2 := by\n    rw [\u2190 <a>log_rpow</a> <a>zero_lt_two</a>, <a>rpow_nat_cast</a>]\n    norm_num", [{"full_name": "Real.log", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [43, 19], "def_end_pos": [43, 22]}, {"full_name": "Real.log", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [43, 19], "def_end_pos": [43, 22]}, {"full_name": "Real.log_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [401, 9], "def_end_pos": [401, 17]}, {"full_name": "zero_lt_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [71, 15], "def_end_pos": [71, 26]}, {"full_name": "Real.rpow_nat_cast", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [357, 9], "def_end_pos": [357, 22]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 log 8 * (69 / 50) \u2264 Real.sqrt (log \u2191N)", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 log 8 * (69 / 50) \u2264 Real.sqrt (log \u2191N)"}, {"tactic": "rw [l8]", "annotated_tactic": ["rw [l8]", []], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 log 8 * (69 / 50) \u2264 Real.sqrt (log \u2191N)", "state_after": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 \u21913 * log 2 * (69 / 50) \u2264 Real.sqrt (log \u2191N)"}, {"tactic": "apply le_sqrt_of_sq_le (le_trans _ this)", "annotated_tactic": ["apply <a>le_sqrt_of_sq_le</a> (<a>le_trans</a> _ this)", [{"full_name": "Real.le_sqrt_of_sq_le", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [326, 9], "def_end_pos": [326, 25]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "state_before": "case refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 \u21913 * log 2 * (69 / 50) \u2264 Real.sqrt (log \u2191N)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 (\u21913 * log 2 * (69 / 50)) ^ 2 \u2264 \u219112 * log 2"}, {"tactic": "rw [mul_right_comm, mul_pow, sq (log 2), \u2190 mul_assoc]", "annotated_tactic": ["rw [<a>mul_right_comm</a>, <a>mul_pow</a>, <a>sq</a> (<a>log</a> 2), \u2190 <a>mul_assoc</a>]", [{"full_name": "mul_right_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [102, 9], "def_end_pos": [102, 23]}, {"full_name": "mul_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 16]}, {"full_name": "sq", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [106, 7], "def_end_pos": [106, 9]}, {"full_name": "Real.log", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [43, 19], "def_end_pos": [43, 22]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 (\u21913 * log 2 * (69 / 50)) ^ 2 \u2264 \u219112 * log 2", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 (\u21913 * (69 / 50)) ^ 2 * log 2 * log 2 \u2264 \u219112 * log 2"}, {"tactic": "apply mul_le_mul_of_nonneg_right _ (log_nonneg one_le_two)", "annotated_tactic": ["apply <a>mul_le_mul_of_nonneg_right</a> _ (<a>log_nonneg</a> <a>one_le_two</a>)", [{"full_name": "mul_le_mul_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [156, 9], "def_end_pos": [156, 35]}, {"full_name": "Real.log_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 19]}, {"full_name": "one_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [50, 7], "def_end_pos": [50, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 (\u21913 * (69 / 50)) ^ 2 * log 2 * log 2 \u2264 \u219112 * log 2", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 (\u21913 * (69 / 50)) ^ 2 * log 2 \u2264 \u219112"}, {"tactic": "rw [\u2190 le_div_iff']", "annotated_tactic": ["rw [\u2190 <a>le_div_iff'</a>]", [{"full_name": "le_div_iff'", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [141, 9], "def_end_pos": [141, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 (\u21913 * (69 / 50)) ^ 2 * log 2 \u2264 \u219112", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 log 2 \u2264 \u219112 / (\u21913 * (69 / 50)) ^ 2\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 0 < (\u21913 * (69 / 50)) ^ 2"}, {"tactic": "exact sq_pos_of_ne_zero _ (by norm_num1)", "annotated_tactic": ["exact <a>sq_pos_of_ne_zero</a> _ (by norm_num1)", [{"full_name": "sq_pos_of_ne_zero", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [656, 9], "def_end_pos": [656, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 0 < (\u21913 * (69 / 50)) ^ 2", "state_after": "no goals"}, {"tactic": "rw [\u2190 log_rpow zero_lt_two, log_le_log, rpow_nat_cast]", "annotated_tactic": ["rw [\u2190 <a>log_rpow</a> <a>zero_lt_two</a>, <a>log_le_log</a>, <a>rpow_nat_cast</a>]", [{"full_name": "Real.log_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [401, 9], "def_end_pos": [401, 17]}, {"full_name": "zero_lt_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [71, 15], "def_end_pos": [71, 26]}, {"full_name": "Real.log_le_log", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [144, 9], "def_end_pos": [144, 19]}, {"full_name": "Real.rpow_nat_cast", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [357, 9], "def_end_pos": [357, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 \u219112 * log 2 \u2264 log \u2191N", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 2 ^ 12 \u2264 \u2191N\n\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < 2 ^ \u219112\n\ncase h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < \u2191N"}, {"tactic": "rw [cast_pos]", "annotated_tactic": ["rw [<a>cast_pos</a>]", [{"full_name": "Nat.cast_pos", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "case h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < \u2191N", "state_after": "case h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < N"}, {"tactic": "exact hN.trans_lt' (by norm_num1)", "annotated_tactic": ["exact hN.trans_lt' (by norm_num1)", []], "state_before": "case h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < N", "state_after": "no goals"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 2 ^ 12 \u2264 \u2191N", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 4096 \u2264 \u2191N"}, {"tactic": "exact_mod_cast hN", "annotated_tactic": ["exact_mod_cast hN", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 4096 \u2264 \u2191N", "state_after": "no goals"}, {"tactic": "exact rpow_pos_of_pos zero_lt_two _", "annotated_tactic": ["exact <a>rpow_pos_of_pos</a> <a>zero_lt_two</a> _", [{"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}, {"full_name": "zero_lt_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [71, 15], "def_end_pos": [71, 26]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < 2 ^ \u219112", "state_after": "no goals"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\n\u22a2 0 < 4096", "state_after": "no goals"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 0 < 8", "state_after": "no goals"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 0 \u2264 69 / 50", "state_after": "no goals"}, {"tactic": "refine' div_pos zero_lt_two _", "annotated_tactic": ["refine' <a>div_pos</a> <a>zero_lt_two</a> _", [{"full_name": "div_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 16]}, {"full_name": "zero_lt_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [71, 15], "def_end_pos": [71, 26]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 0 < 2 / (1 - 2 / rexp 1)", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 0 < 1 - 2 / rexp 1"}, {"tactic": "rw [sub_pos, div_lt_one (exp_pos _)]", "annotated_tactic": ["rw [<a>sub_pos</a>, <a>div_lt_one</a> (<a>exp_pos</a> _)]", [{"full_name": "sub_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [883, 30], "def_end_pos": [883, 37]}, {"full_name": "div_lt_one", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [431, 9], "def_end_pos": [431, 19]}, {"full_name": "Real.exp_pos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1499, 9], "def_end_pos": [1499, 16]}]], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 0 < 1 - 2 / rexp 1", "state_after": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 2 < rexp 1"}, {"tactic": "exact exp_one_gt_d9.trans_le' (by norm_num1)", "annotated_tactic": ["exact exp_one_gt_d9.trans_le' (by norm_num1)", []], "state_before": "case refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 2 < rexp 1", "state_after": "no goals"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 2 \u2264 2.7182818283", "state_after": "no goals"}, {"tactic": "rw [\u2190 log_rpow zero_lt_two, rpow_nat_cast]", "annotated_tactic": ["rw [\u2190 <a>log_rpow</a> <a>zero_lt_two</a>, <a>rpow_nat_cast</a>]", [{"full_name": "Real.log_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [401, 9], "def_end_pos": [401, 17]}, {"full_name": "zero_lt_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [71, 15], "def_end_pos": [71, 26]}, {"full_name": "Real.rpow_nat_cast", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [357, 9], "def_end_pos": [357, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 log 8 = \u21913 * log 2", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 log 8 = log (2 ^ 3)"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\n\u22a2 log 8 = log (2 ^ 3)", "state_after": "no goals"}, {"tactic": "exact log_two_lt_d9.le.trans (by norm_num1)", "annotated_tactic": ["exact log_two_lt_d9.le.trans (by norm_num1)", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 log 2 \u2264 \u219112 / (\u21913 * (69 / 50)) ^ 2", "state_after": "no goals"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 0.6931471808 \u2264 \u219112 / (\u21913 * (69 / 50)) ^ 2", "state_after": "no goals"}, {"tactic": "norm_num1", "annotated_tactic": ["norm_num1", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN : 4096 \u2264 N\nthis : \u219112 * log 2 \u2264 log \u2191N\nl8 : log 8 = \u21913 * log 2\n\u22a2 \u21913 * (69 / 50) \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/BigOperators/Basic.lean", "full_name": "Finset.prod_range_add_div_prod_range", "start": [1256, 1], "end": [1258, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Star/NonUnitalSubalgebra.lean", "full_name": "NonUnitalStarSubalgebra.coe_centralizer", "start": [1057, 1], "end": [1058, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Logic/Equiv/LocalEquiv.lean", "full_name": "LocalEquiv.prod_target", "start": [943, 1], "end": [945, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "full_name": "Multiset.prod_pair", "start": [109, 1], "end": [110, 49], "traced_tactics": [{"tactic": "rw [insert_eq_cons, prod_cons, prod_singleton]", "annotated_tactic": ["rw [<a>insert_eq_cons</a>, <a>prod_cons</a>, <a>prod_singleton</a>]", [{"full_name": "Multiset.insert_eq_cons", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 23]}, {"full_name": "Multiset.prod_cons", "def_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 18]}, {"full_name": "Multiset.prod_singleton", "def_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 23]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : CommMonoid \u03b1\ns t : Multiset \u03b1\na\u271d : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\na b : \u03b1\n\u22a2 prod {a, b} = a * b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.preimage_comp_eq", "start": [167, 1], "end": [168, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "full_name": "MeasurableEmbedding.integrableOn_map_iff", "start": [234, 1], "end": [237, 67], "traced_tactics": [{"tactic": "simp only [IntegrableOn, he.restrict_map, he.integrable_map_iff]", "annotated_tactic": ["simp only [<a>IntegrableOn</a>, he.restrict_map, he.integrable_map_iff]", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf\u271d g : \u03b1 \u2192 E\ns\u271d t : Set \u03b1\n\u03bc\u271d \u03bd : Measure \u03b1\ninst\u271d : MeasurableSpace \u03b2\ne : \u03b1 \u2192 \u03b2\nhe : MeasurableEmbedding e\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b1\ns : Set \u03b2\n\u22a2 IntegrableOn f s \u2194 IntegrableOn (f \u2218 e) (e \u207b\u00b9' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/VectorBundle/Basic.lean", "full_name": "Trivialization.coe_linearMapAt_of_mem", "start": [241, 1], "end": [243, 39], "traced_tactics": [{"tactic": "simp_rw [coe_linearMapAt, if_pos hb]", "annotated_tactic": ["simp_rw [<a>coe_linearMapAt</a>, <a>if_pos</a> hb]", [{"full_name": "Trivialization.coe_linearMapAt", "def_path": "Mathlib/Topology/VectorBundle/Basic.lean", "def_pos": [236, 9], "def_end_pos": [236, 24]}, {"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}]], "state_before": "R : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : TopologicalSpace F\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module R (E x)\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\nhb : b \u2208 e.baseSet\n\u22a2 \u2191(Trivialization.linearMapAt R e b) = fun y => (\u2191e { proj := b, snd := y }).2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Invertible/Defs.lean", "full_name": "invOf_mul_self_assoc'", "start": [117, 1], "end": [118, 44], "traced_tactics": [{"tactic": "rw [\u2190 mul_assoc, invOf_mul_self, one_mul]", "annotated_tactic": ["rw [\u2190 <a>mul_assoc</a>, <a>invOf_mul_self</a>, <a>one_mul</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "invOf_mul_self", "def_path": "Mathlib/Algebra/Invertible/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 23]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na b : \u03b1\nx\u271d : Invertible a\n\u22a2 \u215fa * (a * b) = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Functor/LeftDerived.lean", "full_name": "CategoryTheory.NatTrans.leftDerived_eq", "start": [153, 1], "end": [170, 29], "traced_tactics": [{"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "C : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (leftDerived \u03b1 n).app X =\n    (Functor.leftDerivedObjIso F n P).hom \u226b\n      (homologyFunctor D (ComplexShape.down \u2115) n).map ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) \u226b\n        (Functor.leftDerivedObjIso G n P).inv", "state_after": "C : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (Functor.leftDerivedObjIso F n P).hom \u226b\n      (homologyFunctor D (ComplexShape.down \u2115) n).map ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) \u226b\n        (Functor.leftDerivedObjIso G n P).inv =\n    (leftDerived \u03b1 n).app X"}, {"tactic": "dsimp [NatTrans.leftDerived, Functor.leftDerivedObjIso]", "annotated_tactic": ["dsimp [<a>NatTrans.leftDerived</a>, <a>Functor.leftDerivedObjIso</a>]", [{"full_name": "CategoryTheory.NatTrans.leftDerived", "def_path": "Mathlib/CategoryTheory/Functor/LeftDerived.lean", "def_pos": [129, 5], "def_end_pos": [129, 25]}, {"full_name": "CategoryTheory.Functor.leftDerivedObjIso", "def_path": "Mathlib/CategoryTheory/Functor/LeftDerived.lean", "def_pos": [69, 5], "def_end_pos": [69, 30]}]], "state_before": "C : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (Functor.leftDerivedObjIso F n P).hom \u226b\n      (homologyFunctor D (ComplexShape.down \u2115) n).map ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) \u226b\n        (Functor.leftDerivedObjIso G n P).inv =\n    (leftDerived \u03b1 n).app X", "state_after": "C : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 ((HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom)) \u226b\n        \ud835\udfd9\n          ((HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).obj\n            ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).obj\n              ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex)))) \u226b\n      homology.map\n          (_ :\n            HomologicalComplex.dTo ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n \u226b\n                HomologicalComplex.dFrom ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n =\n              0)\n          (_ :\n            HomologicalComplex.dTo ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n \u226b\n                HomologicalComplex.dFrom ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n =\n              0)\n          (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n)\n          (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n)\n          (_ :\n            (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n).right =\n              (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n).right) \u226b\n        \ud835\udfd9 (HomologicalComplex.homology ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n) \u226b\n          (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n            ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n              ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n                (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as))"}, {"tactic": "simp only [Category.comp_id, Category.id_comp]", "annotated_tactic": ["simp only [<a>Category.comp_id</a>, <a>Category.id_comp</a>]", [{"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [157, 3], "def_end_pos": [157, 10]}, {"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [155, 3], "def_end_pos": [155, 10]}]], "state_before": "C : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 ((HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom)) \u226b\n        \ud835\udfd9\n          ((HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).obj\n            ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).obj\n              ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex)))) \u226b\n      homology.map\n          (_ :\n            HomologicalComplex.dTo ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n \u226b\n                HomologicalComplex.dFrom ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n =\n              0)\n          (_ :\n            HomologicalComplex.dTo ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n \u226b\n                HomologicalComplex.dFrom ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n =\n              0)\n          (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n)\n          (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n)\n          (_ :\n            (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n).right =\n              (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n).right) \u226b\n        \ud835\udfd9 (HomologicalComplex.homology ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n) \u226b\n          (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n            ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n              ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n                (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as))", "state_after": "C : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n        ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n          ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n            (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom)) \u226b\n      homology.map\n          (_ :\n            HomologicalComplex.dTo ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n \u226b\n                HomologicalComplex.dFrom ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n =\n              0)\n          (_ :\n            HomologicalComplex.dTo ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n \u226b\n                HomologicalComplex.dFrom ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n =\n              0)\n          (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n)\n          (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n)\n          (_ :\n            (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n).right =\n              (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n).right) \u226b\n        (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as))"}, {"tactic": "rw [\u2190 homologyFunctor_map, HomotopyCategory.homologyFunctor_map_factors]", "annotated_tactic": ["rw [\u2190 <a>homologyFunctor_map</a>, <a>HomotopyCategory.homologyFunctor_map_factors</a>]", [{"full_name": "homologyFunctor_map", "def_path": "Mathlib/Algebra/Homology/Homology.lean", "def_pos": [289, 3], "def_end_pos": [289, 8]}, {"full_name": "HomotopyCategory.homologyFunctor_map_factors", "def_path": "Mathlib/Algebra/Homology/HomotopyCategory.lean", "def_pos": [170, 9], "def_end_pos": [170, 36]}]], "state_before": "C : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n        ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n          ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n            (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom)) \u226b\n      homology.map\n          (_ :\n            HomologicalComplex.dTo ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n \u226b\n                HomologicalComplex.dFrom ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).obj P.complex) n =\n              0)\n          (_ :\n            HomologicalComplex.dTo ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n \u226b\n                HomologicalComplex.dFrom ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj P.complex) n =\n              0)\n          (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n)\n          (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n)\n          (_ :\n            (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n).right =\n              (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex) n).right) \u226b\n        (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as))", "state_after": "C : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n        ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n          ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n            (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom)) \u226b\n      (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex)) \u226b\n        (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as))"}, {"tactic": "simp only [\u2190 Functor.map_comp]", "annotated_tactic": ["simp only [\u2190 <a>Functor.map_comp</a>]", [{"full_name": "CategoryTheory.Functor.map_comp", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [43, 3], "def_end_pos": [43, 11]}]], "state_before": "C : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n        ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n          ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n            (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom)) \u226b\n      (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex)) \u226b\n        (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n          ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n            ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as))", "state_after": "C : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n            (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n          (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex \u226b\n            (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as))"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "C : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n            (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n          (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex \u226b\n            (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n              (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.down \u2115) n).map\n      ((HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n        ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as))", "state_after": "case e_a\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n      ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n          (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n        (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex \u226b\n          (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n            (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv) =\n    (HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n      ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as)"}, {"tactic": "apply HomotopyCategory.eq_of_homotopy", "annotated_tactic": ["apply <a>HomotopyCategory.eq_of_homotopy</a>", [{"full_name": "HomotopyCategory.eq_of_homotopy", "def_path": "Mathlib/Algebra/Homology/HomotopyCategory.lean", "def_pos": [90, 9], "def_end_pos": [90, 23]}]], "state_before": "case e_a\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n      ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n          (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n        (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex \u226b\n          (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n            (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv) =\n    (HomotopyCategory.quotient D (ComplexShape.down \u2115)).map\n      ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as)", "state_after": "case e_a.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 Homotopy\n    ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n        (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n      (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex \u226b\n        (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n          (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)\n    ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as)"}, {"tactic": "simp only [NatTrans.mapHomologicalComplex_naturality_assoc, \u2190 Functor.map_comp]", "annotated_tactic": ["simp only [<a>NatTrans.mapHomologicalComplex_naturality_assoc</a>, \u2190 <a>Functor.map_comp</a>]", [{"full_name": "CategoryTheory.NatTrans.mapHomologicalComplex_naturality_assoc", "def_path": "Mathlib/Algebra/Homology/Additive.lean", "def_pos": [219, 3], "def_end_pos": [219, 30]}, {"full_name": "CategoryTheory.Functor.map_comp", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [43, 3], "def_end_pos": [43, 11]}]], "state_before": "case e_a.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 Homotopy\n    ((Functor.mapHomologicalComplex F (ComplexShape.down \u2115)).map\n        (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n      (mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app P.complex \u226b\n        (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n          (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv)\n    ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as)", "state_after": "case e_a.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 Homotopy\n    ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as \u226b\n      (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n        ((ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n          (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv))\n    ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as)"}, {"tactic": "apply Homotopy.compLeftId", "annotated_tactic": ["apply <a>Homotopy.compLeftId</a>", [{"full_name": "Homotopy.compLeftId", "def_path": "Mathlib/Algebra/Homology/Homotopy.lean", "def_pos": [232, 5], "def_end_pos": [232, 15]}]], "state_before": "case e_a.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 Homotopy\n    ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as \u226b\n      (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n        ((ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n          (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv))\n    ((mapHomologicalComplex \u03b1 (ComplexShape.down \u2115)).app ((projectiveResolutions C).obj X).as)", "state_after": "case e_a.h.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 Homotopy\n    ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n      ((ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n        (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv))\n    (\ud835\udfd9 ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj ((projectiveResolutions C).obj X).as))"}, {"tactic": "refine' (Functor.mapHomotopy _ (HomotopyEquiv.homotopyHomInvId _) ).trans _", "annotated_tactic": ["refine' (<a>Functor.mapHomotopy</a> _ (<a>HomotopyEquiv.homotopyHomInvId</a> _) ).<a>trans</a> _", [{"full_name": "CategoryTheory.Functor.mapHomotopy", "def_path": "Mathlib/Algebra/Homology/Homotopy.lean", "def_pos": [823, 5], "def_end_pos": [823, 24]}, {"full_name": "HomotopyEquiv.homotopyHomInvId", "def_path": "Mathlib/Algebra/Homology/Homotopy.lean", "def_pos": [733, 3], "def_end_pos": [733, 19]}, {"full_name": "Homotopy.trans", "def_path": "Mathlib/Algebra/Homology/Homotopy.lean", "def_pos": [180, 5], "def_end_pos": [180, 10]}]], "state_before": "case e_a.h.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 Homotopy\n    ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map\n      ((ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).hom \u226b\n        (ProjectiveResolution.homotopyEquiv (projectiveResolution X) P).inv))\n    (\ud835\udfd9 ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj ((projectiveResolutions C).obj X).as))", "state_after": "case e_a.h.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 Homotopy ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map (\ud835\udfd9 ((projectiveResolutions C).obj X).as))\n    (\ud835\udfd9 ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj ((projectiveResolutions C).obj X).as))"}, {"tactic": "apply Homotopy.ofEq", "annotated_tactic": ["apply <a>Homotopy.ofEq</a>", [{"full_name": "Homotopy.ofEq", "def_path": "Mathlib/Algebra/Homology/Homotopy.lean", "def_pos": [157, 5], "def_end_pos": [157, 9]}]], "state_before": "case e_a.h.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 Homotopy ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map (\ud835\udfd9 ((projectiveResolutions C).obj X).as))\n    (\ud835\udfd9 ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj ((projectiveResolutions C).obj X).as))", "state_after": "case e_a.h.h.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map (\ud835\udfd9 ((projectiveResolutions C).obj X).as) =\n    \ud835\udfd9 ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj ((projectiveResolutions C).obj X).as)"}, {"tactic": "simp only [Functor.map_id]", "annotated_tactic": ["simp only [<a>Functor.map_id</a>]", [{"full_name": "CategoryTheory.Functor.map_id", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [41, 3], "def_end_pos": [41, 9]}]], "state_before": "case e_a.h.h.h\nC : Type u\ninst\u271d\u00b9\u00b3 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u00b9\u00b2 : Category.{u_2, u_1} D\ninst\u271d\u00b9\u00b9 : Preadditive C\ninst\u271d\u00b9\u2070 : HasZeroObject C\ninst\u271d\u2079 : HasEqualizers C\ninst\u271d\u2078 : HasImages C\ninst\u271d\u2077 : HasProjectiveResolutions C\ninst\u271d\u2076 : Preadditive D\ninst\u271d\u2075 : HasEqualizers D\ninst\u271d\u2074 : HasCokernels D\ninst\u271d\u00b3 : HasImages D\ninst\u271d\u00b2 : HasImageMaps D\nF G : C \u2964 D\ninst\u271d\u00b9 : Functor.Additive F\ninst\u271d : Functor.Additive G\n\u03b1 : F \u27f6 G\nn : \u2115\nX : C\nP : ProjectiveResolution X\n\u22a2 (Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).map (\ud835\udfd9 ((projectiveResolutions C).obj X).as) =\n    \ud835\udfd9 ((Functor.mapHomologicalComplex G (ComplexShape.down \u2115)).obj ((projectiveResolutions C).obj X).as)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Real/EReal.lean", "full_name": "EReal.coe_toReal_le", "start": [424, 1], "end": [427, 41], "traced_tactics": [{"tactic": "by_cases h' : x = \u22a4", "annotated_tactic": ["by_cases h' : x = \u22a4", []], "state_before": "x : EReal\nh : x \u2260 \u22a5\n\u22a2 \u2191(toReal x) \u2264 x", "state_after": "case pos\nx : EReal\nh : x \u2260 \u22a5\nh' : x = \u22a4\n\u22a2 \u2191(toReal x) \u2264 x\n\ncase neg\nx : EReal\nh : x \u2260 \u22a5\nh' : \u00acx = \u22a4\n\u22a2 \u2191(toReal x) \u2264 x"}, {"tactic": "simp only [h', le_top]", "annotated_tactic": ["simp only [h', <a>le_top</a>]", [{"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [98, 9], "def_end_pos": [98, 15]}]], "state_before": "case pos\nx : EReal\nh : x \u2260 \u22a5\nh' : x = \u22a4\n\u22a2 \u2191(toReal x) \u2264 x", "state_after": "no goals"}, {"tactic": "simp only [le_refl, coe_toReal h' h]", "annotated_tactic": ["simp only [<a>le_refl</a>, <a>coe_toReal</a> h' h]", [{"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 16]}, {"full_name": "EReal.coe_toReal", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [413, 9], "def_end_pos": [413, 19]}]], "state_before": "case neg\nx : EReal\nh : x \u2260 \u22a5\nh' : \u00acx = \u22a4\n\u22a2 \u2191(toReal x) \u2264 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/LinearPMap.lean", "full_name": "LinearPMap.map_smul", "start": [109, 1], "end": [110, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/Lattice.lean", "full_name": "Set.biUnion_lt_succ", "start": [222, 1], "end": [223, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Monotone/Monovary.lean", "full_name": "Antivary.antivaryOn", "start": [62, 11], "end": [63, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Field/Basic.lean", "full_name": "Commute.div_sub_div", "start": [189, 11], "end": [191, 92], "traced_tactics": [{"tactic": "simpa only [mul_neg, neg_div, \u2190 sub_eq_add_neg] using hbc.neg_right.div_add_div hbd hb hd", "annotated_tactic": ["simpa only [<a>mul_neg</a>, <a>neg_div</a>, \u2190 <a>sub_eq_add_neg</a>] using hbc.neg_right.div_add_div hbd hb hd", [{"full_name": "mul_neg", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [294, 9], "def_end_pos": [294, 16]}, {"full_name": "neg_div", "def_path": "Mathlib/Algebra/Field/Basic.lean", "def_pos": [113, 9], "def_end_pos": [113, 16]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [975, 3], "def_end_pos": [975, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : DivisionRing K\na b c d : K\nhbc : Commute b c\nhbd : Commute b d\nhb : b \u2260 0\nhd : d \u2260 0\n\u22a2 a / b - c / d = (a * d - b * c) / (b * d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "full_name": "MeasureTheory.Mem\u2112p.snorm_indicator_le", "start": [388, 1], "end": [398, 38], "traced_tactics": [{"tactic": "have h\u2112p := hf", "annotated_tactic": ["have h\u2112p := hf", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : Mem\u2112p f p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 \u03b4 h\u03b4, \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : Mem\u2112p f p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\n\u22a2 \u2203 \u03b4 h\u03b4, \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "obtain \u27e8\u27e8f', hf', heq\u27e9, _\u27e9 := hf", "annotated_tactic": ["obtain \u27e8\u27e8f', hf', heq\u27e9, _\u27e9 := hf", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nhf : Mem\u2112p f p\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\n\u22a2 \u2203 \u03b4 h\u03b4, \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\nright\u271d : snorm f p \u03bc < \u22a4\nf' : \u03b1 \u2192 \u03b2\nhf' : StronglyMeasurable f'\nheq : f =\u1d50[\u03bc] f'\n\u22a2 \u2203 \u03b4 h\u03b4, \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "obtain \u27e8\u03b4, h\u03b4pos, h\u03b4\u27e9 := (h\u2112p.ae_eq heq).snorm_indicator_le_of_meas \u03bc hp_one hp_top hf' h\u03b5", "annotated_tactic": ["obtain \u27e8\u03b4, h\u03b4pos, h\u03b4\u27e9 := (h\u2112p.ae_eq heq).<a>snorm_indicator_le_of_meas</a> \u03bc hp_one hp_top hf' h\u03b5", [{"full_name": "MeasureTheory.Mem\u2112p.snorm_indicator_le_of_meas", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [377, 9], "def_end_pos": [377, 41]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\nright\u271d : snorm f p \u03bc < \u22a4\nf' : \u03b1 \u2192 \u03b2\nhf' : StronglyMeasurable f'\nheq : f =\u1d50[\u03bc] f'\n\u22a2 \u2203 \u03b4 h\u03b4, \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\nright\u271d : snorm f p \u03bc < \u22a4\nf' : \u03b1 \u2192 \u03b2\nhf' : StronglyMeasurable f'\nheq : f =\u1d50[\u03bc] f'\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f') p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 \u03b4 h\u03b4, \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "refine' \u27e8\u03b4, h\u03b4pos, fun s hs h\u03bcs => _\u27e9", "annotated_tactic": ["refine' \u27e8\u03b4, h\u03b4pos, fun s hs h\u03bcs => _\u27e9", []], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\nright\u271d : snorm f p \u03bc < \u22a4\nf' : \u03b1 \u2192 \u03b2\nhf' : StronglyMeasurable f'\nheq : f =\u1d50[\u03bc] f'\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f') p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 \u03b4 h\u03b4, \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\nright\u271d : snorm f p \u03bc < \u22a4\nf' : \u03b1 \u2192 \u03b2\nhf' : StronglyMeasurable f'\nheq : f =\u1d50[\u03bc] f'\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f') p \u03bc \u2264 ENNReal.ofReal \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\n\u22a2 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "convert h\u03b4 s hs h\u03bcs using 1", "annotated_tactic": ["convert h\u03b4 s hs h\u03bcs using 1", []], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\nright\u271d : snorm f p \u03bc < \u22a4\nf' : \u03b1 \u2192 \u03b2\nhf' : StronglyMeasurable f'\nheq : f =\u1d50[\u03bc] f'\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f') p \u03bc \u2264 ENNReal.ofReal \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\n\u22a2 snorm (Set.indicator s f) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\nright\u271d : snorm f p \u03bc < \u22a4\nf' : \u03b1 \u2192 \u03b2\nhf' : StronglyMeasurable f'\nheq : f =\u1d50[\u03bc] f'\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f') p \u03bc \u2264 ENNReal.ofReal \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\n\u22a2 snorm (Set.indicator s f) p \u03bc = snorm (Set.indicator s f') p \u03bc"}, {"tactic": "rw [snorm_indicator_eq_snorm_restrict hs, snorm_indicator_eq_snorm_restrict hs]", "annotated_tactic": ["rw [<a>snorm_indicator_eq_snorm_restrict</a> hs, <a>snorm_indicator_eq_snorm_restrict</a> hs]", [{"full_name": "MeasureTheory.snorm_indicator_eq_snorm_restrict", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [657, 9], "def_end_pos": [657, 42]}, {"full_name": "MeasureTheory.snorm_indicator_eq_snorm_restrict", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [657, 9], "def_end_pos": [657, 42]}]], "state_before": "case h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\nright\u271d : snorm f p \u03bc < \u22a4\nf' : \u03b1 \u2192 \u03b2\nhf' : StronglyMeasurable f'\nheq : f =\u1d50[\u03bc] f'\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f') p \u03bc \u2264 ENNReal.ofReal \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\n\u22a2 snorm (Set.indicator s f) p \u03bc = snorm (Set.indicator s f') p \u03bc", "state_after": "case h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\nright\u271d : snorm f p \u03bc < \u22a4\nf' : \u03b1 \u2192 \u03b2\nhf' : StronglyMeasurable f'\nheq : f =\u1d50[\u03bc] f'\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f') p \u03bc \u2264 ENNReal.ofReal \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\n\u22a2 snorm f p (Measure.restrict \u03bc s) = snorm f' p (Measure.restrict \u03bc s)"}, {"tactic": "refine' snorm_congr_ae heq.restrict", "annotated_tactic": ["refine' <a>snorm_congr_ae</a> heq.restrict", [{"full_name": "MeasureTheory.snorm_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [549, 9], "def_end_pos": [549, 23]}]], "state_before": "case h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf : \u03b1 \u2192 \u03b2\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u2112p : Mem\u2112p f p\nright\u271d : snorm f p \u03bc < \u22a4\nf' : \u03b1 \u2192 \u03b2\nhf' : StronglyMeasurable f'\nheq : f =\u1d50[\u03bc] f'\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (Set.indicator s f') p \u03bc \u2264 ENNReal.ofReal \u03b5\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2264 ENNReal.ofReal \u03b4\n\u22a2 snorm f p (Measure.restrict \u03bc s) = snorm f' p (Measure.restrict \u03bc s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Lie/IdealOperations.lean", "full_name": "LieSubmodule.lieIdeal_oper_eq_linear_span", "start": [66, 1], "end": [85, 64], "traced_tactics": [{"tactic": "apply le_antisymm", "annotated_tactic": ["apply <a>le_antisymm</a>", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "R : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\n\u22a2 \u2191\u2045I, N\u2046 = Submodule.span R {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}", "state_after": "case a\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\n\u22a2 \u2191\u2045I, N\u2046 \u2264 Submodule.span R {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\n\ncase a\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\n\u22a2 Submodule.span R {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} \u2264 \u2191\u2045I, N\u2046"}, {"tactic": "let s := { m : M | \u2203 (x : \u21a5I) (n : \u21a5N), \u2045(x : L), (n : M)\u2046 = m }", "annotated_tactic": ["let s := { m : M | \u2203 (x : \u21a5I) (n : \u21a5N), \u2045(x : L), (n : M)\u2046 = m }", []], "state_before": "case a\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\n\u22a2 \u2191\u2045I, N\u2046 \u2264 Submodule.span R {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}", "state_after": "case a\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\n\u22a2 \u2191\u2045I, N\u2046 \u2264 Submodule.span R {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}"}, {"tactic": "change _ \u2264 ({ Submodule.span R s with lie_mem := fun hm' => aux _ _ hm' } : LieSubmodule R L M)", "annotated_tactic": ["change _ \u2264 ({ <a>Submodule.span</a> R s with lie_mem := fun hm' => aux _ _ hm' } : <a>LieSubmodule</a> R L M)", [{"full_name": "Submodule.span", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [49, 5], "def_end_pos": [49, 9]}, {"full_name": "LieSubmodule", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [48, 11], "def_end_pos": [48, 23]}]], "state_before": "case a\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\naux : \u2200 (y : L) (m' : M), m' \u2208 Submodule.span R s \u2192 \u2045y, m'\u2046 \u2208 Submodule.span R s\n\u22a2 \u2191\u2045I, N\u2046 \u2264 Submodule.span R {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}", "state_after": "case a\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\naux : \u2200 (y : L) (m' : M), m' \u2208 Submodule.span R s \u2192 \u2045y, m'\u2046 \u2208 Submodule.span R s\n\u22a2 \u2045I, N\u2046 \u2264\n    let src := Submodule.span R s;\n    {\n      toSubmodule :=\n        { toAddSubmonoid := src.toAddSubmonoid,\n          smul_mem' := (_ : \u2200 (c : R) {x : M}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) },\n      lie_mem :=\n        (_ :\n          \u2200 {x : L} {m : M},\n            m \u2208\n                { toAddSubmonoid := src.toAddSubmonoid,\n                        smul_mem' :=\n                          (_ :\n                            \u2200 (c : R) {x : M},\n                              x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n              \u2045x, m\u2046 \u2208 Submodule.span R s) }"}, {"tactic": "rw [lieIdeal_oper_eq_span, lieSpan_le]", "annotated_tactic": ["rw [<a>lieIdeal_oper_eq_span</a>, <a>lieSpan_le</a>]", [{"full_name": "LieSubmodule.lieIdeal_oper_eq_span", "def_path": "Mathlib/Algebra/Lie/IdealOperations.lean", "def_pos": [59, 9], "def_end_pos": [59, 30]}, {"full_name": "LieSubmodule.lieSpan_le", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [688, 9], "def_end_pos": [688, 19]}]], "state_before": "case a\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\naux : \u2200 (y : L) (m' : M), m' \u2208 Submodule.span R s \u2192 \u2045y, m'\u2046 \u2208 Submodule.span R s\n\u22a2 \u2045I, N\u2046 \u2264\n    let src := Submodule.span R s;\n    {\n      toSubmodule :=\n        { toAddSubmonoid := src.toAddSubmonoid,\n          smul_mem' := (_ : \u2200 (c : R) {x : M}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) },\n      lie_mem :=\n        (_ :\n          \u2200 {x : L} {m : M},\n            m \u2208\n                { toAddSubmonoid := src.toAddSubmonoid,\n                        smul_mem' :=\n                          (_ :\n                            \u2200 (c : R) {x : M},\n                              x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n              \u2045x, m\u2046 \u2208 Submodule.span R s) }", "state_after": "case a\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\naux : \u2200 (y : L) (m' : M), m' \u2208 Submodule.span R s \u2192 \u2045y, m'\u2046 \u2208 Submodule.span R s\n\u22a2 {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} \u2286\n    \u2191(let src := Submodule.span R s;\n      {\n        toSubmodule :=\n          { toAddSubmonoid := src.toAddSubmonoid,\n            smul_mem' := (_ : \u2200 (c : R) {x : M}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) },\n        lie_mem :=\n          (_ :\n            \u2200 {x : L} {m : M},\n              m \u2208\n                  { toAddSubmonoid := src.toAddSubmonoid,\n                          smul_mem' :=\n                            (_ :\n                              \u2200 (c : R) {x : M},\n                                x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n                \u2045x, m\u2046 \u2208 Submodule.span R s) })"}, {"tactic": "exact Submodule.subset_span", "annotated_tactic": ["exact <a>Submodule.subset_span</a>", [{"full_name": "Submodule.subset_span", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [62, 9], "def_end_pos": [62, 20]}]], "state_before": "case a\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\naux : \u2200 (y : L) (m' : M), m' \u2208 Submodule.span R s \u2192 \u2045y, m'\u2046 \u2208 Submodule.span R s\n\u22a2 {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} \u2286\n    \u2191(let src := Submodule.span R s;\n      {\n        toSubmodule :=\n          { toAddSubmonoid := src.toAddSubmonoid,\n            smul_mem' := (_ : \u2200 (c : R) {x : M}, x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) },\n        lie_mem :=\n          (_ :\n            \u2200 {x : L} {m : M},\n              m \u2208\n                  { toAddSubmonoid := src.toAddSubmonoid,\n                          smul_mem' :=\n                            (_ :\n                              \u2200 (c : R) {x : M},\n                                x \u2208 src.carrier \u2192 c \u2022 x \u2208 src.carrier) }.toAddSubmonoid.toAddSubsemigroup.carrier \u2192\n                \u2045x, m\u2046 \u2208 Submodule.span R s) })", "state_after": "no goals"}, {"tactic": "intro y m' hm'", "annotated_tactic": ["intro y m' hm'", []], "state_before": "R : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\n\u22a2 \u2200 (y : L) (m' : M), m' \u2208 Submodule.span R s \u2192 \u2045y, m'\u2046 \u2208 Submodule.span R s", "state_after": "R : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\n\u22a2 \u2045y, m'\u2046 \u2208 Submodule.span R s"}, {"tactic": "refine Submodule.span_induction (R := R) (M := M) (s := s)\n  (p := fun m' \u21a6 \u2045y, m'\u2046 \u2208 Submodule.span R s) hm' ?_ ?_ ?_ ?_", "annotated_tactic": ["refine <a>Submodule.span_induction</a> (R := R) (M := M) (s := s)\n        (p := fun m' \u21a6 \u2045y, m'\u2046 \u2208 <a>Submodule.span</a> R s) hm' ?_ ?_ ?_ ?_", [{"full_name": "Submodule.span_induction", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [149, 9], "def_end_pos": [149, 23]}, {"full_name": "Submodule.span", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [49, 5], "def_end_pos": [49, 9]}]], "state_before": "R : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\n\u22a2 \u2045y, m'\u2046 \u2208 Submodule.span R s", "state_after": "case refine_1\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\n\u22a2 \u2200 (x : M), x \u2208 s \u2192 (fun m' => \u2045y, m'\u2046 \u2208 Submodule.span R s) x\n\ncase refine_2\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\n\u22a2 (fun m' => \u2045y, m'\u2046 \u2208 Submodule.span R s) 0\n\ncase refine_3\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\n\u22a2 \u2200 (x y_1 : M),\n    (fun m' => \u2045y, m'\u2046 \u2208 Submodule.span R s) x \u2192\n      (fun m' => \u2045y, m'\u2046 \u2208 Submodule.span R s) y_1 \u2192 (fun m' => \u2045y, m'\u2046 \u2208 Submodule.span R s) (x + y_1)\n\ncase refine_4\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\n\u22a2 \u2200 (a : R) (x : M), (fun m' => \u2045y, m'\u2046 \u2208 Submodule.span R s) x \u2192 (fun m' => \u2045y, m'\u2046 \u2208 Submodule.span R s) (a \u2022 x)"}, {"tactic": "rintro m'' \u27e8x, n, hm''\u27e9", "annotated_tactic": ["rintro m'' \u27e8x, n, hm''\u27e9", []], "state_before": "case refine_1\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\n\u22a2 \u2200 (x : M), x \u2208 s \u2192 (fun m' => \u2045y, m'\u2046 \u2208 Submodule.span R s) x", "state_after": "case refine_1.intro.intro\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\nm'' : M\nx : { x // x \u2208 I }\nn : { x // x \u2208 N }\nhm'' : \u2045\u2191x, \u2191n\u2046 = m''\n\u22a2 \u2045y, m''\u2046 \u2208 Submodule.span R s"}, {"tactic": "rw [\u2190 hm'', leibniz_lie]", "annotated_tactic": ["rw [\u2190 hm'', <a>leibniz_lie</a>]", [{"full_name": "leibniz_lie", "def_path": "Mathlib/Algebra/Lie/Basic.lean", "def_pos": [132, 9], "def_end_pos": [132, 20]}]], "state_before": "case refine_1.intro.intro\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\nm'' : M\nx : { x // x \u2208 I }\nn : { x // x \u2208 N }\nhm'' : \u2045\u2191x, \u2191n\u2046 = m''\n\u22a2 \u2045y, m''\u2046 \u2208 Submodule.span R s", "state_after": "case refine_1.intro.intro\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\nm'' : M\nx : { x // x \u2208 I }\nn : { x // x \u2208 N }\nhm'' : \u2045\u2191x, \u2191n\u2046 = m''\n\u22a2 \u2045\u2045y, \u2191x\u2046, \u2191n\u2046 + \u2045\u2191x, \u2045y, \u2191n\u2046\u2046 \u2208 Submodule.span R s"}, {"tactic": "refine Submodule.add_mem _ ?_ ?_ <;> apply Submodule.subset_span", "annotated_tactic": ["refine <a>Submodule.add_mem</a> _ ?_ ?_ <;> apply <a>Submodule.subset_span</a>", [{"full_name": "Submodule.add_mem", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [239, 19], "def_end_pos": [239, 26]}, {"full_name": "Submodule.subset_span", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [62, 9], "def_end_pos": [62, 20]}]], "state_before": "case refine_1.intro.intro\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\nm'' : M\nx : { x // x \u2208 I }\nn : { x // x \u2208 N }\nhm'' : \u2045\u2191x, \u2191n\u2046 = m''\n\u22a2 \u2045\u2045y, \u2191x\u2046, \u2191n\u2046 + \u2045\u2191x, \u2045y, \u2191n\u2046\u2046 \u2208 Submodule.span R s", "state_after": "case refine_1.intro.intro.refine_1.a\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\nm'' : M\nx : { x // x \u2208 I }\nn : { x // x \u2208 N }\nhm'' : \u2045\u2191x, \u2191n\u2046 = m''\n\u22a2 \u2045\u2045y, \u2191x\u2046, \u2191n\u2046 \u2208 s\n\ncase refine_1.intro.intro.refine_2.a\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\nm'' : M\nx : { x // x \u2208 I }\nn : { x // x \u2208 N }\nhm'' : \u2045\u2191x, \u2191n\u2046 = m''\n\u22a2 \u2045\u2191x, \u2045y, \u2191n\u2046\u2046 \u2208 s"}, {"tactic": "use \u27e8\u2045y, \u2191x\u2046, I.lie_mem x.property\u27e9, n", "annotated_tactic": ["use \u27e8\u2045y, \u2191x\u2046, I.lie_mem x.property\u27e9, n", []], "state_before": "case refine_1.intro.intro.refine_1.a\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\nm'' : M\nx : { x // x \u2208 I }\nn : { x // x \u2208 N }\nhm'' : \u2045\u2191x, \u2191n\u2046 = m''\n\u22a2 \u2045\u2045y, \u2191x\u2046, \u2191n\u2046 \u2208 s", "state_after": "no goals"}, {"tactic": "use x, \u27e8\u2045y, \u2191n\u2046, N.lie_mem n.property\u27e9", "annotated_tactic": ["use x, \u27e8\u2045y, \u2191n\u2046, N.lie_mem n.property\u27e9", []], "state_before": "case refine_1.intro.intro.refine_2.a\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\nm'' : M\nx : { x // x \u2208 I }\nn : { x // x \u2208 N }\nhm'' : \u2045\u2191x, \u2191n\u2046 = m''\n\u22a2 \u2045\u2191x, \u2045y, \u2191n\u2046\u2046 \u2208 s", "state_after": "no goals"}, {"tactic": "simp only [lie_zero, Submodule.zero_mem]", "annotated_tactic": ["simp only [<a>lie_zero</a>, <a>Submodule.zero_mem</a>]", [{"full_name": "lie_zero", "def_path": "Mathlib/Algebra/Lie/Basic.lean", "def_pos": [137, 9], "def_end_pos": [137, 17]}, {"full_name": "Submodule.zero_mem", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [235, 19], "def_end_pos": [235, 27]}]], "state_before": "case refine_2\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\n\u22a2 (fun m' => \u2045y, m'\u2046 \u2208 Submodule.span R s) 0", "state_after": "no goals"}, {"tactic": "intro m\u2081 m\u2082 hm\u2081 hm\u2082", "annotated_tactic": ["intro m\u2081 m\u2082 hm\u2081 hm\u2082", []], "state_before": "case refine_3\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\n\u22a2 \u2200 (x y_1 : M),\n    (fun m' => \u2045y, m'\u2046 \u2208 Submodule.span R s) x \u2192\n      (fun m' => \u2045y, m'\u2046 \u2208 Submodule.span R s) y_1 \u2192 (fun m' => \u2045y, m'\u2046 \u2208 Submodule.span R s) (x + y_1)", "state_after": "case refine_3\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\nm\u2081 m\u2082 : M\nhm\u2081 : \u2045y, m\u2081\u2046 \u2208 Submodule.span R s\nhm\u2082 : \u2045y, m\u2082\u2046 \u2208 Submodule.span R s\n\u22a2 \u2045y, m\u2081 + m\u2082\u2046 \u2208 Submodule.span R s"}, {"tactic": "rw [lie_add]", "annotated_tactic": ["rw [<a>lie_add</a>]", [{"full_name": "lie_add", "def_path": "Mathlib/Algebra/Lie/Basic.lean", "def_pos": [118, 9], "def_end_pos": [118, 16]}]], "state_before": "case refine_3\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\nm\u2081 m\u2082 : M\nhm\u2081 : \u2045y, m\u2081\u2046 \u2208 Submodule.span R s\nhm\u2082 : \u2045y, m\u2082\u2046 \u2208 Submodule.span R s\n\u22a2 \u2045y, m\u2081 + m\u2082\u2046 \u2208 Submodule.span R s", "state_after": "case refine_3\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\nm\u2081 m\u2082 : M\nhm\u2081 : \u2045y, m\u2081\u2046 \u2208 Submodule.span R s\nhm\u2082 : \u2045y, m\u2082\u2046 \u2208 Submodule.span R s\n\u22a2 \u2045y, m\u2081\u2046 + \u2045y, m\u2082\u2046 \u2208 Submodule.span R s"}, {"tactic": "exact Submodule.add_mem _ hm\u2081 hm\u2082", "annotated_tactic": ["exact <a>Submodule.add_mem</a> _ hm\u2081 hm\u2082", [{"full_name": "Submodule.add_mem", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [239, 19], "def_end_pos": [239, 26]}]], "state_before": "case refine_3\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\nm\u2081 m\u2082 : M\nhm\u2081 : \u2045y, m\u2081\u2046 \u2208 Submodule.span R s\nhm\u2082 : \u2045y, m\u2082\u2046 \u2208 Submodule.span R s\n\u22a2 \u2045y, m\u2081\u2046 + \u2045y, m\u2082\u2046 \u2208 Submodule.span R s", "state_after": "no goals"}, {"tactic": "intro t m'' hm''", "annotated_tactic": ["intro t m'' hm''", []], "state_before": "case refine_4\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\n\u22a2 \u2200 (a : R) (x : M), (fun m' => \u2045y, m'\u2046 \u2208 Submodule.span R s) x \u2192 (fun m' => \u2045y, m'\u2046 \u2208 Submodule.span R s) (a \u2022 x)", "state_after": "case refine_4\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\nt : R\nm'' : M\nhm'' : \u2045y, m''\u2046 \u2208 Submodule.span R s\n\u22a2 \u2045y, t \u2022 m''\u2046 \u2208 Submodule.span R s"}, {"tactic": "rw [lie_smul]", "annotated_tactic": ["rw [<a>lie_smul</a>]", [{"full_name": "lie_smul", "def_path": "Mathlib/Algebra/Lie/Basic.lean", "def_pos": [128, 9], "def_end_pos": [128, 17]}]], "state_before": "case refine_4\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\nt : R\nm'' : M\nhm'' : \u2045y, m''\u2046 \u2208 Submodule.span R s\n\u22a2 \u2045y, t \u2022 m''\u2046 \u2208 Submodule.span R s", "state_after": "case refine_4\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\nt : R\nm'' : M\nhm'' : \u2045y, m''\u2046 \u2208 Submodule.span R s\n\u22a2 t \u2022 \u2045y, m''\u2046 \u2208 Submodule.span R s"}, {"tactic": "exact Submodule.smul_mem _ t hm''", "annotated_tactic": ["exact <a>Submodule.smul_mem</a> _ t hm''", [{"full_name": "Submodule.smul_mem", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [243, 9], "def_end_pos": [243, 17]}]], "state_before": "case refine_4\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\ns : Set M := {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m}\ny : L\nm' : M\nhm' : m' \u2208 Submodule.span R s\nt : R\nm'' : M\nhm'' : \u2045y, m''\u2046 \u2208 Submodule.span R s\n\u22a2 t \u2022 \u2045y, m''\u2046 \u2208 Submodule.span R s", "state_after": "no goals"}, {"tactic": "rw [lieIdeal_oper_eq_span]", "annotated_tactic": ["rw [<a>lieIdeal_oper_eq_span</a>]", [{"full_name": "LieSubmodule.lieIdeal_oper_eq_span", "def_path": "Mathlib/Algebra/Lie/IdealOperations.lean", "def_pos": [59, 9], "def_end_pos": [59, 30]}]], "state_before": "case a\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\n\u22a2 Submodule.span R {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} \u2264 \u2191\u2045I, N\u2046", "state_after": "case a\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\n\u22a2 Submodule.span R {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} \u2264 \u2191(lieSpan R L {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m})"}, {"tactic": "apply submodule_span_le_lieSpan", "annotated_tactic": ["apply <a>submodule_span_le_lieSpan</a>", [{"full_name": "LieSubmodule.submodule_span_le_lieSpan", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [683, 9], "def_end_pos": [683, 34]}]], "state_before": "case a\nR : Type u\nL : Type v\nM : Type w\nM\u2082 : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : LieRingModule L M\u2082\ninst\u271d : LieModule R L M\u2082\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN\u2082 : LieSubmodule R L M\u2082\n\u22a2 Submodule.span R {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} \u2264 \u2191(lieSpan R L {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m})", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Prod.lean", "full_name": "Submodule.map_inl", "start": [548, 1], "end": [551, 14], "traced_tactics": [{"tactic": "ext \u27e8x, y\u27e9", "annotated_tactic": ["ext \u27e8x, y\u27e9", []], "state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 map (inl R M M\u2082) p = prod p \u22a5", "state_after": "case h.mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx : M\ny : M\u2082\n\u22a2 (x, y) \u2208 map (inl R M M\u2082) p \u2194 (x, y) \u2208 prod p \u22a5"}, {"tactic": "simp only [and_left_comm, eq_comm, mem_map, Prod.mk.inj_iff, inl_apply, mem_bot, exists_eq_left',\n  mem_prod]", "annotated_tactic": ["simp only [<a>and_left_comm</a>, <a>eq_comm</a>, <a>mem_map</a>, <a>Prod.mk.inj_iff</a>, <a>inl_apply</a>, <a>mem_bot</a>, <a>exists_eq_left'</a>,\n    <a>mem_prod</a>]", [{"full_name": "and_left_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [180, 9], "def_end_pos": [180, 22]}, {"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}, {"full_name": "Submodule.mem_map", "def_path": "Mathlib/LinearAlgebra/Basic.lean", "def_pos": [598, 9], "def_end_pos": [598, 16]}, {"full_name": "Prod.mk.inj_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 19]}, {"full_name": "LinearMap.inl_apply", "def_path": "Mathlib/LinearAlgebra/Prod.lean", "def_pos": [182, 9], "def_end_pos": [182, 18]}, {"full_name": "Submodule.mem_bot", "def_path": "Mathlib/Algebra/Module/Submodule/Lattice.lean", "def_pos": [76, 9], "def_end_pos": [76, 16]}, {"full_name": "exists_eq_left'", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [470, 17], "def_end_pos": [470, 32]}, {"full_name": "Submodule.mem_prod", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [760, 9], "def_end_pos": [760, 17]}]], "state_before": "case h.mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\np : Submodule R M\nq : Submodule R M\u2082\nx : M\ny : M\u2082\n\u22a2 (x, y) \u2208 map (inl R M M\u2082) p \u2194 (x, y) \u2208 prod p \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Convex/Basic.lean", "full_name": "convex_Ioi", "start": [325, 1], "end": [326, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/Units.lean", "full_name": "Int.eq_of_mul_eq_one", "start": [75, 1], "end": [77, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/BigOperators/Basic.lean", "full_name": "List.prod_eq_zero_iff", "start": [392, 1], "end": [396, 56], "traced_tactics": [{"tactic": "induction' L with a L ihL", "annotated_tactic": ["induction' L with a L ihL", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : MonoidWithZero M\u2080\ninst\u271d\u00b9 : Nontrivial M\u2080\ninst\u271d : NoZeroDivisors M\u2080\nL : List M\u2080\n\u22a2 prod L = 0 \u2194 0 \u2208 L", "state_after": "case nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : MonoidWithZero M\u2080\ninst\u271d\u00b9 : Nontrivial M\u2080\ninst\u271d : NoZeroDivisors M\u2080\n\u22a2 prod [] = 0 \u2194 0 \u2208 []\n\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : MonoidWithZero M\u2080\ninst\u271d\u00b9 : Nontrivial M\u2080\ninst\u271d : NoZeroDivisors M\u2080\na : M\u2080\nL : List M\u2080\nihL : prod L = 0 \u2194 0 \u2208 L\n\u22a2 prod (a :: L) = 0 \u2194 0 \u2208 a :: L"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : MonoidWithZero M\u2080\ninst\u271d\u00b9 : Nontrivial M\u2080\ninst\u271d : NoZeroDivisors M\u2080\n\u22a2 prod [] = 0 \u2194 0 \u2208 []", "state_after": "no goals"}, {"tactic": "rw [prod_cons, mul_eq_zero, ihL, mem_cons, eq_comm]", "annotated_tactic": ["rw [<a>prod_cons</a>, <a>mul_eq_zero</a>, ihL, <a>mem_cons</a>, <a>eq_comm</a>]", [{"full_name": "List.prod_cons", "def_path": "Mathlib/Data/List/BigOperators/Basic.lean", "def_pos": [41, 9], "def_end_pos": [41, 18]}, {"full_name": "mul_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [240, 9], "def_end_pos": [240, 20]}, {"full_name": "List.mem_cons", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [62, 17], "def_end_pos": [62, 25]}, {"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "case cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : MonoidWithZero M\u2080\ninst\u271d\u00b9 : Nontrivial M\u2080\ninst\u271d : NoZeroDivisors M\u2080\na : M\u2080\nL : List M\u2080\nihL : prod L = 0 \u2194 0 \u2208 L\n\u22a2 prod (a :: L) = 0 \u2194 0 \u2208 a :: L", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Setoid/Partition.lean", "full_name": "Setoid.mkClasses_classes", "start": [189, 1], "end": [192, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "full_name": "contDiffGroupoid_prod", "start": [626, 1], "end": [640, 13], "traced_tactics": [{"tactic": "cases' he with he he_symm", "annotated_tactic": ["cases' he with he he_symm", []], "state_before": "m n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe : e \u2208 contDiffGroupoid \u22a4 I\nhe' : e' \u2208 contDiffGroupoid \u22a4 I'\n\u22a2 LocalHomeomorph.prod e e' \u2208 contDiffGroupoid \u22a4 (ModelWithCorners.prod I I')", "state_after": "case intro\nm n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 contDiffGroupoid \u22a4 I'\nhe :\n  Pregroupoid.property\n    {\n      property := fun f s =>\n        ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I),\n      comp :=\n        (_ :\n          \u2200 {f g : H \u2192 H} {u v : Set H},\n            ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192\n              ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' v \u2229 range \u2191I) \u2192\n                IsOpen u \u2192\n                  IsOpen v \u2192\n                    IsOpen (u \u2229 f \u207b\u00b9' v) \u2192\n                      (fun f s =>\n                          ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I))\n                            (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I))\n                        (g \u2218 f) (u \u2229 f \u207b\u00b9' v)),\n      id_mem :=\n        (_ : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 id \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' univ \u2229 range \u2191I)),\n      locality :=\n        (_ :\n          \u2200 {f : H \u2192 H} {u : Set H},\n            IsOpen u \u2192\n              (\u2200 (x : H),\n                  x \u2208 u \u2192\n                    \u2203 v,\n                      IsOpen v \u2227\n                        x \u2208 v \u2227\n                          (fun f s =>\n                              ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I))\n                                (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I))\n                            f (u \u2229 v)) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)),\n      congr :=\n        (_ :\n          \u2200 {f g : H \u2192 H} {u : Set H},\n            IsOpen u \u2192\n              (\u2200 (x : H), x \u2208 u \u2192 g x = f x) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192\n                  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)) }\n    (\u2191e) e.source\nhe_symm :\n  Pregroupoid.property\n    {\n      property := fun f s =>\n        ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I),\n      comp :=\n        (_ :\n          \u2200 {f g : H \u2192 H} {u v : Set H},\n            ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192\n              ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' v \u2229 range \u2191I) \u2192\n                IsOpen u \u2192\n                  IsOpen v \u2192\n                    IsOpen (u \u2229 f \u207b\u00b9' v) \u2192\n                      (fun f s =>\n                          ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I))\n                            (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I))\n                        (g \u2218 f) (u \u2229 f \u207b\u00b9' v)),\n      id_mem :=\n        (_ : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 id \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' univ \u2229 range \u2191I)),\n      locality :=\n        (_ :\n          \u2200 {f : H \u2192 H} {u : Set H},\n            IsOpen u \u2192\n              (\u2200 (x : H),\n                  x \u2208 u \u2192\n                    \u2203 v,\n                      IsOpen v \u2227\n                        x \u2208 v \u2227\n                          (fun f s =>\n                              ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I))\n                                (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I))\n                            f (u \u2229 v)) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)),\n      congr :=\n        (_ :\n          \u2200 {f g : H \u2192 H} {u : Set H},\n            IsOpen u \u2192\n              (\u2200 (x : H), x \u2208 u \u2192 g x = f x) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192\n                  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)) }\n    (\u2191(LocalHomeomorph.symm e)) e.target\n\u22a2 LocalHomeomorph.prod e e' \u2208 contDiffGroupoid \u22a4 (ModelWithCorners.prod I I')"}, {"tactic": "cases' he' with he' he'_symm", "annotated_tactic": ["cases' he' with he' he'_symm", []], "state_before": "case intro\nm n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe' : e' \u2208 contDiffGroupoid \u22a4 I'\nhe :\n  Pregroupoid.property\n    {\n      property := fun f s =>\n        ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I),\n      comp :=\n        (_ :\n          \u2200 {f g : H \u2192 H} {u v : Set H},\n            ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192\n              ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' v \u2229 range \u2191I) \u2192\n                IsOpen u \u2192\n                  IsOpen v \u2192\n                    IsOpen (u \u2229 f \u207b\u00b9' v) \u2192\n                      (fun f s =>\n                          ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I))\n                            (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I))\n                        (g \u2218 f) (u \u2229 f \u207b\u00b9' v)),\n      id_mem :=\n        (_ : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 id \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' univ \u2229 range \u2191I)),\n      locality :=\n        (_ :\n          \u2200 {f : H \u2192 H} {u : Set H},\n            IsOpen u \u2192\n              (\u2200 (x : H),\n                  x \u2208 u \u2192\n                    \u2203 v,\n                      IsOpen v \u2227\n                        x \u2208 v \u2227\n                          (fun f s =>\n                              ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I))\n                                (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I))\n                            f (u \u2229 v)) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)),\n      congr :=\n        (_ :\n          \u2200 {f g : H \u2192 H} {u : Set H},\n            IsOpen u \u2192\n              (\u2200 (x : H), x \u2208 u \u2192 g x = f x) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192\n                  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)) }\n    (\u2191e) e.source\nhe_symm :\n  Pregroupoid.property\n    {\n      property := fun f s =>\n        ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I),\n      comp :=\n        (_ :\n          \u2200 {f g : H \u2192 H} {u v : Set H},\n            ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192\n              ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' v \u2229 range \u2191I) \u2192\n                IsOpen u \u2192\n                  IsOpen v \u2192\n                    IsOpen (u \u2229 f \u207b\u00b9' v) \u2192\n                      (fun f s =>\n                          ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I))\n                            (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I))\n                        (g \u2218 f) (u \u2229 f \u207b\u00b9' v)),\n      id_mem :=\n        (_ : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 id \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' univ \u2229 range \u2191I)),\n      locality :=\n        (_ :\n          \u2200 {f : H \u2192 H} {u : Set H},\n            IsOpen u \u2192\n              (\u2200 (x : H),\n                  x \u2208 u \u2192\n                    \u2203 v,\n                      IsOpen v \u2227\n                        x \u2208 v \u2227\n                          (fun f s =>\n                              ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I))\n                                (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I))\n                            f (u \u2229 v)) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)),\n      congr :=\n        (_ :\n          \u2200 {f g : H \u2192 H} {u : Set H},\n            IsOpen u \u2192\n              (\u2200 (x : H), x \u2208 u \u2192 g x = f x) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192\n                  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)) }\n    (\u2191(LocalHomeomorph.symm e)) e.target\n\u22a2 LocalHomeomorph.prod e e' \u2208 contDiffGroupoid \u22a4 (ModelWithCorners.prod I I')", "state_after": "case intro.intro\nm n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe :\n  Pregroupoid.property\n    {\n      property := fun f s =>\n        ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I),\n      comp :=\n        (_ :\n          \u2200 {f g : H \u2192 H} {u v : Set H},\n            ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192\n              ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' v \u2229 range \u2191I) \u2192\n                IsOpen u \u2192\n                  IsOpen v \u2192\n                    IsOpen (u \u2229 f \u207b\u00b9' v) \u2192\n                      (fun f s =>\n                          ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I))\n                            (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I))\n                        (g \u2218 f) (u \u2229 f \u207b\u00b9' v)),\n      id_mem :=\n        (_ : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 id \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' univ \u2229 range \u2191I)),\n      locality :=\n        (_ :\n          \u2200 {f : H \u2192 H} {u : Set H},\n            IsOpen u \u2192\n              (\u2200 (x : H),\n                  x \u2208 u \u2192\n                    \u2203 v,\n                      IsOpen v \u2227\n                        x \u2208 v \u2227\n                          (fun f s =>\n                              ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I))\n                                (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I))\n                            f (u \u2229 v)) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)),\n      congr :=\n        (_ :\n          \u2200 {f g : H \u2192 H} {u : Set H},\n            IsOpen u \u2192\n              (\u2200 (x : H), x \u2208 u \u2192 g x = f x) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192\n                  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)) }\n    (\u2191e) e.source\nhe_symm :\n  Pregroupoid.property\n    {\n      property := fun f s =>\n        ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I),\n      comp :=\n        (_ :\n          \u2200 {f g : H \u2192 H} {u v : Set H},\n            ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192\n              ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' v \u2229 range \u2191I) \u2192\n                IsOpen u \u2192\n                  IsOpen v \u2192\n                    IsOpen (u \u2229 f \u207b\u00b9' v) \u2192\n                      (fun f s =>\n                          ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I))\n                            (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I))\n                        (g \u2218 f) (u \u2229 f \u207b\u00b9' v)),\n      id_mem :=\n        (_ : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 id \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' univ \u2229 range \u2191I)),\n      locality :=\n        (_ :\n          \u2200 {f : H \u2192 H} {u : Set H},\n            IsOpen u \u2192\n              (\u2200 (x : H),\n                  x \u2208 u \u2192\n                    \u2203 v,\n                      IsOpen v \u2227\n                        x \u2208 v \u2227\n                          (fun f s =>\n                              ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I))\n                                (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I))\n                            f (u \u2229 v)) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)),\n      congr :=\n        (_ :\n          \u2200 {f g : H \u2192 H} {u : Set H},\n            IsOpen u \u2192\n              (\u2200 (x : H), x \u2208 u \u2192 g x = f x) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192\n                  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)) }\n    (\u2191(LocalHomeomorph.symm e)) e.target\nhe' :\n  Pregroupoid.property\n    {\n      property := fun f s =>\n        ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' s \u2229 range \u2191I'),\n      comp :=\n        (_ :\n          \u2200 {f g : H' \u2192 H'} {u v : Set H'},\n            ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I') \u2192\n              ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 g \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' v \u2229 range \u2191I') \u2192\n                IsOpen u \u2192\n                  IsOpen v \u2192\n                    IsOpen (u \u2229 f \u207b\u00b9' v) \u2192\n                      (fun f s =>\n                          ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I'))\n                            (\u2191(ModelWithCorners.symm I') \u207b\u00b9' s \u2229 range \u2191I'))\n                        (g \u2218 f) (u \u2229 f \u207b\u00b9' v)),\n      id_mem :=\n        (_ :\n          ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 id \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' univ \u2229 range \u2191I')),\n      locality :=\n        (_ :\n          \u2200 {f : H' \u2192 H'} {u : Set H'},\n            IsOpen u \u2192\n              (\u2200 (x : H'),\n                  x \u2208 u \u2192\n                    \u2203 v,\n                      IsOpen v \u2227\n                        x \u2208 v \u2227\n                          (fun f s =>\n                              ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I'))\n                                (\u2191(ModelWithCorners.symm I') \u207b\u00b9' s \u2229 range \u2191I'))\n                            f (u \u2229 v)) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I')),\n      congr :=\n        (_ :\n          \u2200 {f g : H' \u2192 H'} {u : Set H'},\n            IsOpen u \u2192\n              (\u2200 (x : H'), x \u2208 u \u2192 g x = f x) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I') \u2192\n                  ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 g \u2218 \u2191(ModelWithCorners.symm I'))\n                    (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I')) }\n    (\u2191e') e'.source\nhe'_symm :\n  Pregroupoid.property\n    {\n      property := fun f s =>\n        ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' s \u2229 range \u2191I'),\n      comp :=\n        (_ :\n          \u2200 {f g : H' \u2192 H'} {u v : Set H'},\n            ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I') \u2192\n              ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 g \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' v \u2229 range \u2191I') \u2192\n                IsOpen u \u2192\n                  IsOpen v \u2192\n                    IsOpen (u \u2229 f \u207b\u00b9' v) \u2192\n                      (fun f s =>\n                          ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I'))\n                            (\u2191(ModelWithCorners.symm I') \u207b\u00b9' s \u2229 range \u2191I'))\n                        (g \u2218 f) (u \u2229 f \u207b\u00b9' v)),\n      id_mem :=\n        (_ :\n          ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 id \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' univ \u2229 range \u2191I')),\n      locality :=\n        (_ :\n          \u2200 {f : H' \u2192 H'} {u : Set H'},\n            IsOpen u \u2192\n              (\u2200 (x : H'),\n                  x \u2208 u \u2192\n                    \u2203 v,\n                      IsOpen v \u2227\n                        x \u2208 v \u2227\n                          (fun f s =>\n                              ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I'))\n                                (\u2191(ModelWithCorners.symm I') \u207b\u00b9' s \u2229 range \u2191I'))\n                            f (u \u2229 v)) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I')),\n      congr :=\n        (_ :\n          \u2200 {f g : H' \u2192 H'} {u : Set H'},\n            IsOpen u \u2192\n              (\u2200 (x : H'), x \u2208 u \u2192 g x = f x) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I') \u2192\n                  ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 g \u2218 \u2191(ModelWithCorners.symm I'))\n                    (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I')) }\n    (\u2191(LocalHomeomorph.symm e')) e'.target\n\u22a2 LocalHomeomorph.prod e e' \u2208 contDiffGroupoid \u22a4 (ModelWithCorners.prod I I')"}, {"tactic": "simp only at he he_symm he' he'_symm", "annotated_tactic": ["simp only at he he_symm he' he'_symm", []], "state_before": "case intro.intro\nm n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe :\n  Pregroupoid.property\n    {\n      property := fun f s =>\n        ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I),\n      comp :=\n        (_ :\n          \u2200 {f g : H \u2192 H} {u v : Set H},\n            ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192\n              ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' v \u2229 range \u2191I) \u2192\n                IsOpen u \u2192\n                  IsOpen v \u2192\n                    IsOpen (u \u2229 f \u207b\u00b9' v) \u2192\n                      (fun f s =>\n                          ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I))\n                            (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I))\n                        (g \u2218 f) (u \u2229 f \u207b\u00b9' v)),\n      id_mem :=\n        (_ : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 id \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' univ \u2229 range \u2191I)),\n      locality :=\n        (_ :\n          \u2200 {f : H \u2192 H} {u : Set H},\n            IsOpen u \u2192\n              (\u2200 (x : H),\n                  x \u2208 u \u2192\n                    \u2203 v,\n                      IsOpen v \u2227\n                        x \u2208 v \u2227\n                          (fun f s =>\n                              ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I))\n                                (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I))\n                            f (u \u2229 v)) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)),\n      congr :=\n        (_ :\n          \u2200 {f g : H \u2192 H} {u : Set H},\n            IsOpen u \u2192\n              (\u2200 (x : H), x \u2208 u \u2192 g x = f x) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192\n                  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)) }\n    (\u2191e) e.source\nhe_symm :\n  Pregroupoid.property\n    {\n      property := fun f s =>\n        ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I),\n      comp :=\n        (_ :\n          \u2200 {f g : H \u2192 H} {u v : Set H},\n            ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192\n              ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' v \u2229 range \u2191I) \u2192\n                IsOpen u \u2192\n                  IsOpen v \u2192\n                    IsOpen (u \u2229 f \u207b\u00b9' v) \u2192\n                      (fun f s =>\n                          ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I))\n                            (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I))\n                        (g \u2218 f) (u \u2229 f \u207b\u00b9' v)),\n      id_mem :=\n        (_ : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 id \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' univ \u2229 range \u2191I)),\n      locality :=\n        (_ :\n          \u2200 {f : H \u2192 H} {u : Set H},\n            IsOpen u \u2192\n              (\u2200 (x : H),\n                  x \u2208 u \u2192\n                    \u2203 v,\n                      IsOpen v \u2227\n                        x \u2208 v \u2227\n                          (fun f s =>\n                              ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I))\n                                (\u2191(ModelWithCorners.symm I) \u207b\u00b9' s \u2229 range \u2191I))\n                            f (u \u2229 v)) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)),\n      congr :=\n        (_ :\n          \u2200 {f g : H \u2192 H} {u : Set H},\n            IsOpen u \u2192\n              (\u2200 (x : H), x \u2208 u \u2192 g x = f x) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 f \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I) \u2192\n                  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 g \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' u \u2229 range \u2191I)) }\n    (\u2191(LocalHomeomorph.symm e)) e.target\nhe' :\n  Pregroupoid.property\n    {\n      property := fun f s =>\n        ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' s \u2229 range \u2191I'),\n      comp :=\n        (_ :\n          \u2200 {f g : H' \u2192 H'} {u v : Set H'},\n            ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I') \u2192\n              ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 g \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' v \u2229 range \u2191I') \u2192\n                IsOpen u \u2192\n                  IsOpen v \u2192\n                    IsOpen (u \u2229 f \u207b\u00b9' v) \u2192\n                      (fun f s =>\n                          ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I'))\n                            (\u2191(ModelWithCorners.symm I') \u207b\u00b9' s \u2229 range \u2191I'))\n                        (g \u2218 f) (u \u2229 f \u207b\u00b9' v)),\n      id_mem :=\n        (_ :\n          ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 id \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' univ \u2229 range \u2191I')),\n      locality :=\n        (_ :\n          \u2200 {f : H' \u2192 H'} {u : Set H'},\n            IsOpen u \u2192\n              (\u2200 (x : H'),\n                  x \u2208 u \u2192\n                    \u2203 v,\n                      IsOpen v \u2227\n                        x \u2208 v \u2227\n                          (fun f s =>\n                              ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I'))\n                                (\u2191(ModelWithCorners.symm I') \u207b\u00b9' s \u2229 range \u2191I'))\n                            f (u \u2229 v)) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I')),\n      congr :=\n        (_ :\n          \u2200 {f g : H' \u2192 H'} {u : Set H'},\n            IsOpen u \u2192\n              (\u2200 (x : H'), x \u2208 u \u2192 g x = f x) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I') \u2192\n                  ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 g \u2218 \u2191(ModelWithCorners.symm I'))\n                    (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I')) }\n    (\u2191e') e'.source\nhe'_symm :\n  Pregroupoid.property\n    {\n      property := fun f s =>\n        ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' s \u2229 range \u2191I'),\n      comp :=\n        (_ :\n          \u2200 {f g : H' \u2192 H'} {u v : Set H'},\n            ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I') \u2192\n              ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 g \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' v \u2229 range \u2191I') \u2192\n                IsOpen u \u2192\n                  IsOpen v \u2192\n                    IsOpen (u \u2229 f \u207b\u00b9' v) \u2192\n                      (fun f s =>\n                          ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I'))\n                            (\u2191(ModelWithCorners.symm I') \u207b\u00b9' s \u2229 range \u2191I'))\n                        (g \u2218 f) (u \u2229 f \u207b\u00b9' v)),\n      id_mem :=\n        (_ :\n          ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 id \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' univ \u2229 range \u2191I')),\n      locality :=\n        (_ :\n          \u2200 {f : H' \u2192 H'} {u : Set H'},\n            IsOpen u \u2192\n              (\u2200 (x : H'),\n                  x \u2208 u \u2192\n                    \u2203 v,\n                      IsOpen v \u2227\n                        x \u2208 v \u2227\n                          (fun f s =>\n                              ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I'))\n                                (\u2191(ModelWithCorners.symm I') \u207b\u00b9' s \u2229 range \u2191I'))\n                            f (u \u2229 v)) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I')),\n      congr :=\n        (_ :\n          \u2200 {f g : H' \u2192 H'} {u : Set H'},\n            IsOpen u \u2192\n              (\u2200 (x : H'), x \u2208 u \u2192 g x = f x) \u2192\n                ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 f \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I') \u2192\n                  ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 g \u2218 \u2191(ModelWithCorners.symm I'))\n                    (\u2191(ModelWithCorners.symm I') \u207b\u00b9' u \u2229 range \u2191I')) }\n    (\u2191(LocalHomeomorph.symm e')) e'.target\n\u22a2 LocalHomeomorph.prod e e' \u2208 contDiffGroupoid \u22a4 (ModelWithCorners.prod I I')", "state_after": "case intro.intro\nm n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I)\nhe_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n    (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I)\nhe' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\nhe'_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I'))\n    (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I')\n\u22a2 LocalHomeomorph.prod e e' \u2208 contDiffGroupoid \u22a4 (ModelWithCorners.prod I I')"}, {"tactic": "constructor <;> simp only [LocalEquiv.prod_source, LocalHomeomorph.prod_toLocalEquiv]", "annotated_tactic": ["constructor <;> simp only [<a>LocalEquiv.prod_source</a>, <a>LocalHomeomorph.prod_toLocalEquiv</a>]", [{"full_name": "LocalEquiv.prod_source", "def_path": "Mathlib/Logic/Equiv/LocalEquiv.lean", "def_pos": [937, 9], "def_end_pos": [937, 20]}, {"full_name": "LocalHomeomorph.prod_toLocalEquiv", "def_path": "Mathlib/Topology/LocalHomeomorph.lean", "def_pos": [1016, 31], "def_end_pos": [1016, 43]}]], "state_before": "case intro.intro\nm n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I)\nhe_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n    (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I)\nhe' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\nhe'_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I'))\n    (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I')\n\u22a2 LocalHomeomorph.prod e e' \u2208 contDiffGroupoid \u22a4 (ModelWithCorners.prod I I')", "state_after": "case intro.intro.left\nm n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I)\nhe_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n    (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I)\nhe' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\nhe'_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I'))\n    (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I')\n\u22a2 ContDiffOn \ud835\udd5c \u22a4\n    (\u2191(ModelWithCorners.prod I I') \u2218\n      \u2191(LocalHomeomorph.prod e e') \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')))\n    (\u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')) \u207b\u00b9' e.source \u00d7\u02e2 e'.source \u2229\n      range \u2191(ModelWithCorners.prod I I'))\n\ncase intro.intro.right\nm n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I)\nhe_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n    (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I)\nhe' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\nhe'_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I'))\n    (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I')\n\u22a2 ContDiffOn \ud835\udd5c \u22a4\n    (\u2191(ModelWithCorners.prod I I') \u2218\n      \u2191(LocalHomeomorph.symm (LocalHomeomorph.prod e e')) \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')))\n    (\u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')) \u207b\u00b9' (LocalEquiv.prod e.toLocalEquiv e'.toLocalEquiv).target \u2229\n      range \u2191(ModelWithCorners.prod I I'))"}, {"tactic": "have h3 := ContDiffOn.prod_map he he'", "annotated_tactic": ["have h3 := <a>ContDiffOn.prod_map</a> he he'", [{"full_name": "ContDiffOn.prod_map", "def_path": "Mathlib/Analysis/Calculus/ContDiff.lean", "def_pos": [1638, 9], "def_end_pos": [1638, 28]}]], "state_before": "case intro.intro.left\nm n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I)\nhe_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n    (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I)\nhe' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\nhe'_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I'))\n    (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I')\n\u22a2 ContDiffOn \ud835\udd5c \u22a4\n    (\u2191(ModelWithCorners.prod I I') \u2218\n      \u2191(LocalHomeomorph.prod e e') \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')))\n    (\u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')) \u207b\u00b9' e.source \u00d7\u02e2 e'.source \u2229\n      range \u2191(ModelWithCorners.prod I I'))", "state_after": "case intro.intro.left\nm n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I)\nhe_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n    (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I)\nhe' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\nhe'_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I'))\n    (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I')\nh3 :\n  ContDiffOn \ud835\udd5c \u22a4 (Prod.map (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')))\n    ((\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I) \u00d7\u02e2 (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I'))\n\u22a2 ContDiffOn \ud835\udd5c \u22a4\n    (\u2191(ModelWithCorners.prod I I') \u2218\n      \u2191(LocalHomeomorph.prod e e') \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')))\n    (\u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')) \u207b\u00b9' e.source \u00d7\u02e2 e'.source \u2229\n      range \u2191(ModelWithCorners.prod I I'))"}, {"tactic": "rw [\u2190 I.image_eq, \u2190 I'.image_eq, Set.prod_image_image_eq] at h3", "annotated_tactic": ["rw [\u2190 I.image_eq, \u2190 I'.image_eq, <a>Set.prod_image_image_eq</a>] at h3", [{"full_name": "Set.prod_image_image_eq", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [302, 9], "def_end_pos": [302, 28]}]], "state_before": "case intro.intro.left\nm n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I)\nhe_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n    (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I)\nhe' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\nhe'_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I'))\n    (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I')\nh3 :\n  ContDiffOn \ud835\udd5c \u22a4 (Prod.map (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')))\n    ((\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I) \u00d7\u02e2 (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I'))\n\u22a2 ContDiffOn \ud835\udd5c \u22a4\n    (\u2191(ModelWithCorners.prod I I') \u2218\n      \u2191(LocalHomeomorph.prod e e') \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')))\n    (\u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')) \u207b\u00b9' e.source \u00d7\u02e2 e'.source \u2229\n      range \u2191(ModelWithCorners.prod I I'))", "state_after": "case intro.intro.left\nm n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I)\nhe_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n    (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I)\nhe' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\nhe'_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I'))\n    (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I')\nh3 :\n  ContDiffOn \ud835\udd5c \u22a4 (Prod.map (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')))\n    ((fun p => (\u2191I p.1, \u2191I' p.2)) '' e.source \u00d7\u02e2 e'.source)\n\u22a2 ContDiffOn \ud835\udd5c \u22a4\n    (\u2191(ModelWithCorners.prod I I') \u2218\n      \u2191(LocalHomeomorph.prod e e') \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')))\n    (\u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')) \u207b\u00b9' e.source \u00d7\u02e2 e'.source \u2229\n      range \u2191(ModelWithCorners.prod I I'))"}, {"tactic": "rw [\u2190 (I.prod I').image_eq]", "annotated_tactic": ["rw [\u2190 (I.prod I').<a>image_eq</a>]", [{"full_name": "ModelWithCorners.image_eq", "def_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "def_pos": [293, 19], "def_end_pos": [293, 27]}]], "state_before": "case intro.intro.left\nm n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I)\nhe_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n    (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I)\nhe' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\nhe'_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I'))\n    (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I')\nh3 :\n  ContDiffOn \ud835\udd5c \u22a4 (Prod.map (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')))\n    ((fun p => (\u2191I p.1, \u2191I' p.2)) '' e.source \u00d7\u02e2 e'.source)\n\u22a2 ContDiffOn \ud835\udd5c \u22a4\n    (\u2191(ModelWithCorners.prod I I') \u2218\n      \u2191(LocalHomeomorph.prod e e') \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')))\n    (\u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')) \u207b\u00b9' e.source \u00d7\u02e2 e'.source \u2229\n      range \u2191(ModelWithCorners.prod I I'))", "state_after": "case intro.intro.left\nm n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I)\nhe_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n    (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I)\nhe' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\nhe'_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I'))\n    (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I')\nh3 :\n  ContDiffOn \ud835\udd5c \u22a4 (Prod.map (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')))\n    ((fun p => (\u2191I p.1, \u2191I' p.2)) '' e.source \u00d7\u02e2 e'.source)\n\u22a2 ContDiffOn \ud835\udd5c \u22a4\n    (\u2191(ModelWithCorners.prod I I') \u2218\n      \u2191(LocalHomeomorph.prod e e') \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')))\n    (\u2191(ModelWithCorners.prod I I') '' e.source \u00d7\u02e2 e'.source)"}, {"tactic": "exact h3", "annotated_tactic": ["exact h3", []], "state_before": "case intro.intro.left\nm n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I)\nhe_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n    (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I)\nhe' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\nhe'_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I'))\n    (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I')\nh3 :\n  ContDiffOn \ud835\udd5c \u22a4 (Prod.map (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')))\n    ((fun p => (\u2191I p.1, \u2191I' p.2)) '' e.source \u00d7\u02e2 e'.source)\n\u22a2 ContDiffOn \ud835\udd5c \u22a4\n    (\u2191(ModelWithCorners.prod I I') \u2218\n      \u2191(LocalHomeomorph.prod e e') \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')))\n    (\u2191(ModelWithCorners.prod I I') '' e.source \u00d7\u02e2 e'.source)", "state_after": "no goals"}, {"tactic": "have h3 := ContDiffOn.prod_map he_symm he'_symm", "annotated_tactic": ["have h3 := <a>ContDiffOn.prod_map</a> he_symm he'_symm", [{"full_name": "ContDiffOn.prod_map", "def_path": "Mathlib/Analysis/Calculus/ContDiff.lean", "def_pos": [1638, 9], "def_end_pos": [1638, 28]}]], "state_before": "case intro.intro.right\nm n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I)\nhe_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n    (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I)\nhe' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\nhe'_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I'))\n    (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I')\n\u22a2 ContDiffOn \ud835\udd5c \u22a4\n    (\u2191(ModelWithCorners.prod I I') \u2218\n      \u2191(LocalHomeomorph.symm (LocalHomeomorph.prod e e')) \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')))\n    (\u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')) \u207b\u00b9' (LocalEquiv.prod e.toLocalEquiv e'.toLocalEquiv).target \u2229\n      range \u2191(ModelWithCorners.prod I I'))", "state_after": "case intro.intro.right\nm n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I)\nhe_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n    (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I)\nhe' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\nhe'_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I'))\n    (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I')\nh3 :\n  ContDiffOn \ud835\udd5c \u22a4\n    (Prod.map (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n      (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I')))\n    ((\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I) \u00d7\u02e2 (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I'))\n\u22a2 ContDiffOn \ud835\udd5c \u22a4\n    (\u2191(ModelWithCorners.prod I I') \u2218\n      \u2191(LocalHomeomorph.symm (LocalHomeomorph.prod e e')) \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')))\n    (\u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')) \u207b\u00b9' (LocalEquiv.prod e.toLocalEquiv e'.toLocalEquiv).target \u2229\n      range \u2191(ModelWithCorners.prod I I'))"}, {"tactic": "rw [\u2190 I.image_eq, \u2190 I'.image_eq, Set.prod_image_image_eq] at h3", "annotated_tactic": ["rw [\u2190 I.image_eq, \u2190 I'.image_eq, <a>Set.prod_image_image_eq</a>] at h3", [{"full_name": "Set.prod_image_image_eq", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [302, 9], "def_end_pos": [302, 28]}]], "state_before": "case intro.intro.right\nm n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I)\nhe_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n    (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I)\nhe' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\nhe'_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I'))\n    (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I')\nh3 :\n  ContDiffOn \ud835\udd5c \u22a4\n    (Prod.map (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n      (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I')))\n    ((\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I) \u00d7\u02e2 (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I'))\n\u22a2 ContDiffOn \ud835\udd5c \u22a4\n    (\u2191(ModelWithCorners.prod I I') \u2218\n      \u2191(LocalHomeomorph.symm (LocalHomeomorph.prod e e')) \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')))\n    (\u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')) \u207b\u00b9' (LocalEquiv.prod e.toLocalEquiv e'.toLocalEquiv).target \u2229\n      range \u2191(ModelWithCorners.prod I I'))", "state_after": "case intro.intro.right\nm n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I)\nhe_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n    (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I)\nhe' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\nhe'_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I'))\n    (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I')\nh3 :\n  ContDiffOn \ud835\udd5c \u22a4\n    (Prod.map (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n      (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I')))\n    ((fun p => (\u2191I p.1, \u2191I' p.2)) '' e.target \u00d7\u02e2 e'.target)\n\u22a2 ContDiffOn \ud835\udd5c \u22a4\n    (\u2191(ModelWithCorners.prod I I') \u2218\n      \u2191(LocalHomeomorph.symm (LocalHomeomorph.prod e e')) \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')))\n    (\u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')) \u207b\u00b9' (LocalEquiv.prod e.toLocalEquiv e'.toLocalEquiv).target \u2229\n      range \u2191(ModelWithCorners.prod I I'))"}, {"tactic": "rw [\u2190 (I.prod I').image_eq]", "annotated_tactic": ["rw [\u2190 (I.prod I').<a>image_eq</a>]", [{"full_name": "ModelWithCorners.image_eq", "def_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "def_pos": [293, 19], "def_end_pos": [293, 27]}]], "state_before": "case intro.intro.right\nm n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I)\nhe_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n    (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I)\nhe' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\nhe'_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I'))\n    (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I')\nh3 :\n  ContDiffOn \ud835\udd5c \u22a4\n    (Prod.map (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n      (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I')))\n    ((fun p => (\u2191I p.1, \u2191I' p.2)) '' e.target \u00d7\u02e2 e'.target)\n\u22a2 ContDiffOn \ud835\udd5c \u22a4\n    (\u2191(ModelWithCorners.prod I I') \u2218\n      \u2191(LocalHomeomorph.symm (LocalHomeomorph.prod e e')) \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')))\n    (\u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')) \u207b\u00b9' (LocalEquiv.prod e.toLocalEquiv e'.toLocalEquiv).target \u2229\n      range \u2191(ModelWithCorners.prod I I'))", "state_after": "case intro.intro.right\nm n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I)\nhe_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n    (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I)\nhe' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\nhe'_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I'))\n    (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I')\nh3 :\n  ContDiffOn \ud835\udd5c \u22a4\n    (Prod.map (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n      (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I')))\n    ((fun p => (\u2191I p.1, \u2191I' p.2)) '' e.target \u00d7\u02e2 e'.target)\n\u22a2 ContDiffOn \ud835\udd5c \u22a4\n    (\u2191(ModelWithCorners.prod I I') \u2218\n      \u2191(LocalHomeomorph.symm (LocalHomeomorph.prod e e')) \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')))\n    (\u2191(ModelWithCorners.prod I I') '' (LocalEquiv.prod e.toLocalEquiv e'.toLocalEquiv).target)"}, {"tactic": "exact h3", "annotated_tactic": ["exact h3", []], "state_before": "case intro.intro.right\nm n : \u2115\u221e\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u2074 : TopologicalSpace H\nI\u271d : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\nE' : Type u_5\nH' : Type u_6\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d : TopologicalSpace H'\nI : ModelWithCorners \ud835\udd5c E H\nI' : ModelWithCorners \ud835\udd5c E' H'\ne : LocalHomeomorph H H\ne' : LocalHomeomorph H' H'\nhe : ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191e \u2218 \u2191(ModelWithCorners.symm I)) (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.source \u2229 range \u2191I)\nhe_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n    (\u2191(ModelWithCorners.symm I) \u207b\u00b9' e.target \u2229 range \u2191I)\nhe' : ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191e' \u2218 \u2191(ModelWithCorners.symm I')) (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.source \u2229 range \u2191I')\nhe'_symm :\n  ContDiffOn \ud835\udd5c \u22a4 (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I'))\n    (\u2191(ModelWithCorners.symm I') \u207b\u00b9' e'.target \u2229 range \u2191I')\nh3 :\n  ContDiffOn \ud835\udd5c \u22a4\n    (Prod.map (\u2191I \u2218 \u2191(LocalHomeomorph.symm e) \u2218 \u2191(ModelWithCorners.symm I))\n      (\u2191I' \u2218 \u2191(LocalHomeomorph.symm e') \u2218 \u2191(ModelWithCorners.symm I')))\n    ((fun p => (\u2191I p.1, \u2191I' p.2)) '' e.target \u00d7\u02e2 e'.target)\n\u22a2 ContDiffOn \ud835\udd5c \u22a4\n    (\u2191(ModelWithCorners.prod I I') \u2218\n      \u2191(LocalHomeomorph.symm (LocalHomeomorph.prod e e')) \u2218 \u2191(ModelWithCorners.symm (ModelWithCorners.prod I I')))\n    (\u2191(ModelWithCorners.prod I I') '' (LocalEquiv.prod e.toLocalEquiv e'.toLocalEquiv).target)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/WittVector/Truncated.lean", "full_name": "TruncatedWittVector.ext_iff", "start": [90, 1], "end": [91, 30], "traced_tactics": [{"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\nx y : TruncatedWittVector p n R\nh : x = y\ni : Fin n\n\u22a2 coeff i x = coeff i y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "full_name": "Nat.recDiagAux_succ_succ", "start": [58, 1], "end": [62, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/InnerProductSpace/TwoDim.lean", "full_name": "Orientation.rightAngleRotationAux\u2081_rightAngleRotationAux\u2081", "start": [239, 1], "end": [246, 21], "traced_tactics": [{"tactic": "apply ext_inner_left \u211d", "annotated_tactic": ["apply <a>ext_inner_left</a> \u211d", [{"full_name": "ext_inner_left", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [687, 9], "def_end_pos": [687, 23]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\n\u22a2 \u2191(rightAngleRotationAux\u2081 o) (\u2191(rightAngleRotationAux\u2081 o) x) = -x", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\n\u22a2 \u2200 (v : (fun x => E) (\u2191(rightAngleRotationAux\u2081 o) x)),\n    inner v (\u2191(rightAngleRotationAux\u2081 o) (\u2191(rightAngleRotationAux\u2081 o) x)) = inner v (-x)"}, {"tactic": "intro y", "annotated_tactic": ["intro y", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\n\u22a2 \u2200 (v : (fun x => E) (\u2191(rightAngleRotationAux\u2081 o) x)),\n    inner v (\u2191(rightAngleRotationAux\u2081 o) (\u2191(rightAngleRotationAux\u2081 o) x)) = inner v (-x)", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\ny : (fun x => E) (\u2191(rightAngleRotationAux\u2081 o) x)\n\u22a2 inner y (\u2191(rightAngleRotationAux\u2081 o) (\u2191(rightAngleRotationAux\u2081 o) x)) = inner y (-x)"}, {"tactic": "have : \u27eao.rightAngleRotationAux\u2081 y, o.rightAngleRotationAux\u2081 x\u27eb = \u27eay, x\u27eb :=\n  LinearIsometry.inner_map_map o.rightAngleRotationAux\u2082 y x", "annotated_tactic": ["have : \u27eao.rightAngleRotationAux\u2081 y, o.rightAngleRotationAux\u2081 x\u27eb = \u27eay, x\u27eb :=\n    <a>LinearIsometry.inner_map_map</a> o.rightAngleRotationAux\u2082 y x", [{"full_name": "LinearIsometry.inner_map_map", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [1272, 9], "def_end_pos": [1272, 37]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\ny : (fun x => E) (\u2191(rightAngleRotationAux\u2081 o) x)\n\u22a2 inner y (\u2191(rightAngleRotationAux\u2081 o) (\u2191(rightAngleRotationAux\u2081 o) x)) = inner y (-x)", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\ny : (fun x => E) (\u2191(rightAngleRotationAux\u2081 o) x)\nthis : inner (\u2191(rightAngleRotationAux\u2081 o) y) (\u2191(rightAngleRotationAux\u2081 o) x) = inner y x\n\u22a2 inner y (\u2191(rightAngleRotationAux\u2081 o) (\u2191(rightAngleRotationAux\u2081 o) x)) = inner y (-x)"}, {"tactic": "rw [o.inner_rightAngleRotationAux\u2081_right, \u2190 o.inner_rightAngleRotationAux\u2081_left, this,\n  inner_neg_right]", "annotated_tactic": ["rw [o.inner_rightAngleRotationAux\u2081_right, \u2190 o.inner_rightAngleRotationAux\u2081_left, this,\n    <a>inner_neg_right</a>]", [{"full_name": "inner_neg_right", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [635, 9], "def_end_pos": [635, 24]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx : E\ny : (fun x => E) (\u2191(rightAngleRotationAux\u2081 o) x)\nthis : inner (\u2191(rightAngleRotationAux\u2081 o) y) (\u2191(rightAngleRotationAux\u2081 o) x) = inner y x\n\u22a2 inner y (\u2191(rightAngleRotationAux\u2081 o) (\u2191(rightAngleRotationAux\u2081 o) x)) = inner y (-x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/WithBot.lean", "full_name": "WithTop.coe_mono", "start": [1147, 1], "end": [1148, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Basis.lean", "full_name": "Basis.equiv_trans", "start": [693, 1], "end": [695, 26], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nR : Type u_3\nR\u2082 : Type u_4\nK : Type u_5\nM : Type u_6\nM' : Type u_7\nM'' : Type u_8\nV : Type u\nV' : Type u_9\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid M'\ninst\u271d\u00b2 : Module R M'\nb b\u2081 : Basis \u03b9 R M\ni\u271d : \u03b9\nc : R\nx : M\nb' : Basis \u03b9' R M'\ne\u271d : \u03b9 \u2243 \u03b9'\ninst\u271d\u00b9 : AddCommMonoid M''\ninst\u271d : Module R M''\n\u03b9'' : Type u_10\nb'' : Basis \u03b9'' R M''\ne : \u03b9 \u2243 \u03b9'\ne' : \u03b9' \u2243 \u03b9''\ni : \u03b9\n\u22a2 \u2191(LinearEquiv.trans (Basis.equiv b b' e) (Basis.equiv b' b'' e')) (\u2191b i) = \u2191(Basis.equiv b b'' (e.trans e')) (\u2191b i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Group/Pi.lean", "full_name": "Function.update_inv", "start": [638, 1], "end": [640, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Basic.lean", "full_name": "Set.Icc_union_Ico_eq_Ico", "start": [1662, 1], "end": [1665, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/LocallyConvex/Basic.lean", "full_name": "balanced_iInter", "start": [195, 1], "end": [196, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/ModelTheory/Semantics.lean", "full_name": "FirstOrder.Language.BoundedFormula.realize_rel", "start": [301, 1], "end": [303, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Halting.lean", "full_name": "Nat.Partrec'.tail", "start": [319, 1], "end": [321, 56], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "n : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : Partrec' f\nv : Vector \u2115 (succ n)\n\u22a2 (mOfFn fun i => (\u2191fun v => Vector.get v (Fin.succ i)) v) >>= f = f (Vector.tail v)", "state_after": "n : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : Partrec' f\nv : Vector \u2115 (succ n)\n\u22a2 f (ofFn fun i => Vector.get v (Fin.succ i)) = f (Vector.tail v)"}, {"tactic": "rw [\u2190 ofFn_get v.tail]", "annotated_tactic": ["rw [\u2190 <a>ofFn_get</a> v.tail]", [{"full_name": "Vector.ofFn_get", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [155, 9], "def_end_pos": [155, 17]}]], "state_before": "n : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : Partrec' f\nv : Vector \u2115 (succ n)\n\u22a2 f (ofFn fun i => Vector.get v (Fin.succ i)) = f (Vector.tail v)", "state_after": "n : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : Partrec' f\nv : Vector \u2115 (succ n)\n\u22a2 f (ofFn fun i => Vector.get v (Fin.succ i)) = f (ofFn (Vector.get (Vector.tail v)))"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "n : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : Partrec' f\nv : Vector \u2115 (succ n)\n\u22a2 f (ofFn fun i => Vector.get v (Fin.succ i)) = f (ofFn (Vector.get (Vector.tail v)))", "state_after": "case e_a.e_a\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : Partrec' f\nv : Vector \u2115 (succ n)\n\u22a2 (fun i => Vector.get v (Fin.succ i)) = Vector.get (Vector.tail v)"}, {"tactic": "funext i", "annotated_tactic": ["funext i", []], "state_before": "case e_a.e_a\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : Partrec' f\nv : Vector \u2115 (succ n)\n\u22a2 (fun i => Vector.get v (Fin.succ i)) = Vector.get (Vector.tail v)", "state_after": "case e_a.e_a.h\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : Partrec' f\nv : Vector \u2115 (succ n)\ni : Fin n\n\u22a2 Vector.get v (Fin.succ i) = Vector.get (Vector.tail v) i"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case e_a.e_a.h\nn : \u2115\nf : Vector \u2115 n \u2192. \u2115\nhf : Partrec' f\nv : Vector \u2115 (succ n)\ni : Fin n\n\u22a2 Vector.get v (Fin.succ i) = Vector.get (Vector.tail v) i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.biInter_eq_iInter", "start": [1015, 1], "end": [1017, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Instances.lean", "full_name": "Set.Icc.coe_pow", "start": [129, 1], "end": [130, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Int/SuccPred.lean", "full_name": "Int.covby_iff_succ_eq", "start": [76, 11], "end": [77, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Lie/Free.lean", "full_name": "FreeLieAlgebra.liftAux_map_mul", "start": [197, 1], "end": [199, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Congruence.lean", "full_name": "Con.comap_rel", "start": [673, 1], "end": [675, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Lie/Semisimple.lean", "full_name": "LieAlgebra.center_eq_bot_of_semisimple", "start": [79, 1], "end": [80, 72], "traced_tactics": [{"tactic": "rw [isSemisimple_iff_no_abelian_ideals] at h", "annotated_tactic": ["rw [<a>isSemisimple_iff_no_abelian_ideals</a>] at h", [{"full_name": "LieAlgebra.isSemisimple_iff_no_abelian_ideals", "def_path": "Mathlib/Algebra/Lie/Semisimple.lean", "def_pos": [69, 9], "def_end_pos": [69, 43]}]], "state_before": "R : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nh : IsSemisimple R L\n\u22a2 center R L = \u22a5", "state_after": "R : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nh : \u2200 (I : LieIdeal R L), IsLieAbelian { x // x \u2208 \u2191I } \u2192 I = \u22a5\n\u22a2 center R L = \u22a5"}, {"tactic": "apply h", "annotated_tactic": ["apply h", []], "state_before": "R : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nh : \u2200 (I : LieIdeal R L), IsLieAbelian { x // x \u2208 \u2191I } \u2192 I = \u22a5\n\u22a2 center R L = \u22a5", "state_after": "case a\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nh : \u2200 (I : LieIdeal R L), IsLieAbelian { x // x \u2208 \u2191I } \u2192 I = \u22a5\n\u22a2 IsLieAbelian { x // x \u2208 \u2191(center R L) }"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "case a\nR : Type u\nL : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nh : \u2200 (I : LieIdeal R L), IsLieAbelian { x // x \u2208 \u2191I } \u2192 I = \u22a5\n\u22a2 IsLieAbelian { x // x \u2208 \u2191(center R L) }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/MetricSpace/Infsep.lean", "full_name": "Set.relatively_discrete_of_finite", "start": [302, 1], "end": [305, 30], "traced_tactics": [{"tactic": "rw [\u2190 einfsep_pos]", "annotated_tactic": ["rw [\u2190 <a>einfsep_pos</a>]", [{"full_name": "Set.einfsep_pos", "def_path": "Mathlib/Topology/MetricSpace/Infsep.lean", "def_pos": [60, 9], "def_end_pos": [60, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : EMetricSpace \u03b1\nx y z : \u03b1\ns t : Set \u03b1\nC : \u211d\u22650\u221e\nsC : Set \u211d\u22650\u221e\ninst\u271d : Finite \u2191s\n\u22a2 \u2203 C _hC, \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 x \u2260 y \u2192 C \u2264 edist x y", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : EMetricSpace \u03b1\nx y z : \u03b1\ns t : Set \u03b1\nC : \u211d\u22650\u221e\nsC : Set \u211d\u22650\u221e\ninst\u271d : Finite \u2191s\n\u22a2 0 < einfsep s"}, {"tactic": "exact einfsep_pos_of_finite", "annotated_tactic": ["exact <a>einfsep_pos_of_finite</a>", [{"full_name": "Set.einfsep_pos_of_finite", "def_path": "Mathlib/Topology/MetricSpace/Infsep.lean", "def_pos": [293, 9], "def_end_pos": [293, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : EMetricSpace \u03b1\nx y z : \u03b1\ns t : Set \u03b1\nC : \u211d\u22650\u221e\nsC : Set \u211d\u22650\u221e\ninst\u271d : Finite \u2191s\n\u22a2 0 < einfsep s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Multiset/Fold.lean", "full_name": "Multiset.fold_bind", "start": [83, 1], "end": [88, 85], "traced_tactics": [{"tactic": "induction' s using Multiset.induction_on with a ha ih", "annotated_tactic": ["induction' s using <a>Multiset.induction_on</a> with a ha ih", [{"full_name": "Multiset.induction_on", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [160, 19], "def_end_pos": [160, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nhc : IsCommutative \u03b1 op\nha : IsAssociative \u03b1 op\n\u03b9 : Type u_3\ns : Multiset \u03b9\nt : \u03b9 \u2192 Multiset \u03b1\nb : \u03b9 \u2192 \u03b1\nb\u2080 : \u03b1\n\u22a2 fold op (fold op b\u2080 (map b s)) (bind s t) = fold op b\u2080 (map (fun i => fold op (b i) (t i)) s)", "state_after": "case empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nhc : IsCommutative \u03b1 op\nha : IsAssociative \u03b1 op\n\u03b9 : Type u_3\nt : \u03b9 \u2192 Multiset \u03b1\nb : \u03b9 \u2192 \u03b1\nb\u2080 : \u03b1\n\u22a2 fold op (fold op b\u2080 (map b 0)) (bind 0 t) = fold op b\u2080 (map (fun i => fold op (b i) (t i)) 0)\n\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nhc : IsCommutative \u03b1 op\nha\u271d : IsAssociative \u03b1 op\n\u03b9 : Type u_3\nt : \u03b9 \u2192 Multiset \u03b1\nb : \u03b9 \u2192 \u03b1\nb\u2080 : \u03b1\na : \u03b9\nha : Multiset \u03b9\nih : fold op (fold op b\u2080 (map b ha)) (bind ha t) = fold op b\u2080 (map (fun i => fold op (b i) (t i)) ha)\n\u22a2 fold op (fold op b\u2080 (map b (a ::\u2098 ha))) (bind (a ::\u2098 ha) t) =\n    fold op b\u2080 (map (fun i => fold op (b i) (t i)) (a ::\u2098 ha))"}, {"tactic": "rw [zero_bind, map_zero, map_zero, fold_zero]", "annotated_tactic": ["rw [<a>zero_bind</a>, <a>map_zero</a>, <a>map_zero</a>, <a>fold_zero</a>]", [{"full_name": "Multiset.zero_bind", "def_path": "Mathlib/Data/Multiset/Bind.lean", "def_pos": [104, 9], "def_end_pos": [104, 18]}, {"full_name": "Multiset.map_zero", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1171, 9], "def_end_pos": [1171, 17]}, {"full_name": "Multiset.map_zero", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1171, 9], "def_end_pos": [1171, 17]}, {"full_name": "Multiset.fold_zero", "def_path": "Mathlib/Data/Multiset/Fold.lean", "def_pos": [55, 9], "def_end_pos": [55, 18]}]], "state_before": "case empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nhc : IsCommutative \u03b1 op\nha : IsAssociative \u03b1 op\n\u03b9 : Type u_3\nt : \u03b9 \u2192 Multiset \u03b1\nb : \u03b9 \u2192 \u03b1\nb\u2080 : \u03b1\n\u22a2 fold op (fold op b\u2080 (map b 0)) (bind 0 t) = fold op b\u2080 (map (fun i => fold op (b i) (t i)) 0)", "state_after": "no goals"}, {"tactic": "rw [cons_bind, map_cons, map_cons, fold_cons_left, fold_cons_left, fold_add, ih]", "annotated_tactic": ["rw [<a>cons_bind</a>, <a>map_cons</a>, <a>map_cons</a>, <a>fold_cons_left</a>, <a>fold_cons_left</a>, <a>fold_add</a>, ih]", [{"full_name": "Multiset.cons_bind", "def_path": "Mathlib/Data/Multiset/Bind.lean", "def_pos": [109, 9], "def_end_pos": [109, 18]}, {"full_name": "Multiset.map_cons", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1176, 9], "def_end_pos": [1176, 17]}, {"full_name": "Multiset.map_cons", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1176, 9], "def_end_pos": [1176, 17]}, {"full_name": "Multiset.fold_cons_left", "def_path": "Mathlib/Data/Multiset/Fold.lean", "def_pos": [60, 9], "def_end_pos": [60, 23]}, {"full_name": "Multiset.fold_cons_left", "def_path": "Mathlib/Data/Multiset/Fold.lean", "def_pos": [60, 9], "def_end_pos": [60, 23]}, {"full_name": "Multiset.fold_add", "def_path": "Mathlib/Data/Multiset/Fold.lean", "def_pos": [76, 9], "def_end_pos": [76, 17]}]], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nop : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nhc : IsCommutative \u03b1 op\nha\u271d : IsAssociative \u03b1 op\n\u03b9 : Type u_3\nt : \u03b9 \u2192 Multiset \u03b1\nb : \u03b9 \u2192 \u03b1\nb\u2080 : \u03b1\na : \u03b9\nha : Multiset \u03b9\nih : fold op (fold op b\u2080 (map b ha)) (bind ha t) = fold op b\u2080 (map (fun i => fold op (b i) (t i)) ha)\n\u22a2 fold op (fold op b\u2080 (map b (a ::\u2098 ha))) (bind (a ::\u2098 ha) t) =\n    fold op b\u2080 (map (fun i => fold op (b i) (t i)) (a ::\u2098 ha))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Sites/SheafOfTypes.lean", "full_name": "CategoryTheory.Presieve.extend_agrees", "start": [199, 1], "end": [206, 38], "traced_tactics": [{"tactic": "have h := (le_generate R Y hf).choose_spec", "annotated_tactic": ["have h := (<a>le_generate</a> R Y hf).<a>choose_spec</a>", [{"full_name": "CategoryTheory.Sieve.le_generate", "def_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "def_pos": [426, 9], "def_end_pos": [426, 20]}, {"full_name": "Exists.choose_spec", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [445, 9], "def_end_pos": [445, 27]}]], "state_before": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y \u27f6 X\nhf : R f\n\u22a2 FamilyOfElements.sieveExtend x f (_ : f \u2208 (generate R).arrows) = x f hf", "state_after": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y \u27f6 X\nhf : R f\nh : \u2203 h g, R g \u2227 h \u226b g = f\n\u22a2 FamilyOfElements.sieveExtend x f (_ : f \u2208 (generate R).arrows) = x f hf"}, {"tactic": "unfold FamilyOfElements.sieveExtend", "annotated_tactic": ["unfold <a>FamilyOfElements.sieveExtend</a>", [{"full_name": "CategoryTheory.Presieve.FamilyOfElements.sieveExtend", "def_path": "Mathlib/CategoryTheory/Sites/SheafOfTypes.lean", "def_pos": [184, 19], "def_end_pos": [184, 47]}]], "state_before": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y \u27f6 X\nhf : R f\nh : \u2203 h g, R g \u2227 h \u226b g = f\n\u22a2 FamilyOfElements.sieveExtend x f (_ : f \u2208 (generate R).arrows) = x f hf", "state_after": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y \u27f6 X\nhf : R f\nh : \u2203 h g, R g \u2227 h \u226b g = f\n\u22a2 P.map (Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f)).op\n      (x (Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f) \u226b g = f))\n        (_ : R (Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f) \u226b g = f)))) =\n    x f hf"}, {"tactic": "rw [t h.choose (\ud835\udfd9 _) _ hf _]", "annotated_tactic": ["rw [t h.choose (\ud835\udfd9 _) _ hf _]", []], "state_before": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y \u27f6 X\nhf : R f\nh : \u2203 h g, R g \u2227 h \u226b g = f\n\u22a2 P.map (Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f)).op\n      (x (Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f) \u226b g = f))\n        (_ : R (Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f) \u226b g = f)))) =\n    x f hf", "state_after": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y \u27f6 X\nhf : R f\nh : \u2203 h g, R g \u2227 h \u226b g = f\n\u22a2 P.map (\ud835\udfd9 Y).op (x f hf) = x f hf\n\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y \u27f6 X\nhf : R f\nh : \u2203 h g, R g \u2227 h \u226b g = f\n\u22a2 Exists.choose h \u226b Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f) \u226b g = f) = \ud835\udfd9 Y \u226b f"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y \u27f6 X\nhf : R f\nh : \u2203 h g, R g \u2227 h \u226b g = f\n\u22a2 P.map (\ud835\udfd9 Y).op (x f hf) = x f hf", "state_after": "no goals"}, {"tactic": "rw [id_comp]", "annotated_tactic": ["rw [<a>id_comp</a>]", [{"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [155, 3], "def_end_pos": [155, 10]}]], "state_before": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y \u27f6 X\nhf : R f\nh : \u2203 h g, R g \u2227 h \u226b g = f\n\u22a2 Exists.choose h \u226b Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f) \u226b g = f) = \ud835\udfd9 Y \u226b f", "state_after": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y \u27f6 X\nhf : R f\nh : \u2203 h g, R g \u2227 h \u226b g = f\n\u22a2 Exists.choose h \u226b Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f) \u226b g = f) = f"}, {"tactic": "exact h.choose_spec.choose_spec.2", "annotated_tactic": ["exact h.choose_spec.choose_spec.2", []], "state_before": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nP Q U : C\u1d52\u1d56 \u2964 Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J\u2082 : GrothendieckTopology C\nx : FamilyOfElements P R\nt : FamilyOfElements.Compatible x\nf : Y \u27f6 X\nhf : R f\nh : \u2203 h g, R g \u2227 h \u226b g = f\n\u22a2 Exists.choose h \u226b Exists.choose (_ : \u2203 g, R g \u2227 Exists.choose (_ : \u2203 h g, R g \u2227 h \u226b g = f) \u226b g = f) = f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Logic/Equiv/Basic.lean", "full_name": "Equiv.emptySum_apply_inr", "start": [467, 1], "end": [468, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Order/NhdsSet.lean", "full_name": "hasBasis_nhdsSet_Ici_Ioi", "start": [189, 1], "end": [191, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Pi.lean", "full_name": "Set.image_mulSingle_Ioc_left", "start": [237, 1], "end": [239, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Logic/Equiv/LocalEquiv.lean", "full_name": "LocalEquiv.prod_source", "start": [937, 1], "end": [939, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Connected/TotallyDisconnected.lean", "full_name": "isTotallySeparated_singleton", "start": [185, 1], "end": [186, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/LocallyConvex/Bounded.lean", "full_name": "NormedSpace.isVonNBounded_iff", "start": [284, 1], "end": [294, 68], "traced_tactics": [{"tactic": "rw [Metric.isBounded_iff_subset_closedBall (0 : E)]", "annotated_tactic": ["rw [<a>Metric.isBounded_iff_subset_closedBall</a> (0 : E)]", [{"full_name": "Metric.isBounded_iff_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [2381, 9], "def_end_pos": [2381, 40]}]], "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 Bornology.IsVonNBounded \ud835\udd5c s \u2194 Bornology.IsBounded s", "state_after": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 Bornology.IsVonNBounded \ud835\udd5c s \u2194 \u2203 r, s \u2286 Metric.closedBall 0 r"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 Bornology.IsVonNBounded \ud835\udd5c s \u2194 \u2203 r, s \u2286 Metric.closedBall 0 r", "state_after": "case mp\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 Bornology.IsVonNBounded \ud835\udd5c s \u2192 \u2203 r, s \u2286 Metric.closedBall 0 r\n\ncase mpr\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 (\u2203 r, s \u2286 Metric.closedBall 0 r) \u2192 Bornology.IsVonNBounded \ud835\udd5c s"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mp\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 Bornology.IsVonNBounded \ud835\udd5c s \u2192 \u2203 r, s \u2286 Metric.closedBall 0 r", "state_after": "case mp\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nh : Bornology.IsVonNBounded \ud835\udd5c s\n\u22a2 \u2203 r, s \u2286 Metric.closedBall 0 r"}, {"tactic": "rcases h (Metric.ball_mem_nhds 0 zero_lt_one) with \u27e8\u03c1, h\u03c1, h\u03c1ball\u27e9", "annotated_tactic": ["rcases h (<a>Metric.ball_mem_nhds</a> 0 <a>zero_lt_one</a>) with \u27e8\u03c1, h\u03c1, h\u03c1ball\u27e9", [{"full_name": "Metric.ball_mem_nhds", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [1021, 9], "def_end_pos": [1021, 22]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "case mp\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nh : Bornology.IsVonNBounded \ud835\udd5c s\n\u22a2 \u2203 r, s \u2286 Metric.closedBall 0 r", "state_after": "case mp.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nh : Bornology.IsVonNBounded \ud835\udd5c s\n\u03c1 : \u211d\nh\u03c1 : 0 < \u03c1\nh\u03c1ball : \u2200 (a : \ud835\udd5c), \u03c1 \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 Metric.ball 0 1\n\u22a2 \u2203 r, s \u2286 Metric.closedBall 0 r"}, {"tactic": "rcases NormedField.exists_lt_norm \ud835\udd5c \u03c1 with \u27e8a, ha\u27e9", "annotated_tactic": ["rcases <a>NormedField.exists_lt_norm</a> \ud835\udd5c \u03c1 with \u27e8a, ha\u27e9", [{"full_name": "NormedField.exists_lt_norm", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [726, 9], "def_end_pos": [726, 23]}]], "state_before": "case mp.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nh : Bornology.IsVonNBounded \ud835\udd5c s\n\u03c1 : \u211d\nh\u03c1 : 0 < \u03c1\nh\u03c1ball : \u2200 (a : \ud835\udd5c), \u03c1 \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 Metric.ball 0 1\n\u22a2 \u2203 r, s \u2286 Metric.closedBall 0 r", "state_after": "case mp.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nh : Bornology.IsVonNBounded \ud835\udd5c s\n\u03c1 : \u211d\nh\u03c1 : 0 < \u03c1\nh\u03c1ball : \u2200 (a : \ud835\udd5c), \u03c1 \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 Metric.ball 0 1\na : \ud835\udd5c\nha : \u03c1 < \u2016a\u2016\n\u22a2 \u2203 r, s \u2286 Metric.closedBall 0 r"}, {"tactic": "specialize h\u03c1ball a ha.le", "annotated_tactic": ["specialize h\u03c1ball a ha.le", []], "state_before": "case mp.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nh : Bornology.IsVonNBounded \ud835\udd5c s\n\u03c1 : \u211d\nh\u03c1 : 0 < \u03c1\nh\u03c1ball : \u2200 (a : \ud835\udd5c), \u03c1 \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 Metric.ball 0 1\na : \ud835\udd5c\nha : \u03c1 < \u2016a\u2016\n\u22a2 \u2203 r, s \u2286 Metric.closedBall 0 r", "state_after": "case mp.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nh : Bornology.IsVonNBounded \ud835\udd5c s\n\u03c1 : \u211d\nh\u03c1 : 0 < \u03c1\na : \ud835\udd5c\nha : \u03c1 < \u2016a\u2016\nh\u03c1ball : s \u2286 a \u2022 Metric.ball 0 1\n\u22a2 \u2203 r, s \u2286 Metric.closedBall 0 r"}, {"tactic": "rw [\u2190 ball_normSeminorm \ud835\udd5c E, Seminorm.smul_ball_zero (norm_pos_iff.1 <| h\u03c1.trans ha),\n  ball_normSeminorm, mul_one] at h\u03c1ball", "annotated_tactic": ["rw [\u2190 <a>ball_normSeminorm</a> \ud835\udd5c E, <a>Seminorm.smul_ball_zero</a> (<a>norm_pos_iff</a>.1 <| h\u03c1.trans ha),\n      <a>ball_normSeminorm</a>, <a>mul_one</a>] at h\u03c1ball", [{"full_name": "ball_normSeminorm", "def_path": "Mathlib/Analysis/Seminorm.lean", "def_pos": [1405, 9], "def_end_pos": [1405, 26]}, {"full_name": "Seminorm.smul_ball_zero", "def_path": "Mathlib/Analysis/Seminorm.lean", "def_pos": [1007, 9], "def_end_pos": [1007, 23]}, {"full_name": "norm_pos_iff", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [2030, 30], "def_end_pos": [2030, 42]}, {"full_name": "ball_normSeminorm", "def_path": "Mathlib/Analysis/Seminorm.lean", "def_pos": [1405, 9], "def_end_pos": [1405, 26]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "case mp.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nh : Bornology.IsVonNBounded \ud835\udd5c s\n\u03c1 : \u211d\nh\u03c1 : 0 < \u03c1\na : \ud835\udd5c\nha : \u03c1 < \u2016a\u2016\nh\u03c1ball : s \u2286 a \u2022 Metric.ball 0 1\n\u22a2 \u2203 r, s \u2286 Metric.closedBall 0 r", "state_after": "case mp.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nh : Bornology.IsVonNBounded \ud835\udd5c s\n\u03c1 : \u211d\nh\u03c1 : 0 < \u03c1\na : \ud835\udd5c\nha : \u03c1 < \u2016a\u2016\nh\u03c1ball : s \u2286 Metric.ball 0 \u2016a\u2016\n\u22a2 \u2203 r, s \u2286 Metric.closedBall 0 r"}, {"tactic": "exact \u27e8\u2016a\u2016, h\u03c1ball.trans Metric.ball_subset_closedBall\u27e9", "annotated_tactic": ["exact \u27e8\u2016a\u2016, h\u03c1ball.trans <a>Metric.ball_subset_closedBall</a>\u27e9", [{"full_name": "Metric.ball_subset_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [534, 9], "def_end_pos": [534, 31]}]], "state_before": "case mp.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\nh : Bornology.IsVonNBounded \ud835\udd5c s\n\u03c1 : \u211d\nh\u03c1 : 0 < \u03c1\na : \ud835\udd5c\nha : \u03c1 < \u2016a\u2016\nh\u03c1ball : s \u2286 Metric.ball 0 \u2016a\u2016\n\u22a2 \u2203 r, s \u2286 Metric.closedBall 0 r", "state_after": "no goals"}, {"tactic": "exact fun \u27e8C, hC\u27e9 => (isVonNBounded_closedBall \ud835\udd5c E C).subset hC", "annotated_tactic": ["exact fun \u27e8C, hC\u27e9 => (<a>isVonNBounded_closedBall</a> \ud835\udd5c E C).<a>subset</a> hC", [{"full_name": "NormedSpace.isVonNBounded_closedBall", "def_path": "Mathlib/Analysis/LocallyConvex/Bounded.lean", "def_pos": [279, 9], "def_end_pos": [279, 33]}, {"full_name": "Bornology.IsVonNBounded.subset", "def_path": "Mathlib/Analysis/LocallyConvex/Bounded.lean", "def_pos": [89, 9], "def_end_pos": [89, 29]}]], "state_before": "case mpr\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\ns : Set E\n\u22a2 (\u2203 r, s \u2286 Metric.closedBall 0 r) \u2192 Bornology.IsVonNBounded \ud835\udd5c s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "full_name": "List.get?_inj", "start": [638, 1], "end": [653, 39], "traced_tactics": [{"tactic": "induction xs generalizing i j with\n| nil => cases h\u2080\n| cons x xs ih =>\n  match i, j with\n  | 0, 0 => rfl\n  | i+1, j+1 => simp; cases h\u2081 with\n    | cons ha h\u2081 => exact ih (Nat.lt_of_succ_lt_succ h\u2080) h\u2081 h\u2082\n  | i+1, 0 => ?_ | 0, j+1 => ?_\n  all_goals\n    simp at h\u2082\n    cases h\u2081; rename_i h' h\n    have := h x ?_ rfl; cases this\n    rw [mem_iff_get?]\n  exact \u27e8_, h\u2082\u27e9; exact \u27e8_ , h\u2082.symm\u27e9", "annotated_tactic": ["induction xs generalizing i j with\n  | <a>nil</a> => cases h\u2080\n  | <a>cons</a> x xs ih =>\n    match i, j with\n    | 0, 0 => rfl\n    | i+1, j+1 => simp; cases h\u2081 with\n      | <a>cons</a> ha h\u2081 => exact ih (<a>Nat.lt_of_succ_lt_succ</a> h\u2080) h\u2081 h\u2082\n    | i+1, 0 => ?_ | 0, j+1 => ?_\n    all_goals\n      simp at h\u2082\n      cases h\u2081; rename_i h' h\n      have := h x ?_ <a>rfl</a>; cases this\n      rw [<a>mem_iff_get?</a>]\n    exact \u27e8_, h\u2082\u27e9; exact \u27e8_ , h\u2082.symm\u27e9", [{"full_name": "List.nil", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2184, 5], "def_end_pos": [2184, 8]}, {"full_name": "List.cons", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2187, 5], "def_end_pos": [2187, 9]}, {"full_name": "List.Pairwise.cons", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1123, 5], "def_end_pos": [1123, 9]}, {"full_name": "Nat.lt_of_succ_lt_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [307, 9], "def_end_pos": [307, 27]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "List.mem_iff_get?", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [626, 9], "def_end_pos": [626, 21]}]], "state_before": "i : Nat\n\u03b1\u271d : Type u_1\nxs : List \u03b1\u271d\nj : Nat\nh\u2080 : i < length xs\nh\u2081 : Nodup xs\nh\u2082 : get? xs i = get? xs j\n\u22a2 i = j", "state_after": "no goals"}, {"tactic": "cases h\u2080", "annotated_tactic": ["cases h\u2080", []], "state_before": "case nil\n\u03b1\u271d : Type u_1\ni j : Nat\nh\u2080 : i < length []\nh\u2081 : Nodup []\nh\u2082 : get? [] i = get? [] j\n\u22a2 i = j", "state_after": "no goals"}, {"tactic": "match i, j with\n| 0, 0 => rfl\n| i+1, j+1 => simp; cases h\u2081 with\n  | cons ha h\u2081 => exact ih (Nat.lt_of_succ_lt_succ h\u2080) h\u2081 h\u2082\n| i+1, 0 => ?_ | 0, j+1 => ?_", "annotated_tactic": ["match i, j with\n    | 0, 0 => rfl\n    | i+1, j+1 => simp; cases h\u2081 with\n      | <a>cons</a> ha h\u2081 => exact ih (<a>Nat.lt_of_succ_lt_succ</a> h\u2080) h\u2081 h\u2082\n    | i+1, 0 => ?_ | 0, j+1 => ?_", [{"full_name": "List.Pairwise.cons", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1123, 5], "def_end_pos": [1123, 9]}, {"full_name": "Nat.lt_of_succ_lt_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [307, 9], "def_end_pos": [307, 27]}]], "state_before": "case cons\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni j : Nat\nh\u2080 : i < length (x :: xs)\nh\u2081 : Nodup (x :: xs)\nh\u2082 : get? (x :: xs) i = get? (x :: xs) j\n\u22a2 i = j", "state_after": "case cons.refine_1\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni\u271d j : Nat\nh\u2081 : Nodup (x :: xs)\ni : Nat\nh\u2080 : i + 1 < length (x :: xs)\nh\u2082 : get? (x :: xs) (i + 1) = get? (x :: xs) 0\n\u22a2 i + 1 = 0\n\ncase cons.refine_2\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni j\u271d : Nat\nh\u2081 : Nodup (x :: xs)\nj : Nat\nh\u2080 : 0 < length (x :: xs)\nh\u2082 : get? (x :: xs) 0 = get? (x :: xs) (j + 1)\n\u22a2 0 = j + 1"}, {"tactic": "all_goals\n  simp at h\u2082\n  cases h\u2081; rename_i h' h\n  have := h x ?_ rfl; cases this\n  rw [mem_iff_get?]", "annotated_tactic": ["all_goals\n      simp at h\u2082\n      cases h\u2081; rename_i h' h\n      have := h x ?_ <a>rfl</a>; cases this\n      rw [<a>mem_iff_get?</a>]", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "List.mem_iff_get?", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [626, 9], "def_end_pos": [626, 21]}]], "state_before": "case cons.refine_1\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni\u271d j : Nat\nh\u2081 : Nodup (x :: xs)\ni : Nat\nh\u2080 : i + 1 < length (x :: xs)\nh\u2082 : get? (x :: xs) (i + 1) = get? (x :: xs) 0\n\u22a2 i + 1 = 0\n\ncase cons.refine_2\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni j\u271d : Nat\nh\u2081 : Nodup (x :: xs)\nj : Nat\nh\u2080 : 0 < length (x :: xs)\nh\u2082 : get? (x :: xs) 0 = get? (x :: xs) (j + 1)\n\u22a2 0 = j + 1", "state_after": "case cons.refine_1.cons.refine_1\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni\u271d j i : Nat\nh\u2080 : i + 1 < length (x :: xs)\nh\u2082 : get? xs i = some x\nh' : Pairwise (fun x x_1 => x \u2260 x_1) xs\nh : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a'\n\u22a2 \u2203 n, get? xs n = some x\n\ncase cons.refine_2.cons.refine_1\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni j\u271d j : Nat\nh\u2080 : 0 < length (x :: xs)\nh\u2082 : some x = get? xs j\nh' : Pairwise (fun x x_1 => x \u2260 x_1) xs\nh : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a'\n\u22a2 \u2203 n, get? xs n = some x"}, {"tactic": "exact \u27e8_, h\u2082\u27e9", "annotated_tactic": ["exact \u27e8_, h\u2082\u27e9", []], "state_before": "case cons.refine_1.cons.refine_1\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni\u271d j i : Nat\nh\u2080 : i + 1 < length (x :: xs)\nh\u2082 : get? xs i = some x\nh' : Pairwise (fun x x_1 => x \u2260 x_1) xs\nh : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a'\n\u22a2 \u2203 n, get? xs n = some x\n\ncase cons.refine_2.cons.refine_1\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni j\u271d j : Nat\nh\u2080 : 0 < length (x :: xs)\nh\u2082 : some x = get? xs j\nh' : Pairwise (fun x x_1 => x \u2260 x_1) xs\nh : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a'\n\u22a2 \u2203 n, get? xs n = some x", "state_after": "case cons.refine_2.cons.refine_1\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni j\u271d j : Nat\nh\u2080 : 0 < length (x :: xs)\nh\u2082 : some x = get? xs j\nh' : Pairwise (fun x x_1 => x \u2260 x_1) xs\nh : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a'\n\u22a2 \u2203 n, get? xs n = some x"}, {"tactic": "exact \u27e8_ , h\u2082.symm\u27e9", "annotated_tactic": ["exact \u27e8_ , h\u2082.symm\u27e9", []], "state_before": "case cons.refine_2.cons.refine_1\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni j\u271d j : Nat\nh\u2080 : 0 < length (x :: xs)\nh\u2082 : some x = get? xs j\nh' : Pairwise (fun x x_1 => x \u2260 x_1) xs\nh : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a'\n\u22a2 \u2203 n, get? xs n = some x", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni j : Nat\nh\u2081 : Nodup (x :: xs)\nh\u2080 : 0 < length (x :: xs)\nh\u2082 : get? (x :: xs) 0 = get? (x :: xs) 0\n\u22a2 0 = 0", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni\u271d j\u271d : Nat\nh\u2081 : Nodup (x :: xs)\ni j : Nat\nh\u2080 : i + 1 < length (x :: xs)\nh\u2082 : get? (x :: xs) (i + 1) = get? (x :: xs) (j + 1)\n\u22a2 i + 1 = j + 1", "state_after": "\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni\u271d j\u271d : Nat\nh\u2081 : Nodup (x :: xs)\ni j : Nat\nh\u2080 : i + 1 < length (x :: xs)\nh\u2082 : get? (x :: xs) (i + 1) = get? (x :: xs) (j + 1)\n\u22a2 i = j"}, {"tactic": "cases h\u2081 with\n| cons ha h\u2081 => exact ih (Nat.lt_of_succ_lt_succ h\u2080) h\u2081 h\u2082", "annotated_tactic": ["cases h\u2081 with\n      | <a>cons</a> ha h\u2081 => exact ih (<a>Nat.lt_of_succ_lt_succ</a> h\u2080) h\u2081 h\u2082", [{"full_name": "List.Pairwise.cons", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [1123, 5], "def_end_pos": [1123, 9]}, {"full_name": "Nat.lt_of_succ_lt_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [307, 9], "def_end_pos": [307, 27]}]], "state_before": "\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni\u271d j\u271d : Nat\nh\u2081 : Nodup (x :: xs)\ni j : Nat\nh\u2080 : i + 1 < length (x :: xs)\nh\u2082 : get? (x :: xs) (i + 1) = get? (x :: xs) (j + 1)\n\u22a2 i = j", "state_after": "no goals"}, {"tactic": "exact ih (Nat.lt_of_succ_lt_succ h\u2080) h\u2081 h\u2082", "annotated_tactic": ["exact ih (<a>Nat.lt_of_succ_lt_succ</a> h\u2080) h\u2081 h\u2082", [{"full_name": "Nat.lt_of_succ_lt_succ", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [307, 9], "def_end_pos": [307, 27]}]], "state_before": "case cons\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni\u271d j\u271d i j : Nat\nh\u2080 : i + 1 < length (x :: xs)\nh\u2082 : get? (x :: xs) (i + 1) = get? (x :: xs) (j + 1)\nh\u2081 : Pairwise (fun x x_1 => x \u2260 x_1) xs\nha : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a'\n\u22a2 i = j", "state_after": "no goals"}, {"tactic": "simp at h\u2082", "annotated_tactic": ["simp at h\u2082", []], "state_before": "case cons.refine_2\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni j\u271d : Nat\nh\u2081 : Nodup (x :: xs)\nj : Nat\nh\u2080 : 0 < length (x :: xs)\nh\u2082 : get? (x :: xs) 0 = get? (x :: xs) (j + 1)\n\u22a2 0 = j + 1", "state_after": "case cons.refine_2\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni j\u271d : Nat\nh\u2081 : Nodup (x :: xs)\nj : Nat\nh\u2080 : 0 < length (x :: xs)\nh\u2082 : some x = get? xs j\n\u22a2 0 = j + 1"}, {"tactic": "cases h\u2081", "annotated_tactic": ["cases h\u2081", []], "state_before": "case cons.refine_2\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni j\u271d : Nat\nh\u2081 : Nodup (x :: xs)\nj : Nat\nh\u2080 : 0 < length (x :: xs)\nh\u2082 : some x = get? xs j\n\u22a2 0 = j + 1", "state_after": "case cons.refine_2.cons\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni j\u271d j : Nat\nh\u2080 : 0 < length (x :: xs)\nh\u2082 : some x = get? xs j\na\u271d\u00b9 : Pairwise (fun x x_1 => x \u2260 x_1) xs\na\u271d : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a'\n\u22a2 0 = j + 1"}, {"tactic": "rename_i h' h", "annotated_tactic": ["rename_i h' h", []], "state_before": "case cons.refine_2.cons\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni j\u271d j : Nat\nh\u2080 : 0 < length (x :: xs)\nh\u2082 : some x = get? xs j\na\u271d\u00b9 : Pairwise (fun x x_1 => x \u2260 x_1) xs\na\u271d : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a'\n\u22a2 0 = j + 1", "state_after": "case cons.refine_2.cons\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni j\u271d j : Nat\nh\u2080 : 0 < length (x :: xs)\nh\u2082 : some x = get? xs j\nh' : Pairwise (fun x x_1 => x \u2260 x_1) xs\nh : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a'\n\u22a2 0 = j + 1"}, {"tactic": "have := h x ?_ rfl", "annotated_tactic": ["have := h x ?_ <a>rfl</a>", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case cons.refine_2.cons\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni j\u271d j : Nat\nh\u2080 : 0 < length (x :: xs)\nh\u2082 : some x = get? xs j\nh' : Pairwise (fun x x_1 => x \u2260 x_1) xs\nh : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a'\n\u22a2 0 = j + 1", "state_after": "case cons.refine_2.cons.refine_2\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni j\u271d j : Nat\nh\u2080 : 0 < length (x :: xs)\nh\u2082 : some x = get? xs j\nh' : Pairwise (fun x x_1 => x \u2260 x_1) xs\nh : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a'\nthis : False\n\u22a2 0 = j + 1\n\ncase cons.refine_2.cons.refine_1\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni j\u271d j : Nat\nh\u2080 : 0 < length (x :: xs)\nh\u2082 : some x = get? xs j\nh' : Pairwise (fun x x_1 => x \u2260 x_1) xs\nh : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a'\n\u22a2 x \u2208 xs"}, {"tactic": "cases this", "annotated_tactic": ["cases this", []], "state_before": "case cons.refine_2.cons.refine_2\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni j\u271d j : Nat\nh\u2080 : 0 < length (x :: xs)\nh\u2082 : some x = get? xs j\nh' : Pairwise (fun x x_1 => x \u2260 x_1) xs\nh : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a'\nthis : False\n\u22a2 0 = j + 1\n\ncase cons.refine_2.cons.refine_1\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni j\u271d j : Nat\nh\u2080 : 0 < length (x :: xs)\nh\u2082 : some x = get? xs j\nh' : Pairwise (fun x x_1 => x \u2260 x_1) xs\nh : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a'\n\u22a2 x \u2208 xs", "state_after": "case cons.refine_2.cons.refine_1\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni j\u271d j : Nat\nh\u2080 : 0 < length (x :: xs)\nh\u2082 : some x = get? xs j\nh' : Pairwise (fun x x_1 => x \u2260 x_1) xs\nh : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a'\n\u22a2 x \u2208 xs"}, {"tactic": "rw [mem_iff_get?]", "annotated_tactic": ["rw [<a>mem_iff_get?</a>]", [{"full_name": "List.mem_iff_get?", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [626, 9], "def_end_pos": [626, 21]}]], "state_before": "case cons.refine_2.cons.refine_1\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni j\u271d j : Nat\nh\u2080 : 0 < length (x :: xs)\nh\u2082 : some x = get? xs j\nh' : Pairwise (fun x x_1 => x \u2260 x_1) xs\nh : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a'\n\u22a2 x \u2208 xs", "state_after": "case cons.refine_2.cons.refine_1\n\u03b1\u271d : Type u_1\nx : \u03b1\u271d\nxs : List \u03b1\u271d\nih : \u2200 {i j : Nat}, i < length xs \u2192 Nodup xs \u2192 get? xs i = get? xs j \u2192 i = j\ni j\u271d j : Nat\nh\u2080 : 0 < length (x :: xs)\nh\u2082 : some x = get? xs j\nh' : Pairwise (fun x x_1 => x \u2260 x_1) xs\nh : \u2200 (a' : \u03b1\u271d), a' \u2208 xs \u2192 x \u2260 a'\n\u22a2 \u2203 n, get? xs n = some x"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean", "full_name": "DedekindDomain.ProdAdicCompletions.IsFiniteAdele.zero", "start": [208, 1], "end": [218, 23], "traced_tactics": [{"tactic": "rw [IsFiniteAdele, Filter.eventually_cofinite]", "annotated_tactic": ["rw [<a>IsFiniteAdele</a>, <a>Filter.eventually_cofinite</a>]", [{"full_name": "DedekindDomain.ProdAdicCompletions.IsFiniteAdele", "def_path": "Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean", "def_pos": [183, 5], "def_end_pos": [183, 18]}, {"full_name": "Filter.eventually_cofinite", "def_path": "Mathlib/Order/Filter/Cofinite.lean", "def_pos": [45, 9], "def_end_pos": [45, 28]}]], "state_before": "R : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\n\u22a2 IsFiniteAdele 0", "state_after": "R : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\n\u22a2 Set.Finite {x | \u00acOfNat.ofNat 0 x \u2208 adicCompletionIntegers K x}"}, {"tactic": "have h_empty :\n  {v : HeightOneSpectrum R | \u00ac(0 : v.adicCompletion K) \u2208 v.adicCompletionIntegers K} = \u2205 := by\n  ext v; rw [mem_empty_iff_false, iff_false_iff]; intro hv\n  rw [mem_setOf] at hv; apply hv; rw [mem_adicCompletionIntegers]\n  have h_zero : (Valued.v (0 : v.adicCompletion K) : WithZero (Multiplicative \u2124)) = 0 :=\n    Valued.v.map_zero'\n  rw [h_zero]; exact zero_le_one' _", "annotated_tactic": ["have h_empty :\n    {v : <a>HeightOneSpectrum</a> R | \u00ac(0 : v.adicCompletion K) \u2208 v.adicCompletionIntegers K} = \u2205 := by\n    ext v; rw [<a>mem_empty_iff_false</a>, <a>iff_false_iff</a>]; intro hv\n    rw [<a>mem_setOf</a>] at hv; apply hv; rw [<a>mem_adicCompletionIntegers</a>]\n    have h_zero : (<a>Valued.v</a> (0 : v.adicCompletion K) : <a>WithZero</a> (<a>Multiplicative</a> \u2124)) = 0 :=\n      Valued.v.map_zero'\n    rw [h_zero]; exact <a>zero_le_one'</a> _", [{"full_name": "IsDedekindDomain.HeightOneSpectrum", "def_path": "Mathlib/RingTheory/DedekindDomain/Ideal.lean", "def_pos": [978, 11], "def_end_pos": [978, 28]}, {"full_name": "Set.mem_empty_iff_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [562, 9], "def_end_pos": [562, 28]}, {"full_name": "iff_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [201, 9], "def_end_pos": [201, 22]}, {"full_name": "Set.mem_setOf", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [259, 9], "def_end_pos": [259, 18]}, {"full_name": "IsDedekindDomain.HeightOneSpectrum.mem_adicCompletionIntegers", "def_path": "Mathlib/RingTheory/DedekindDomain/AdicValuation.lean", "def_pos": [378, 9], "def_end_pos": [378, 35]}, {"full_name": "Valued.v", "def_path": "Mathlib/Topology/Algebra/Valuation.lean", "def_pos": [94, 3], "def_end_pos": [94, 4]}, {"full_name": "WithZero", "def_path": "Mathlib/Algebra/Group/WithOne/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 14]}, {"full_name": "Multiplicative", "def_path": "Mathlib/Algebra/Group/TypeTags.lean", "def_pos": [47, 5], "def_end_pos": [47, 19]}, {"full_name": "zero_le_one'", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [31, 7], "def_end_pos": [31, 19]}]], "state_before": "R : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\n\u22a2 Set.Finite {x | \u00acOfNat.ofNat 0 x \u2208 adicCompletionIntegers K x}", "state_after": "R : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nh_empty : {v | \u00ac0 \u2208 adicCompletionIntegers K v} = \u2205\n\u22a2 Set.Finite {x | \u00acOfNat.ofNat 0 x \u2208 adicCompletionIntegers K x}"}, {"tactic": "convert finite_empty", "annotated_tactic": ["convert <a>finite_empty</a>", [{"full_name": "Set.finite_empty", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [837, 9], "def_end_pos": [837, 21]}]], "state_before": "R : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\nh_empty : {v | \u00ac0 \u2208 adicCompletionIntegers K v} = \u2205\n\u22a2 Set.Finite {x | \u00acOfNat.ofNat 0 x \u2208 adicCompletionIntegers K x}", "state_after": "no goals"}, {"tactic": "ext v", "annotated_tactic": ["ext v", []], "state_before": "R : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv : HeightOneSpectrum R\n\u22a2 {v | \u00ac0 \u2208 adicCompletionIntegers K v} = \u2205", "state_after": "case h\nR : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d v : HeightOneSpectrum R\n\u22a2 v \u2208 {v | \u00ac0 \u2208 adicCompletionIntegers K v} \u2194 v \u2208 \u2205"}, {"tactic": "rw [mem_empty_iff_false, iff_false_iff]", "annotated_tactic": ["rw [<a>mem_empty_iff_false</a>, <a>iff_false_iff</a>]", [{"full_name": "Set.mem_empty_iff_false", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [562, 9], "def_end_pos": [562, 28]}, {"full_name": "iff_false_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [201, 9], "def_end_pos": [201, 22]}]], "state_before": "case h\nR : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d v : HeightOneSpectrum R\n\u22a2 v \u2208 {v | \u00ac0 \u2208 adicCompletionIntegers K v} \u2194 v \u2208 \u2205", "state_after": "case h\nR : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d v : HeightOneSpectrum R\n\u22a2 \u00acv \u2208 {v | \u00ac0 \u2208 adicCompletionIntegers K v}"}, {"tactic": "intro hv", "annotated_tactic": ["intro hv", []], "state_before": "case h\nR : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d v : HeightOneSpectrum R\n\u22a2 \u00acv \u2208 {v | \u00ac0 \u2208 adicCompletionIntegers K v}", "state_after": "case h\nR : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d v : HeightOneSpectrum R\nhv : v \u2208 {v | \u00ac0 \u2208 adicCompletionIntegers K v}\n\u22a2 False"}, {"tactic": "rw [mem_setOf] at hv", "annotated_tactic": ["rw [<a>mem_setOf</a>] at hv", [{"full_name": "Set.mem_setOf", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [259, 9], "def_end_pos": [259, 18]}]], "state_before": "case h\nR : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d v : HeightOneSpectrum R\nhv : v \u2208 {v | \u00ac0 \u2208 adicCompletionIntegers K v}\n\u22a2 False", "state_after": "case h\nR : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d v : HeightOneSpectrum R\nhv : \u00ac0 \u2208 adicCompletionIntegers K v\n\u22a2 False"}, {"tactic": "apply hv", "annotated_tactic": ["apply hv", []], "state_before": "case h\nR : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d v : HeightOneSpectrum R\nhv : \u00ac0 \u2208 adicCompletionIntegers K v\n\u22a2 False", "state_after": "case h\nR : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d v : HeightOneSpectrum R\nhv : \u00ac0 \u2208 adicCompletionIntegers K v\n\u22a2 0 \u2208 adicCompletionIntegers K v"}, {"tactic": "rw [mem_adicCompletionIntegers]", "annotated_tactic": ["rw [<a>mem_adicCompletionIntegers</a>]", [{"full_name": "IsDedekindDomain.HeightOneSpectrum.mem_adicCompletionIntegers", "def_path": "Mathlib/RingTheory/DedekindDomain/AdicValuation.lean", "def_pos": [378, 9], "def_end_pos": [378, 35]}]], "state_before": "case h\nR : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d v : HeightOneSpectrum R\nhv : \u00ac0 \u2208 adicCompletionIntegers K v\n\u22a2 0 \u2208 adicCompletionIntegers K v", "state_after": "case h\nR : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d v : HeightOneSpectrum R\nhv : \u00ac0 \u2208 adicCompletionIntegers K v\n\u22a2 \u2191Valued.v 0 \u2264 1"}, {"tactic": "have h_zero : (Valued.v (0 : v.adicCompletion K) : WithZero (Multiplicative \u2124)) = 0 :=\n  Valued.v.map_zero'", "annotated_tactic": ["have h_zero : (<a>Valued.v</a> (0 : v.adicCompletion K) : <a>WithZero</a> (<a>Multiplicative</a> \u2124)) = 0 :=\n      Valued.v.map_zero'", [{"full_name": "Valued.v", "def_path": "Mathlib/Topology/Algebra/Valuation.lean", "def_pos": [94, 3], "def_end_pos": [94, 4]}, {"full_name": "WithZero", "def_path": "Mathlib/Algebra/Group/WithOne/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 14]}, {"full_name": "Multiplicative", "def_path": "Mathlib/Algebra/Group/TypeTags.lean", "def_pos": [47, 5], "def_end_pos": [47, 19]}]], "state_before": "case h\nR : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d v : HeightOneSpectrum R\nhv : \u00ac0 \u2208 adicCompletionIntegers K v\n\u22a2 \u2191Valued.v 0 \u2264 1", "state_after": "case h\nR : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d v : HeightOneSpectrum R\nhv : \u00ac0 \u2208 adicCompletionIntegers K v\nh_zero : \u2191Valued.v 0 = 0\n\u22a2 \u2191Valued.v 0 \u2264 1"}, {"tactic": "rw [h_zero]", "annotated_tactic": ["rw [h_zero]", []], "state_before": "case h\nR : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d v : HeightOneSpectrum R\nhv : \u00ac0 \u2208 adicCompletionIntegers K v\nh_zero : \u2191Valued.v 0 = 0\n\u22a2 \u2191Valued.v 0 \u2264 1", "state_after": "case h\nR : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d v : HeightOneSpectrum R\nhv : \u00ac0 \u2208 adicCompletionIntegers K v\nh_zero : \u2191Valued.v 0 = 0\n\u22a2 0 \u2264 1"}, {"tactic": "exact zero_le_one' _", "annotated_tactic": ["exact <a>zero_le_one'</a> _", [{"full_name": "zero_le_one'", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [31, 7], "def_end_pos": [31, 19]}]], "state_before": "case h\nR : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : IsDedekindDomain R\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\nv\u271d v : HeightOneSpectrum R\nhv : \u00ac0 \u2208 adicCompletionIntegers K v\nh_zero : \u2191Valued.v 0 = 0\n\u22a2 0 \u2264 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.tendsto_atTop_add_nonneg_right", "start": [685, 1], "end": [687, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Order/Field/Basic.lean", "full_name": "inv_le_inv", "start": [256, 1], "end": [257, 74], "traced_tactics": [{"tactic": "rw [\u2190 one_div, div_le_iff ha, \u2190 div_eq_inv_mul, le_div_iff hb, one_mul]", "annotated_tactic": ["rw [\u2190 <a>one_div</a>, <a>div_le_iff</a> ha, \u2190 <a>div_eq_inv_mul</a>, <a>le_div_iff</a> hb, <a>one_mul</a>]", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "div_le_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [144, 9], "def_end_pos": [144, 19]}, {"full_name": "div_eq_inv_mul", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [492, 9], "def_end_pos": [492, 23]}, {"full_name": "le_div_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [132, 9], "def_end_pos": [132, 19]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedSemifield \u03b1\na b c d e : \u03b1\nm n : \u2124\nha : 0 < a\nhb : 0 < b\n\u22a2 a\u207b\u00b9 \u2264 b\u207b\u00b9 \u2194 b \u2264 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Convex/Quasiconvex.lean", "full_name": "QuasiconcaveOn.inf", "start": [113, 1], "end": [115, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Constructions.lean", "full_name": "isOpenMap_ofDual", "start": [171, 1], "end": [171, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Group/Basic.lean", "full_name": "div_self'", "start": [735, 1], "end": [735, 81], "traced_tactics": [{"tactic": "rw [div_eq_mul_inv, mul_right_inv a]", "annotated_tactic": ["rw [<a>div_eq_mul_inv</a>, <a>mul_right_inv</a> a]", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}, {"full_name": "mul_right_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1135, 9], "def_end_pos": [1135, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nG : Type u_3\ninst\u271d : Group G\na\u271d b c d a : G\n\u22a2 a / a = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.snorm_indicator_const\u2080", "start": [594, 1], "end": [608, 17], "traced_tactics": [{"tactic": "congr 2", "annotated_tactic": ["congr 2", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc\u271d : E\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03b1\nc : G\nhs : NullMeasurableSet s\nhp : p \u2260 0\nhp_top : p \u2260 \u22a4\nhp_pos : 0 < ENNReal.toReal p\n\u22a2 (\u222b\u207b (x : \u03b1), \u2191\u2016Set.indicator s (fun x => c) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p) =\n    (\u222b\u207b (x : \u03b1), Set.indicator s (fun x => \u2191\u2016c\u2016\u208a ^ ENNReal.toReal p) x \u2202\u03bc) ^ (1 / ENNReal.toReal p)", "state_after": "case e_a.e_f\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc\u271d : E\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03b1\nc : G\nhs : NullMeasurableSet s\nhp : p \u2260 0\nhp_top : p \u2260 \u22a4\nhp_pos : 0 < ENNReal.toReal p\n\u22a2 (fun x => \u2191\u2016Set.indicator s (fun x => c) x\u2016\u208a ^ ENNReal.toReal p) = fun x =>\n    Set.indicator s (fun x => \u2191\u2016c\u2016\u208a ^ ENNReal.toReal p) x"}, {"tactic": "refine (Set.comp_indicator_const c (fun x : G \u21a6 (\u2016x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal) ?_)", "annotated_tactic": ["refine (<a>Set.comp_indicator_const</a> c (fun x : G \u21a6 (\u2016x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal) ?_)", [{"full_name": "Set.comp_indicator_const", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [269, 3], "def_end_pos": [269, 14]}]], "state_before": "case e_a.e_f\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc\u271d : E\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03b1\nc : G\nhs : NullMeasurableSet s\nhp : p \u2260 0\nhp_top : p \u2260 \u22a4\nhp_pos : 0 < ENNReal.toReal p\n\u22a2 (fun x => \u2191\u2016Set.indicator s (fun x => c) x\u2016\u208a ^ ENNReal.toReal p) = fun x =>\n    Set.indicator s (fun x => \u2191\u2016c\u2016\u208a ^ ENNReal.toReal p) x", "state_after": "case e_a.e_f\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc\u271d : E\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03b1\nc : G\nhs : NullMeasurableSet s\nhp : p \u2260 0\nhp_top : p \u2260 \u22a4\nhp_pos : 0 < ENNReal.toReal p\n\u22a2 (fun x => \u2191\u2016x\u2016\u208a ^ ENNReal.toReal p) 0 = 0"}, {"tactic": "simp [hp_pos]", "annotated_tactic": ["simp [hp_pos]", []], "state_before": "case e_a.e_f\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc\u271d : E\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03b1\nc : G\nhs : NullMeasurableSet s\nhp : p \u2260 0\nhp_top : p \u2260 \u22a4\nhp_pos : 0 < ENNReal.toReal p\n\u22a2 (fun x => \u2191\u2016x\u2016\u208a ^ ENNReal.toReal p) 0 = 0", "state_after": "no goals"}, {"tactic": "rw [lintegral_indicator_const\u2080 hs, ENNReal.mul_rpow_of_nonneg, \u2190 ENNReal.rpow_mul,\n  mul_one_div_cancel hp_pos.ne', ENNReal.rpow_one]", "annotated_tactic": ["rw [<a>lintegral_indicator_const\u2080</a> hs, <a>ENNReal.mul_rpow_of_nonneg</a>, \u2190 <a>ENNReal.rpow_mul</a>,\n        <a>mul_one_div_cancel</a> hp_pos.ne', <a>ENNReal.rpow_one</a>]", [{"full_name": "MeasureTheory.lintegral_indicator_const\u2080", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [783, 9], "def_end_pos": [783, 35]}, {"full_name": "ENNReal.mul_rpow_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [594, 9], "def_end_pos": [594, 27]}, {"full_name": "ENNReal.rpow_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [532, 9], "def_end_pos": [532, 17]}, {"full_name": "mul_one_div_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [75, 9], "def_end_pos": [75, 27]}, {"full_name": "ENNReal.rpow_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [450, 9], "def_end_pos": [450, 17]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc\u271d : E\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03b1\nc : G\nhs : NullMeasurableSet s\nhp : p \u2260 0\nhp_top : p \u2260 \u22a4\nhp_pos : 0 < ENNReal.toReal p\n\u22a2 (\u222b\u207b (x : \u03b1), Set.indicator s (fun x => \u2191\u2016c\u2016\u208a ^ ENNReal.toReal p) x \u2202\u03bc) ^ (1 / ENNReal.toReal p) =\n    \u2191\u2016c\u2016\u208a * \u2191\u2191\u03bc s ^ (1 / ENNReal.toReal p)", "state_after": "case hz\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc\u271d : E\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03b1\nc : G\nhs : NullMeasurableSet s\nhp : p \u2260 0\nhp_top : p \u2260 \u22a4\nhp_pos : 0 < ENNReal.toReal p\n\u22a2 0 \u2264 1 / ENNReal.toReal p"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "case hz\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedAddCommGroup G\nc\u271d : E\nf : \u03b1 \u2192 E\nhf : AEStronglyMeasurable f \u03bc\ns : Set \u03b1\nc : G\nhs : NullMeasurableSet s\nhp : p \u2260 0\nhp_top : p \u2260 \u22a4\nhp_pos : 0 < ENNReal.toReal p\n\u22a2 0 \u2264 1 / ENNReal.toReal p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/BigOperators/Finprod.lean", "full_name": "finprod_false", "start": [209, 1], "end": [210, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Subsemigroup/Basic.lean", "full_name": "Subsemigroup.coe_bot", "start": [209, 1], "end": [210, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean", "full_name": "MvPolynomial.NewtonIdentities.disjUnion_filter_pairs_eq_pairs", "start": [197, 9], "end": [206, 8], "traced_tactics": [{"tactic": "simp only [disjUnion_eq_union, Finset.ext_iff, pairs, filter_filter, mem_filter]", "annotated_tactic": ["simp only [<a>disjUnion_eq_union</a>, <a>Finset.ext_iff</a>, <a>pairs</a>, <a>filter_filter</a>, <a>mem_filter</a>]", [{"full_name": "Finset.disjUnion_eq_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1378, 9], "def_end_pos": [1378, 27]}, {"full_name": "Finset.ext_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [234, 9], "def_end_pos": [234, 16]}, {"full_name": "_private.Mathlib.RingTheory.MvPolynomial.NewtonIdentities.0.MvPolynomial.NewtonIdentities.pairs", "def_path": "Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean", "def_pos": [60, 13], "def_end_pos": [60, 18]}, {"full_name": "Finset.filter_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2725, 9], "def_end_pos": [2725, 22]}, {"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2708, 9], "def_end_pos": [2708, 19]}]], "state_before": "\u03c3 : Type u_1\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : DecidableEq \u03c3\nR : Type u_2\ninst\u271d : CommRing R\nk : \u2115\n\u22a2 disjUnion (filter (fun t => card t.1 < k) (MvPolynomial.NewtonIdentities.pairs \u03c3 k))\n      (filter (fun t => card t.1 = k) (MvPolynomial.NewtonIdentities.pairs \u03c3 k))\n      (_ :\n        Disjoint (filter (fun t => card t.1 < k) (MvPolynomial.NewtonIdentities.pairs \u03c3 k))\n          (filter (fun t => card t.1 = k) (MvPolynomial.NewtonIdentities.pairs \u03c3 k))) =\n    MvPolynomial.NewtonIdentities.pairs \u03c3 k", "state_after": "\u03c3 : Type u_1\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : DecidableEq \u03c3\nR : Type u_2\ninst\u271d : CommRing R\nk : \u2115\n\u22a2 \u2200 (a : Finset \u03c3 \u00d7 \u03c3),\n    a \u2208\n        filter (fun a => (card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 < k) univ \u222a\n          filter (fun a => (card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 = k) univ \u2194\n      a \u2208 univ \u2227 card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)"}, {"tactic": "intro a", "annotated_tactic": ["intro a", []], "state_before": "\u03c3 : Type u_1\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : DecidableEq \u03c3\nR : Type u_2\ninst\u271d : CommRing R\nk : \u2115\n\u22a2 \u2200 (a : Finset \u03c3 \u00d7 \u03c3),\n    a \u2208\n        filter (fun a => (card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 < k) univ \u222a\n          filter (fun a => (card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 = k) univ \u2194\n      a \u2208 univ \u2227 card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)", "state_after": "\u03c3 : Type u_1\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : DecidableEq \u03c3\nR : Type u_2\ninst\u271d : CommRing R\nk : \u2115\na : Finset \u03c3 \u00d7 \u03c3\n\u22a2 a \u2208\n      filter (fun a => (card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 < k) univ \u222a\n        filter (fun a => (card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 = k) univ \u2194\n    a \u2208 univ \u2227 card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)"}, {"tactic": "rw [\u2190 filter_or, mem_filter]", "annotated_tactic": ["rw [\u2190 <a>filter_or</a>, <a>mem_filter</a>]", [{"full_name": "Finset.filter_or", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2906, 9], "def_end_pos": [2906, 18]}, {"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2708, 9], "def_end_pos": [2708, 19]}]], "state_before": "\u03c3 : Type u_1\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : DecidableEq \u03c3\nR : Type u_2\ninst\u271d : CommRing R\nk : \u2115\na : Finset \u03c3 \u00d7 \u03c3\n\u22a2 a \u2208\n      filter (fun a => (card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 < k) univ \u222a\n        filter (fun a => (card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 = k) univ \u2194\n    a \u2208 univ \u2227 card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)", "state_after": "\u03c3 : Type u_1\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : DecidableEq \u03c3\nR : Type u_2\ninst\u271d : CommRing R\nk : \u2115\na : Finset \u03c3 \u00d7 \u03c3\n\u22a2 a \u2208 univ \u2227\n      ((card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 < k \u2228\n        (card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 = k) \u2194\n    a \u2208 univ \u2227 card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)"}, {"tactic": "refine' \u27e8fun ha \u21a6 by tauto, fun ha \u21a6 _\u27e9", "annotated_tactic": ["refine' \u27e8fun ha \u21a6 by tauto, fun ha \u21a6 _\u27e9", []], "state_before": "\u03c3 : Type u_1\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : DecidableEq \u03c3\nR : Type u_2\ninst\u271d : CommRing R\nk : \u2115\na : Finset \u03c3 \u00d7 \u03c3\n\u22a2 a \u2208 univ \u2227\n      ((card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 < k \u2228\n        (card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 = k) \u2194\n    a \u2208 univ \u2227 card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)", "state_after": "\u03c3 : Type u_1\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : DecidableEq \u03c3\nR : Type u_2\ninst\u271d : CommRing R\nk : \u2115\na : Finset \u03c3 \u00d7 \u03c3\nha : a \u2208 univ \u2227 card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)\n\u22a2 a \u2208 univ \u2227\n    ((card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 < k \u2228\n      (card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 = k)"}, {"tactic": "have hacard := le_iff_lt_or_eq.mp ha.2.1", "annotated_tactic": ["have hacard := le_iff_lt_or_eq.mp ha.2.1", []], "state_before": "\u03c3 : Type u_1\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : DecidableEq \u03c3\nR : Type u_2\ninst\u271d : CommRing R\nk : \u2115\na : Finset \u03c3 \u00d7 \u03c3\nha : a \u2208 univ \u2227 card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)\n\u22a2 a \u2208 univ \u2227\n    ((card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 < k \u2228\n      (card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 = k)", "state_after": "\u03c3 : Type u_1\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : DecidableEq \u03c3\nR : Type u_2\ninst\u271d : CommRing R\nk : \u2115\na : Finset \u03c3 \u00d7 \u03c3\nha : a \u2208 univ \u2227 card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)\nhacard : card a.1 < k \u2228 card a.1 = k\n\u22a2 a \u2208 univ \u2227\n    ((card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 < k \u2228\n      (card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 = k)"}, {"tactic": "tauto", "annotated_tactic": ["tauto", []], "state_before": "\u03c3 : Type u_1\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : DecidableEq \u03c3\nR : Type u_2\ninst\u271d : CommRing R\nk : \u2115\na : Finset \u03c3 \u00d7 \u03c3\nha : a \u2208 univ \u2227 card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)\nhacard : card a.1 < k \u2228 card a.1 = k\n\u22a2 a \u2208 univ \u2227\n    ((card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 < k \u2228\n      (card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 = k)", "state_after": "no goals"}, {"tactic": "tauto", "annotated_tactic": ["tauto", []], "state_before": "\u03c3 : Type u_1\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : DecidableEq \u03c3\nR : Type u_2\ninst\u271d : CommRing R\nk : \u2115\na : Finset \u03c3 \u00d7 \u03c3\nha :\n  a \u2208 univ \u2227\n    ((card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 < k \u2228\n      (card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)) \u2227 card a.1 = k)\n\u22a2 a \u2208 univ \u2227 card a.1 \u2264 k \u2227 (card a.1 = k \u2192 a.2 \u2208 a.1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "full_name": "SimpleGraph.Walk.darts_dropUntil_subset", "start": [1231, 1], "end": [1234, 18], "traced_tactics": [{"tactic": "rw [\u2190 take_spec p h, darts_append, List.mem_append]", "annotated_tactic": ["rw [\u2190 <a>take_spec</a> p h, <a>darts_append</a>, <a>List.mem_append</a>]", [{"full_name": "SimpleGraph.Walk.take_spec", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [1142, 9], "def_end_pos": [1142, 18]}, {"full_name": "SimpleGraph.Walk.darts_append", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [704, 9], "def_end_pos": [704, 21]}, {"full_name": "List.mem_append", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [137, 17], "def_end_pos": [137, 27]}]], "state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst\u271d : DecidableEq V\nu v w : V\np : Walk G v w\nh : u \u2208 support p\nx : Dart G\nhx : x \u2208 darts (dropUntil p u h)\n\u22a2 x \u2208 darts p", "state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst\u271d : DecidableEq V\nu v w : V\np : Walk G v w\nh : u \u2208 support p\nx : Dart G\nhx : x \u2208 darts (dropUntil p u h)\n\u22a2 x \u2208 darts (takeUntil p u h) \u2228 x \u2208 darts (dropUntil p u h)"}, {"tactic": "exact Or.inr hx", "annotated_tactic": ["exact <a>Or.inr</a> hx", [{"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}]], "state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst\u271d : DecidableEq V\nu v w : V\np : Walk G v w\nh : u \u2208 support p\nx : Dart G\nhx : x \u2208 darts (dropUntil p u h)\n\u22a2 x \u2208 darts (takeUntil p u h) \u2228 x \u2208 darts (dropUntil p u h)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean", "full_name": "CategoryTheory.Limits.terminal.comp_from", "start": [382, 1], "end": [383, 54], "traced_tactics": [{"tactic": "aesop", "annotated_tactic": ["aesop", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\ninst\u271d : HasTerminal C\nP Q : C\nf : P \u27f6 Q\n\u22a2 f \u226b from Q = from P", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.sup_eq_lsub_or_sup_succ_eq_lsub", "start": [1614, 1], "end": [1618, 67], "traced_tactics": [{"tactic": "cases' eq_or_lt_of_le (sup_le_lsub.{_, v} f) with h h", "annotated_tactic": ["cases' <a>eq_or_lt_of_le</a> (<a>sup_le_lsub</a>.{_, v} f) with h h", [{"full_name": "eq_or_lt_of_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [414, 9], "def_end_pos": [414, 23]}, {"full_name": "Ordinal.sup_le_lsub", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [1605, 9], "def_end_pos": [1605, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b9 : Type u\nf : \u03b9 \u2192 Ordinal.{max u v}\n\u22a2 sup f = lsub f \u2228 succ (sup f) = lsub f", "state_after": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b9 : Type u\nf : \u03b9 \u2192 Ordinal.{max u v}\nh : sup f = lsub f\n\u22a2 sup f = lsub f \u2228 succ (sup f) = lsub f\n\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b9 : Type u\nf : \u03b9 \u2192 Ordinal.{max u v}\nh : sup f < lsub f\n\u22a2 sup f = lsub f \u2228 succ (sup f) = lsub f"}, {"tactic": "exact Or.inl h", "annotated_tactic": ["exact <a>Or.inl</a> h", [{"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}]], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b9 : Type u\nf : \u03b9 \u2192 Ordinal.{max u v}\nh : sup f = lsub f\n\u22a2 sup f = lsub f \u2228 succ (sup f) = lsub f", "state_after": "no goals"}, {"tactic": "exact Or.inr ((succ_le_of_lt h).antisymm (lsub_le_sup_succ f))", "annotated_tactic": ["exact <a>Or.inr</a> ((<a>succ_le_of_lt</a> h).<a>antisymm</a> (<a>lsub_le_sup_succ</a> f))", [{"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "Order.succ_le_of_lt", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [221, 9], "def_end_pos": [221, 22]}, {"full_name": "LE.le.antisymm", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [128, 7], "def_end_pos": [128, 21]}, {"full_name": "Ordinal.lsub_le_sup_succ", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [1609, 9], "def_end_pos": [1609, 25]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\n\u03b9 : Type u\nf : \u03b9 \u2192 Ordinal.{max u v}\nh : sup f < lsub f\n\u22a2 sup f = lsub f \u2228 succ (sup f) = lsub f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/SymmDiff.lean", "full_name": "bihimp_right_surjective", "start": [672, 1], "end": [673, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean", "full_name": "PrimeSpectrum.comap_singleton_isClosed_of_isIntegral", "start": [627, 1], "end": [632, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Sites/Surjective.lean", "full_name": "CategoryTheory.isLocallySurjective_iff_whisker_forget", "start": [97, 1], "end": [100, 6], "traced_tactics": [{"tactic": "simp only [isLocallySurjective_iff_imagePresheaf_sheafify_eq_top]", "annotated_tactic": ["simp only [<a>isLocallySurjective_iff_imagePresheaf_sheafify_eq_top</a>]", [{"full_name": "CategoryTheory.isLocallySurjective_iff_imagePresheaf_sheafify_eq_top", "def_path": "Mathlib/CategoryTheory/Sites/Surjective.lean", "def_pos": [74, 9], "def_end_pos": [74, 62]}]], "state_before": "C : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\n\u22a2 IsLocallySurjective J f \u2194 IsLocallySurjective J (whiskerRight f (forget A))", "state_after": "C : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\n\u22a2 Subpresheaf.sheafify J (imagePresheaf (whiskerRight f (forget A))) = \u22a4 \u2194\n    Subpresheaf.sheafify J (imagePresheaf (whiskerRight (whiskerRight f (forget A)) (forget (Type w')))) = \u22a4"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "C : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\n\u22a2 Subpresheaf.sheafify J (imagePresheaf (whiskerRight f (forget A))) = \u22a4 \u2194\n    Subpresheaf.sheafify J (imagePresheaf (whiskerRight (whiskerRight f (forget A)) (forget (Type w')))) = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.insert_subset_insert", "start": [1197, 1], "end": [1198, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/InnerProductSpace/Projection.lean", "full_name": "orthogonalProjection_norm_le", "start": [630, 1], "end": [631, 51], "traced_tactics": [{"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\n\u22a2 0 \u2264 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.aestronglyMeasurable_one", "start": [1161, 1], "end": [1163, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/Choose/Basic.lean", "full_name": "Nat.choose_two_right", "start": [98, 1], "end": [102, 20], "traced_tactics": [{"tactic": "induction' n with n ih", "annotated_tactic": ["induction' n with n ih", []], "state_before": "n : \u2115\n\u22a2 choose n 2 = n * (n - 1) / 2", "state_after": "case zero\n\n\u22a2 choose zero 2 = zero * (zero - 1) / 2\n\ncase succ\nn : \u2115\nih : choose n 2 = n * (n - 1) / 2\n\u22a2 choose (succ n) 2 = succ n * (succ n - 1) / 2"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case zero\n\n\u22a2 choose zero 2 = zero * (zero - 1) / 2", "state_after": "no goals"}, {"tactic": "rw [triangle_succ n, choose, ih]", "annotated_tactic": ["rw [<a>triangle_succ</a> n, <a>choose</a>, ih]", [{"full_name": "Nat.triangle_succ", "def_path": "Mathlib/Data/Nat/Choose/Basic.lean", "def_pos": [92, 9], "def_end_pos": [92, 22]}, {"full_name": "Nat.choose", "def_path": "Mathlib/Data/Nat/Choose/Basic.lean", "def_pos": [46, 5], "def_end_pos": [46, 11]}]], "state_before": "case succ\nn : \u2115\nih : choose n 2 = n * (n - 1) / 2\n\u22a2 choose (succ n) 2 = succ n * (succ n - 1) / 2", "state_after": "case succ\nn : \u2115\nih : choose n 2 = n * (n - 1) / 2\n\u22a2 choose n 1 + n * (n - 1) / 2 = n * (n - 1) / 2 + n"}, {"tactic": "simp [add_comm]", "annotated_tactic": ["simp [<a>add_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "case succ\nn : \u2115\nih : choose n 2 = n * (n - 1) / 2\n\u22a2 choose n 1 + n * (n - 1) / 2 = n * (n - 1) / 2 + n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Subring/Pointwise.lean", "full_name": "Subring.pointwise_smul_subset_iff", "start": [128, 1], "end": [129, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Maps.lean", "full_name": "IsOpenMap.nhds_le", "start": [379, 1], "end": [380, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/BilinearForm.lean", "full_name": "BilinForm.comp_id_id", "start": [658, 1], "end": [660, 6], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB : BilinForm R M\n\u22a2 comp B LinearMap.id LinearMap.id = B", "state_after": "case H\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB : BilinForm R M\nx\u271d y\u271d : M\n\u22a2 bilin (comp B LinearMap.id LinearMap.id) x\u271d y\u271d = bilin B x\u271d y\u271d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case H\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nB : BilinForm R M\nx\u271d y\u271d : M\n\u22a2 bilin (comp B LinearMap.id LinearMap.id) x\u271d y\u271d = bilin B x\u271d y\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecialFunctions/Integrals.lean", "full_name": "integral_sin_sq_mul_cos", "start": [770, 1], "end": [773, 31], "traced_tactics": [{"tactic": "have := @integral_sin_pow_mul_cos_pow_odd a b 2 0", "annotated_tactic": ["have := @<a>integral_sin_pow_mul_cos_pow_odd</a> a b 2 0", [{"full_name": "integral_sin_pow_mul_cos_pow_odd", "def_path": "Mathlib/Analysis/SpecialFunctions/Integrals.lean", "def_pos": [748, 9], "def_end_pos": [748, 41]}]], "state_before": "a b : \u211d\nn : \u2115\n\u22a2 \u222b (x : \u211d) in a..b, sin x ^ 2 * cos x = (sin b ^ 3 - sin a ^ 3) / 3", "state_after": "a b : \u211d\nn : \u2115\nthis : \u222b (x : \u211d) in a..b, sin x ^ 2 * cos x ^ (2 * 0 + 1) = \u222b (u : \u211d) in sin a..sin b, u ^ 2 * (1 - u ^ 2) ^ 0\n\u22a2 \u222b (x : \u211d) in a..b, sin x ^ 2 * cos x = (sin b ^ 3 - sin a ^ 3) / 3"}, {"tactic": "norm_num at this", "annotated_tactic": ["norm_num at this", []], "state_before": "a b : \u211d\nn : \u2115\nthis : \u222b (x : \u211d) in a..b, sin x ^ 2 * cos x ^ (2 * 0 + 1) = \u222b (u : \u211d) in sin a..sin b, u ^ 2 * (1 - u ^ 2) ^ 0\n\u22a2 \u222b (x : \u211d) in a..b, sin x ^ 2 * cos x = (sin b ^ 3 - sin a ^ 3) / 3", "state_after": "a b : \u211d\nn : \u2115\nthis : \u222b (x : \u211d) in a..b, sin x ^ 2 * cos x = (sin b ^ 3 - sin a ^ 3) / 3\n\u22a2 \u222b (x : \u211d) in a..b, sin x ^ 2 * cos x = (sin b ^ 3 - sin a ^ 3) / 3"}, {"tactic": "exact this", "annotated_tactic": ["exact this", []], "state_before": "a b : \u211d\nn : \u2115\nthis : \u222b (x : \u211d) in a..b, sin x ^ 2 * cos x = (sin b ^ 3 - sin a ^ 3) / 3\n\u22a2 \u222b (x : \u211d) in a..b, sin x ^ 2 * cos x = (sin b ^ 3 - sin a ^ 3) / 3", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.nontrivial_iff_exists_lt", "start": [2541, 1], "end": [2543, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Monotone.lean", "full_name": "Monotone.Ici", "start": [37, 11], "end": [38, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Compactness/SigmaCompact.lean", "full_name": "countable_cover_nhdsWithin_of_sigma_compact", "start": [292, 1], "end": [302, 43], "traced_tactics": [{"tactic": "simp only [nhdsWithin, mem_inf_principal] at hf", "annotated_tactic": ["simp only [<a>nhdsWithin</a>, <a>mem_inf_principal</a>] at hf", [{"full_name": "nhdsWithin", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [838, 5], "def_end_pos": [838, 15]}, {"full_name": "Filter.mem_inf_principal", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1013, 7], "def_end_pos": [1013, 24]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\nf : \u03b1 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsClosed s\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2208 \ud835\udcdd[s] x\n\u22a2 \u2203 t x, Set.Countable t \u2227 s \u2286 \u22c3 x \u2208 t, f x", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\nf : \u03b1 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsClosed s\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x} \u2208 \ud835\udcdd x\n\u22a2 \u2203 t x, Set.Countable t \u2227 s \u2286 \u22c3 x \u2208 t, f x"}, {"tactic": "choose t ht hsub using fun n =>\n  ((isCompact_compactCovering \u03b1 n).inter_right hs).elim_nhds_subcover _ fun x hx => hf x hx.right", "annotated_tactic": ["choose t ht hsub using fun n =>\n    ((<a>isCompact_compactCovering</a> \u03b1 n).<a>inter_right</a> hs).<a>elim_nhds_subcover</a> _ fun x hx => hf x hx.right", [{"full_name": "isCompact_compactCovering", "def_path": "Mathlib/Topology/Compactness/SigmaCompact.lean", "def_pos": [201, 9], "def_end_pos": [201, 34]}, {"full_name": "IsCompact.inter_right", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [81, 9], "def_end_pos": [81, 30]}, {"full_name": "IsCompact.elim_nhds_subcover", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [176, 9], "def_end_pos": [176, 37]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\nf : \u03b1 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsClosed s\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x} \u2208 \ud835\udcdd x\n\u22a2 \u2203 t x, Set.Countable t \u2227 s \u2286 \u22c3 x \u2208 t, f x", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\nf : \u03b1 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsClosed s\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x} \u2208 \ud835\udcdd x\nt : \u2115 \u2192 Finset \u03b1\nht : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t n \u2192 x \u2208 compactCovering \u03b1 n \u2229 s\nhsub : \u2200 (n : \u2115), compactCovering \u03b1 n \u2229 s \u2286 \u22c3 x \u2208 t n, {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x}\n\u22a2 \u2203 t x, Set.Countable t \u2227 s \u2286 \u22c3 x \u2208 t, f x"}, {"tactic": "refine'\n  \u27e8\u22c3 n, (t n : Set \u03b1), iUnion_subset fun n x hx => (ht n x hx).2,\n    countable_iUnion fun n => (t n).countable_toSet, fun x hx => mem_iUnion\u2082.2 _\u27e9", "annotated_tactic": ["refine'\n    \u27e8\u22c3 n, (t n : <a>Set</a> \u03b1), <a>iUnion_subset</a> fun n x hx => (ht n x hx).2,\n      <a>countable_iUnion</a> fun n => (t n).<a>countable_toSet</a>, fun x hx => <a>mem_iUnion\u2082</a>.2 _\u27e9", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Set.iUnion_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [390, 9], "def_end_pos": [390, 22]}, {"full_name": "Set.countable_iUnion", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [185, 9], "def_end_pos": [185, 25]}, {"full_name": "Finset.countable_toSet", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [314, 9], "def_end_pos": [314, 31]}, {"full_name": "Set.mem_iUnion\u2082", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [212, 9], "def_end_pos": [212, 20]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\nf : \u03b1 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsClosed s\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x} \u2208 \ud835\udcdd x\nt : \u2115 \u2192 Finset \u03b1\nht : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t n \u2192 x \u2208 compactCovering \u03b1 n \u2229 s\nhsub : \u2200 (n : \u2115), compactCovering \u03b1 n \u2229 s \u2286 \u22c3 x \u2208 t n, {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x}\n\u22a2 \u2203 t x, Set.Countable t \u2227 s \u2286 \u22c3 x \u2208 t, f x", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\nf : \u03b1 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsClosed s\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x} \u2208 \ud835\udcdd x\nt : \u2115 \u2192 Finset \u03b1\nht : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t n \u2192 x \u2208 compactCovering \u03b1 n \u2229 s\nhsub : \u2200 (n : \u2115), compactCovering \u03b1 n \u2229 s \u2286 \u22c3 x \u2208 t n, {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x}\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2203 i j, x \u2208 f i"}, {"tactic": "rcases exists_mem_compactCovering x with \u27e8n, hn\u27e9", "annotated_tactic": ["rcases <a>exists_mem_compactCovering</a> x with \u27e8n, hn\u27e9", [{"full_name": "exists_mem_compactCovering", "def_path": "Mathlib/Topology/Compactness/SigmaCompact.lean", "def_pos": [217, 9], "def_end_pos": [217, 35]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\nf : \u03b1 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsClosed s\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x} \u2208 \ud835\udcdd x\nt : \u2115 \u2192 Finset \u03b1\nht : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t n \u2192 x \u2208 compactCovering \u03b1 n \u2229 s\nhsub : \u2200 (n : \u2115), compactCovering \u03b1 n \u2229 s \u2286 \u22c3 x \u2208 t n, {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x}\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2203 i j, x \u2208 f i", "state_after": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\nf : \u03b1 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsClosed s\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x} \u2208 \ud835\udcdd x\nt : \u2115 \u2192 Finset \u03b1\nht : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t n \u2192 x \u2208 compactCovering \u03b1 n \u2229 s\nhsub : \u2200 (n : \u2115), compactCovering \u03b1 n \u2229 s \u2286 \u22c3 x \u2208 t n, {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x}\nx : \u03b1\nhx : x \u2208 s\nn : \u2115\nhn : x \u2208 compactCovering \u03b1 n\n\u22a2 \u2203 i j, x \u2208 f i"}, {"tactic": "rcases mem_iUnion\u2082.1 (hsub n \u27e8hn, hx\u27e9) with \u27e8y, hyt : y \u2208 t n, hyf : x \u2208 s \u2192 x \u2208 f y\u27e9", "annotated_tactic": ["rcases <a>mem_iUnion\u2082</a>.1 (hsub n \u27e8hn, hx\u27e9) with \u27e8y, hyt : y \u2208 t n, hyf : x \u2208 s \u2192 x \u2208 f y\u27e9", [{"full_name": "Set.mem_iUnion\u2082", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [212, 9], "def_end_pos": [212, 20]}]], "state_before": "case intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\nf : \u03b1 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsClosed s\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x} \u2208 \ud835\udcdd x\nt : \u2115 \u2192 Finset \u03b1\nht : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t n \u2192 x \u2208 compactCovering \u03b1 n \u2229 s\nhsub : \u2200 (n : \u2115), compactCovering \u03b1 n \u2229 s \u2286 \u22c3 x \u2208 t n, {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x}\nx : \u03b1\nhx : x \u2208 s\nn : \u2115\nhn : x \u2208 compactCovering \u03b1 n\n\u22a2 \u2203 i j, x \u2208 f i", "state_after": "case intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\nf : \u03b1 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsClosed s\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x} \u2208 \ud835\udcdd x\nt : \u2115 \u2192 Finset \u03b1\nht : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t n \u2192 x \u2208 compactCovering \u03b1 n \u2229 s\nhsub : \u2200 (n : \u2115), compactCovering \u03b1 n \u2229 s \u2286 \u22c3 x \u2208 t n, {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x}\nx : \u03b1\nhx : x \u2208 s\nn : \u2115\nhn : x \u2208 compactCovering \u03b1 n\ny : \u03b1\nhyt : y \u2208 t n\nhyf : x \u2208 s \u2192 x \u2208 f y\n\u22a2 \u2203 i j, x \u2208 f i"}, {"tactic": "exact \u27e8y, mem_iUnion.2 \u27e8n, hyt\u27e9, hyf hx\u27e9", "annotated_tactic": ["exact \u27e8y, <a>mem_iUnion</a>.2 \u27e8n, hyt\u27e9, hyf hx\u27e9", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\ninst\u271d : SigmaCompactSpace \u03b1\nf : \u03b1 \u2192 Set \u03b1\ns : Set \u03b1\nhs : IsClosed s\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x} \u2208 \ud835\udcdd x\nt : \u2115 \u2192 Finset \u03b1\nht : \u2200 (n : \u2115) (x : \u03b1), x \u2208 t n \u2192 x \u2208 compactCovering \u03b1 n \u2229 s\nhsub : \u2200 (n : \u2115), compactCovering \u03b1 n \u2229 s \u2286 \u22c3 x \u2208 t n, {x_1 | x_1 \u2208 s \u2192 x_1 \u2208 f x}\nx : \u03b1\nhx : x \u2208 s\nn : \u2115\nhn : x \u2208 compactCovering \u03b1 n\ny : \u03b1\nhyt : y \u2208 t n\nhyf : x \u2208 s \u2192 x \u2208 f y\n\u22a2 \u2203 i j, x \u2208 f i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "measurableSet_region_between_oc", "start": [489, 1], "end": [497, 26], "traced_tactics": [{"tactic": "dsimp only [regionBetween, Ioc, mem_setOf_eq, setOf_and]", "annotated_tactic": ["dsimp only [<a>regionBetween</a>, <a>Ioc</a>, <a>mem_setOf_eq</a>, <a>setOf_and</a>]", [{"full_name": "regionBetween", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [466, 5], "def_end_pos": [466, 18]}, {"full_name": "Set.Ioc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [69, 5], "def_end_pos": [69, 8]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "Set.setOf_and", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [300, 9], "def_end_pos": [300, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 MeasurableSet {p | p.1 \u2208 s \u2227 p.2 \u2208 Ioc (f p.1) (g p.1)}", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 MeasurableSet ({a | a.1 \u2208 s} \u2229 {a | a.2 \u2208 {a_1 | f a.1 < a_1} \u2229 {a_1 | a_1 \u2264 g a.1}})"}, {"tactic": "refine'\n  MeasurableSet.inter _\n    ((measurableSet_lt (hf.rst.immp measurable_fst) measurable_snd).inter\n      (measurableSet_le measurable_snd (hg.comp measurable_fst)))", "annotated_tactic": ["refine'\n    <a>MeasurableSet.inter</a> _\n      ((<a>measurableSet_lt</a> (hf.rst.immp <a>measurable_fst</a>) <a>measurable_snd</a>).<a>inter</a>\n        (<a>measurableSet_le</a> <a>measurable_snd</a> (hg.comp <a>measurable_fst</a>)))", [{"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "measurableSet_lt", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [616, 9], "def_end_pos": [616, 25]}, {"full_name": "measurable_fst", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [692, 9], "def_end_pos": [692, 23]}, {"full_name": "measurable_snd", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [698, 9], "def_end_pos": [698, 23]}, {"full_name": "MeasurableSet.inter", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [198, 19], "def_end_pos": [198, 38]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 25]}, {"full_name": "measurable_snd", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [698, 9], "def_end_pos": [698, 23]}, {"full_name": "measurable_fst", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [692, 9], "def_end_pos": [692, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 MeasurableSet ({a | a.1 \u2208 s} \u2229 {a | a.2 \u2208 {a_1 | f a.1 < a_1} \u2229 {a_1 | a_1 \u2264 g a.1}})", "state_after": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 MeasurableSet {a | a.1 \u2208 s}"}, {"tactic": "exact measurable_fst hs", "annotated_tactic": ["exact <a>measurable_fst</a> hs", [{"full_name": "measurable_fst", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [692, 9], "def_end_pos": [692, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 MeasurableSet {a | a.1 \u2208 s}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Hom/Ring/Defs.lean", "full_name": "RingHom.congr_fun", "start": [518, 1], "end": [519, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "iInf_mono", "start": [905, 1], "end": [906, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Instances/ENNReal.lean", "full_name": "ENNReal.exists_countable_dense_no_zero_top", "start": [682, 1], "end": [687, 88], "traced_tactics": [{"tactic": "obtain \u27e8s, s_count, s_dense, hs\u27e9 :\n  \u2203 s : Set \u211d\u22650\u221e, s.Countable \u2227 Dense s \u2227 (\u2200 x, IsBot x \u2192 x \u2209 s) \u2227 \u2200 x, IsTop x \u2192 x \u2209 s :=\n  exists_countable_dense_no_bot_top \u211d\u22650\u221e", "annotated_tactic": ["obtain \u27e8s, s_count, s_dense, hs\u27e9 :\n    \u2203 s : <a>Set</a> \u211d\u22650\u221e, s.Countable \u2227 <a>Dense</a> s \u2227 (\u2200 x, <a>IsBot</a> x \u2192 x \u2209 s) \u2227 \u2200 x, <a>IsTop</a> x \u2192 x \u2209 s :=\n    <a>exists_countable_dense_no_bot_top</a> \u211d\u22650\u221e", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Dense", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [616, 5], "def_end_pos": [616, 10]}, {"full_name": "IsBot", "def_path": "Mathlib/Order/Max.lean", "def_pos": [187, 5], "def_end_pos": [187, 10]}, {"full_name": "IsTop", "def_path": "Mathlib/Order/Max.lean", "def_pos": [195, 5], "def_end_pos": [195, 10]}, {"full_name": "exists_countable_dense_no_bot_top", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [2691, 9], "def_end_pos": [2691, 42]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\n\u22a2 \u2203 s, Set.Countable s \u2227 Dense s \u2227 \u00ac0 \u2208 s \u2227 \u00ac\u22a4 \u2208 s", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns\u271d s : Set \u211d\u22650\u221e\ns_count : Set.Countable s\ns_dense : Dense s\nhs : (\u2200 (x : \u211d\u22650\u221e), IsBot x \u2192 \u00acx \u2208 s) \u2227 \u2200 (x : \u211d\u22650\u221e), IsTop x \u2192 \u00acx \u2208 s\n\u22a2 \u2203 s, Set.Countable s \u2227 Dense s \u2227 \u00ac0 \u2208 s \u2227 \u00ac\u22a4 \u2208 s"}, {"tactic": "exact \u27e8s, s_count, s_dense, fun h => hs.1 0 (by simp) h, fun h => hs.2 \u221e (by simp) h\u27e9", "annotated_tactic": ["exact \u27e8s, s_count, s_dense, fun h => hs.1 0 (by simp) h, fun h => hs.2 \u221e (by simp) h\u27e9", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns\u271d s : Set \u211d\u22650\u221e\ns_count : Set.Countable s\ns_dense : Dense s\nhs : (\u2200 (x : \u211d\u22650\u221e), IsBot x \u2192 \u00acx \u2208 s) \u2227 \u2200 (x : \u211d\u22650\u221e), IsTop x \u2192 \u00acx \u2208 s\n\u22a2 \u2203 s, Set.Countable s \u2227 Dense s \u2227 \u00ac0 \u2208 s \u2227 \u00ac\u22a4 \u2208 s", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns\u271d s : Set \u211d\u22650\u221e\ns_count : Set.Countable s\ns_dense : Dense s\nhs : (\u2200 (x : \u211d\u22650\u221e), IsBot x \u2192 \u00acx \u2208 s) \u2227 \u2200 (x : \u211d\u22650\u221e), IsTop x \u2192 \u00acx \u2208 s\nh : 0 \u2208 s\n\u22a2 IsBot 0", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns\u271d s : Set \u211d\u22650\u221e\ns_count : Set.Countable s\ns_dense : Dense s\nhs : (\u2200 (x : \u211d\u22650\u221e), IsBot x \u2192 \u00acx \u2208 s) \u2227 \u2200 (x : \u211d\u22650\u221e), IsTop x \u2192 \u00acx \u2208 s\nh : \u22a4 \u2208 s\n\u22a2 IsTop \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/BigOperators/Fin.lean", "full_name": "Fin.prod_univ_five", "start": [130, 1], "end": [133, 6], "traced_tactics": [{"tactic": "rw [prod_univ_castSucc, prod_univ_four]", "annotated_tactic": ["rw [<a>prod_univ_castSucc</a>, <a>prod_univ_four</a>]", [{"full_name": "Fin.prod_univ_castSucc", "def_path": "Mathlib/Algebra/BigOperators/Fin.lean", "def_pos": [91, 9], "def_end_pos": [91, 27]}, {"full_name": "Fin.prod_univ_four", "def_path": "Mathlib/Algebra/BigOperators/Fin.lean", "def_pos": [123, 9], "def_end_pos": [123, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nf : Fin 5 \u2192 \u03b2\n\u22a2 \u220f i : Fin 5, f i = f 0 * f 1 * f 2 * f 3 * f 4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nf : Fin 5 \u2192 \u03b2\n\u22a2 f (castSucc 0) * f (castSucc 1) * f (castSucc 2) * f (castSucc 3) * f (last 4) = f 0 * f 1 * f 2 * f 3 * f 4"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nf : Fin 5 \u2192 \u03b2\n\u22a2 f (castSucc 0) * f (castSucc 1) * f (castSucc 2) * f (castSucc 3) * f (last 4) = f 0 * f 1 * f 2 * f 3 * f 4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Group/Defs.lean", "full_name": "Group.toDivInvMonoid_injective", "start": [1187, 1], "end": [1188, 76], "traced_tactics": [{"tactic": "rintro \u27e8\u27e9 \u27e8\u27e9 \u27e8\u27e9", "annotated_tactic": ["rintro \u27e8\u27e9 \u27e8\u27e9 \u27e8\u27e9", []], "state_before": "G\u271d : Type u_1\nG : Type u_2\n\u22a2 Injective (@toDivInvMonoid G)", "state_after": "case mk.mk.refl\nG\u271d : Type u_1\nG : Type u_2\ntoDivInvMonoid\u271d : DivInvMonoid G\nmul_left_inv\u271d\u00b9 mul_left_inv\u271d : \u2200 (a : G), a\u207b\u00b9 * a = 1\n\u22a2 mk mul_left_inv\u271d\u00b9 = mk mul_left_inv\u271d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case mk.mk.refl\nG\u271d : Type u_1\nG : Type u_2\ntoDivInvMonoid\u271d : DivInvMonoid G\nmul_left_inv\u271d\u00b9 mul_left_inv\u271d : \u2200 (a : G), a\u207b\u00b9 * a = 1\n\u22a2 mk mul_left_inv\u271d\u00b9 = mk mul_left_inv\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/Perfection.lean", "full_name": "PerfectionMap.hom_ext", "start": [335, 1], "end": [338, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Sylow.lean", "full_name": "Sylow.characteristic_of_normal", "start": [727, 1], "end": [734, 7], "traced_tactics": [{"tactic": "haveI := Sylow.subsingleton_of_normal P h", "annotated_tactic": ["haveI := <a>Sylow.subsingleton_of_normal</a> P h", [{"full_name": "Sylow.subsingleton_of_normal", "def_path": "Mathlib/GroupTheory/Sylow.lean", "def_pos": [713, 9], "def_end_pos": [713, 31]}]], "state_before": "G : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b2 : Group G\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nh : Normal \u2191P\n\u22a2 Characteristic \u2191P", "state_after": "G : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b2 : Group G\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nh : Normal \u2191P\nthis : Subsingleton (Sylow p G)\n\u22a2 Characteristic \u2191P"}, {"tactic": "rw [characteristic_iff_map_eq]", "annotated_tactic": ["rw [<a>characteristic_iff_map_eq</a>]", [{"full_name": "Subgroup.characteristic_iff_map_eq", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [2047, 9], "def_end_pos": [2047, 34]}]], "state_before": "G : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b2 : Group G\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nh : Normal \u2191P\nthis : Subsingleton (Sylow p G)\n\u22a2 Characteristic \u2191P", "state_after": "G : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b2 : Group G\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nh : Normal \u2191P\nthis : Subsingleton (Sylow p G)\n\u22a2 \u2200 (\u03d5 : G \u2243* G), Subgroup.map (MulEquiv.toMonoidHom \u03d5) \u2191P = \u2191P"}, {"tactic": "intro \u03a6", "annotated_tactic": ["intro \u03a6", []], "state_before": "G : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b2 : Group G\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nh : Normal \u2191P\nthis : Subsingleton (Sylow p G)\n\u22a2 \u2200 (\u03d5 : G \u2243* G), Subgroup.map (MulEquiv.toMonoidHom \u03d5) \u2191P = \u2191P", "state_after": "G : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b2 : Group G\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nh : Normal \u2191P\nthis : Subsingleton (Sylow p G)\n\u03a6 : G \u2243* G\n\u22a2 Subgroup.map (MulEquiv.toMonoidHom \u03a6) \u2191P = \u2191P"}, {"tactic": "show (\u03a6 \u2022 P).toSubgroup = P.toSubgroup", "annotated_tactic": ["show (\u03a6 \u2022 P).toSubgroup = P.toSubgroup", []], "state_before": "G : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b2 : Group G\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nh : Normal \u2191P\nthis : Subsingleton (Sylow p G)\n\u03a6 : G \u2243* G\n\u22a2 Subgroup.map (MulEquiv.toMonoidHom \u03a6) \u2191P = \u2191P", "state_after": "G : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b2 : Group G\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nh : Normal \u2191P\nthis : Subsingleton (Sylow p G)\n\u03a6 : G \u2243* G\n\u22a2 \u2191(\u03a6 \u2022 P) = \u2191P"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "G : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b2 : Group G\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nh : Normal \u2191P\nthis : Subsingleton (Sylow p G)\n\u03a6 : G \u2243* G\n\u22a2 \u2191(\u03a6 \u2022 P) = \u2191P", "state_after": "case e_self\nG : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b2 : Group G\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nh : Normal \u2191P\nthis : Subsingleton (Sylow p G)\n\u03a6 : G \u2243* G\n\u22a2 \u03a6 \u2022 P = P"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case e_self\nG : Type u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d\u00b2 : Group G\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Finite (Sylow p G)\nP : Sylow p G\nh : Normal \u2191P\nthis : Subsingleton (Sylow p G)\n\u03a6 : G \u2243* G\n\u22a2 \u03a6 \u2022 P = P", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean", "full_name": "CategoryTheory.MonoidalOfChosenFiniteProducts.triangle", "start": [295, 1], "end": [300, 62], "traced_tactics": [{"tactic": "dsimp [tensorHom]", "annotated_tactic": ["dsimp [<a>tensorHom</a>]", [{"full_name": "CategoryTheory.MonoidalOfChosenFiniteProducts.tensorHom", "def_path": "Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean", "def_pos": [257, 5], "def_end_pos": [257, 14]}]], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y : C\n\u22a2 (BinaryFan.associatorOfLimitCone \u212c X \ud835\udcaf.cone.pt Y).hom \u226b\n      tensorHom \u212c (\ud835\udfd9 X) (BinaryFan.leftUnitor \ud835\udcaf.isLimit (\u212c \ud835\udcaf.cone.pt Y).isLimit).hom =\n    tensorHom \u212c (BinaryFan.rightUnitor \ud835\udcaf.isLimit (\u212c X \ud835\udcaf.cone.pt).isLimit).hom (\ud835\udfd9 Y)", "state_after": "C : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y : C\n\u22a2 (BinaryFan.associatorOfLimitCone \u212c X \ud835\udcaf.cone.pt Y).hom \u226b\n      IsLimit.lift (\u212c X Y).isLimit\n        (BinaryFan.mk (BinaryFan.fst (\u212c X (\u212c \ud835\udcaf.cone.pt Y).cone.pt).cone \u226b \ud835\udfd9 X)\n          (BinaryFan.snd (\u212c X (\u212c \ud835\udcaf.cone.pt Y).cone.pt).cone \u226b BinaryFan.snd (\u212c \ud835\udcaf.cone.pt Y).cone)) =\n    IsLimit.lift (\u212c X Y).isLimit\n      (BinaryFan.mk (BinaryFan.fst (\u212c (\u212c X \ud835\udcaf.cone.pt).cone.pt Y).cone \u226b BinaryFan.fst (\u212c X \ud835\udcaf.cone.pt).cone)\n        (BinaryFan.snd (\u212c (\u212c X \ud835\udcaf.cone.pt).cone.pt Y).cone \u226b \ud835\udfd9 Y))"}, {"tactic": "apply IsLimit.hom_ext (\u212c _ _).isLimit", "annotated_tactic": ["apply <a>IsLimit.hom_ext</a> (\u212c _ _).<a>isLimit</a>", [{"full_name": "CategoryTheory.Limits.IsLimit.hom_ext", "def_path": "Mathlib/CategoryTheory/Limits/IsLimit.lean", "def_pos": [231, 9], "def_end_pos": [231, 16]}, {"full_name": "CategoryTheory.Limits.LimitCone.isLimit", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [81, 3], "def_end_pos": [81, 10]}]], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y : C\n\u22a2 (BinaryFan.associatorOfLimitCone \u212c X \ud835\udcaf.cone.pt Y).hom \u226b\n      IsLimit.lift (\u212c X Y).isLimit\n        (BinaryFan.mk (BinaryFan.fst (\u212c X (\u212c \ud835\udcaf.cone.pt Y).cone.pt).cone \u226b \ud835\udfd9 X)\n          (BinaryFan.snd (\u212c X (\u212c \ud835\udcaf.cone.pt Y).cone.pt).cone \u226b BinaryFan.snd (\u212c \ud835\udcaf.cone.pt Y).cone)) =\n    IsLimit.lift (\u212c X Y).isLimit\n      (BinaryFan.mk (BinaryFan.fst (\u212c (\u212c X \ud835\udcaf.cone.pt).cone.pt Y).cone \u226b BinaryFan.fst (\u212c X \ud835\udcaf.cone.pt).cone)\n        (BinaryFan.snd (\u212c (\u212c X \ud835\udcaf.cone.pt).cone.pt Y).cone \u226b \ud835\udfd9 Y))", "state_after": "C : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y : C\n\u22a2 \u2200 (j : Discrete WalkingPair),\n    ((BinaryFan.associatorOfLimitCone \u212c X \ud835\udcaf.cone.pt Y).hom \u226b\n          IsLimit.lift (\u212c X Y).isLimit\n            (BinaryFan.mk (BinaryFan.fst (\u212c X (\u212c \ud835\udcaf.cone.pt Y).cone.pt).cone \u226b \ud835\udfd9 X)\n              (BinaryFan.snd (\u212c X (\u212c \ud835\udcaf.cone.pt Y).cone.pt).cone \u226b BinaryFan.snd (\u212c \ud835\udcaf.cone.pt Y).cone))) \u226b\n        (\u212c X Y).cone.\u03c0.app j =\n      IsLimit.lift (\u212c X Y).isLimit\n          (BinaryFan.mk (BinaryFan.fst (\u212c (\u212c X \ud835\udcaf.cone.pt).cone.pt Y).cone \u226b BinaryFan.fst (\u212c X \ud835\udcaf.cone.pt).cone)\n            (BinaryFan.snd (\u212c (\u212c X \ud835\udcaf.cone.pt).cone.pt Y).cone \u226b \ud835\udfd9 Y)) \u226b\n        (\u212c X Y).cone.\u03c0.app j"}, {"tactic": "rintro \u27e8\u27e8\u27e9\u27e9 <;> simp", "annotated_tactic": ["rintro \u27e8\u27e8\u27e9\u27e9 <;> simp", []], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d : C\n\ud835\udcaf : LimitCone (Functor.empty C)\n\u212c : (X Y : C) \u2192 LimitCone (pair X Y)\nX Y : C\n\u22a2 \u2200 (j : Discrete WalkingPair),\n    ((BinaryFan.associatorOfLimitCone \u212c X \ud835\udcaf.cone.pt Y).hom \u226b\n          IsLimit.lift (\u212c X Y).isLimit\n            (BinaryFan.mk (BinaryFan.fst (\u212c X (\u212c \ud835\udcaf.cone.pt Y).cone.pt).cone \u226b \ud835\udfd9 X)\n              (BinaryFan.snd (\u212c X (\u212c \ud835\udcaf.cone.pt Y).cone.pt).cone \u226b BinaryFan.snd (\u212c \ud835\udcaf.cone.pt Y).cone))) \u226b\n        (\u212c X Y).cone.\u03c0.app j =\n      IsLimit.lift (\u212c X Y).isLimit\n          (BinaryFan.mk (BinaryFan.fst (\u212c (\u212c X \ud835\udcaf.cone.pt).cone.pt Y).cone \u226b BinaryFan.fst (\u212c X \ud835\udcaf.cone.pt).cone)\n            (BinaryFan.snd (\u212c (\u212c X \ud835\udcaf.cone.pt).cone.pt Y).cone \u226b \ud835\udfd9 Y)) \u226b\n        (\u212c X Y).cone.\u03c0.app j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/WellFoundedSet.lean", "full_name": "Set.wellFoundedOn_singleton", "start": [527, 1], "end": [528, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/MaxPowDiv.lean", "full_name": "Nat.maxPowDiv.base_mul_eq_succ", "start": [72, 1], "end": [79, 30], "traced_tactics": [{"tactic": "have : 0 < p := lt_trans (b := 1) (by simp) hp", "annotated_tactic": ["have : 0 < p := <a>lt_trans</a> (b := 1) (by simp) hp", [{"full_name": "lt_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [90, 9], "def_end_pos": [90, 17]}]], "state_before": "p n : \u2115\nhp : 1 < p\nhn : 0 < n\n\u22a2 maxPowDiv p (p * n) = maxPowDiv p n + 1", "state_after": "p n : \u2115\nhp : 1 < p\nhn : 0 < n\nthis : 0 < p\n\u22a2 maxPowDiv p (p * n) = maxPowDiv p n + 1"}, {"tactic": "dsimp [maxPowDiv]", "annotated_tactic": ["dsimp [<a>maxPowDiv</a>]", [{"full_name": "Nat.maxPowDiv", "def_path": "Mathlib/Data/Nat/MaxPowDiv.lean", "def_pos": [30, 5], "def_end_pos": [30, 14]}]], "state_before": "p n : \u2115\nhp : 1 < p\nhn : 0 < n\nthis : 0 < p\n\u22a2 maxPowDiv p (p * n) = maxPowDiv p n + 1", "state_after": "p n : \u2115\nhp : 1 < p\nhn : 0 < n\nthis : 0 < p\n\u22a2 go 0 p (p * n) = go 0 p n + 1"}, {"tactic": "rw [maxPowDiv.go_eq, if_pos, mul_div_right _ this]", "annotated_tactic": ["rw [<a>maxPowDiv.go_eq</a>, <a>if_pos</a>, <a>mul_div_right</a> _ this]", [{"full_name": "Nat.maxPowDiv.go_eq", "def_path": "Mathlib/Data/Nat/MaxPowDiv.lean", "def_pos": [45, 9], "def_end_pos": [45, 14]}, {"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}, {"full_name": "Nat.mul_div_right", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [623, 17], "def_end_pos": [623, 30]}]], "state_before": "p n : \u2115\nhp : 1 < p\nhn : 0 < n\nthis : 0 < p\n\u22a2 go 0 p (p * n) = go 0 p n + 1", "state_after": "p n : \u2115\nhp : 1 < p\nhn : 0 < n\nthis : 0 < p\n\u22a2 go (0 + 1) p n = go 0 p n + 1\n\ncase hc\np n : \u2115\nhp : 1 < p\nhn : 0 < n\nthis : 0 < p\n\u22a2 1 < p \u2227 0 < p * n \u2227 p * n % p = 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "p n : \u2115\nhp : 1 < p\nhn : 0 < n\n\u22a2 0 < 1", "state_after": "no goals"}, {"tactic": "apply go_succ", "annotated_tactic": ["apply <a>go_succ</a>", [{"full_name": "Nat.maxPowDiv.go_succ", "def_path": "Mathlib/Data/Nat/MaxPowDiv.lean", "def_pos": [51, 9], "def_end_pos": [51, 16]}]], "state_before": "p n : \u2115\nhp : 1 < p\nhn : 0 < n\nthis : 0 < p\n\u22a2 go (0 + 1) p n = go 0 p n + 1", "state_after": "no goals"}, {"tactic": "refine \u27e8hp, ?_, by simp\u27e9", "annotated_tactic": ["refine \u27e8hp, ?_, by simp\u27e9", []], "state_before": "case hc\np n : \u2115\nhp : 1 < p\nhn : 0 < n\nthis : 0 < p\n\u22a2 1 < p \u2227 0 < p * n \u2227 p * n % p = 0", "state_after": "case hc\np n : \u2115\nhp : 1 < p\nhn : 0 < n\nthis : 0 < p\n\u22a2 0 < p * n"}, {"tactic": "apply Nat.mul_pos this hn", "annotated_tactic": ["apply <a>Nat.mul_pos</a> this hn", [{"full_name": "Nat.mul_pos", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [455, 19], "def_end_pos": [455, 26]}]], "state_before": "case hc\np n : \u2115\nhp : 1 < p\nhn : 0 < n\nthis : 0 < p\n\u22a2 0 < p * n", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "p n : \u2115\nhp : 1 < p\nhn : 0 < n\nthis : 0 < p\n\u22a2 p * n % p = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Normed/Field/UnitBall.lean", "full_name": "coe_mul_unitClosedBall", "start": [76, 1], "end": [78, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/UpperLower/Basic.lean", "full_name": "upperClosure_empty", "start": [1505, 1], "end": [1506, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/ContinuousFunction/Compact.lean", "full_name": "BoundedContinuousFunction.dist_mkOfCompact", "start": [112, 1], "end": [114, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/CompleteBooleanAlgebra.lean", "full_name": "disjoint_sSup_iff", "start": [261, 1], "end": [262, 54], "traced_tactics": [{"tactic": "simpa only [disjoint_comm] using @sSup_disjoint_iff", "annotated_tactic": ["simpa only [<a>disjoint_comm</a>] using @<a>sSup_disjoint_iff</a>", [{"full_name": "disjoint_comm", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [45, 9], "def_end_pos": [45, 22]}, {"full_name": "sSup_disjoint_iff", "def_path": "Mathlib/Order/CompleteBooleanAlgebra.lean", "def_pos": [257, 9], "def_end_pos": [257, 26]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\n\u03ba : \u03b9 \u2192 Sort w'\ninst\u271d : Frame \u03b1\ns\u271d t : Set \u03b1\na b : \u03b1\ns : Set \u03b1\n\u22a2 Disjoint a (sSup s) \u2194 \u2200 (b : \u03b1), b \u2208 s \u2192 Disjoint a b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/IsomorphismClasses.lean", "full_name": "CategoryTheory.Groupoid.isIsomorphic_iff_nonempty_hom", "start": [63, 1], "end": [65, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Convex/Cone/Dual.lean", "full_name": "ConvexCone.hyperplane_separation_of_nonempty_of_isClosed_of_nmem", "start": [163, 1], "end": [193, 49], "traced_tactics": [{"tactic": "obtain \u27e8z, hzK, infi\u27e9 := exists_norm_eq_iInf_of_complete_convex ne hc.isComplete K.convex b", "annotated_tactic": ["obtain \u27e8z, hzK, infi\u27e9 := <a>exists_norm_eq_iInf_of_complete_convex</a> ne hc.isComplete K.convex b", [{"full_name": "exists_norm_eq_iInf_of_complete_convex", "def_path": "Mathlib/Analysis/InnerProductSpace/Projection.lean", "def_pos": [75, 9], "def_end_pos": [75, 47]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\n\u22a2 \u2203 y, (\u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x y) \u2227 inner y b < 0", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\n\u22a2 \u2203 y, (\u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x y) \u2227 inner y b < 0"}, {"tactic": "have hinner := (norm_eq_iInf_iff_real_inner_le_zero K.convex hzK).1 infi", "annotated_tactic": ["have hinner := (<a>norm_eq_iInf_iff_real_inner_le_zero</a> K.convex hzK).1 infi", [{"full_name": "norm_eq_iInf_iff_real_inner_le_zero", "def_path": "Mathlib/Analysis/InnerProductSpace/Projection.lean", "def_pos": [201, 9], "def_end_pos": [201, 44]}]], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\n\u22a2 \u2203 y, (\u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x y) \u2227 inner y b < 0", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\n\u22a2 \u2203 y, (\u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x y) \u2227 inner y b < 0"}, {"tactic": "use z - b", "annotated_tactic": ["use z - b", []], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\n\u22a2 \u2203 y, (\u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x y) \u2227 inner y b < 0", "state_after": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\n\u22a2 (\u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x (z - b)) \u2227 inner (z - b) b < 0"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\n\u22a2 (\u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x (z - b)) \u2227 inner (z - b) b < 0", "state_after": "case h.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\n\u22a2 \u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x (z - b)\n\ncase h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\n\u22a2 inner (z - b) b < 0"}, {"tactic": "rintro x hxK", "annotated_tactic": ["rintro x hxK", []], "state_before": "case h.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\n\u22a2 \u2200 (x : H), x \u2208 K \u2192 0 \u2264 inner x (z - b)", "state_after": "case h.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nx : H\nhxK : x \u2208 K\n\u22a2 0 \u2264 inner x (z - b)"}, {"tactic": "specialize hinner _ (K.add_mem hxK hzK)", "annotated_tactic": ["specialize hinner _ (K.add_mem hxK hzK)", []], "state_before": "case h.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nx : H\nhxK : x \u2208 K\n\u22a2 0 \u2264 inner x (z - b)", "state_after": "case h.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nx : H\nhxK : x \u2208 K\nhinner : inner (b - z) (x + z - z) \u2264 0\n\u22a2 0 \u2264 inner x (z - b)"}, {"tactic": "rwa [add_sub_cancel, real_inner_comm, \u2190 neg_nonneg, neg_eq_neg_one_mul, \u2190 real_inner_smul_right,\n  neg_smul, one_smul, neg_sub] at hinner", "annotated_tactic": ["rwa [<a>add_sub_cancel</a>, <a>real_inner_comm</a>, \u2190 <a>neg_nonneg</a>, <a>neg_eq_neg_one_mul</a>, \u2190 <a>real_inner_smul_right</a>,\n      <a>neg_smul</a>, <a>one_smul</a>, <a>neg_sub</a>] at hinner", [{"full_name": "add_sub_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [739, 30], "def_end_pos": [739, 44]}, {"full_name": "real_inner_comm", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [442, 9], "def_end_pos": [442, 24]}, {"full_name": "neg_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [671, 24], "def_end_pos": [671, 34]}, {"full_name": "neg_eq_neg_one_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 27]}, {"full_name": "real_inner_smul_right", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [487, 9], "def_end_pos": [487, 30]}, {"full_name": "neg_smul", "def_path": "Mathlib/Algebra/Module/Basic.lean", "def_pos": [278, 9], "def_end_pos": [278, 17]}, {"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}, {"full_name": "neg_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [400, 3], "def_end_pos": [400, 14]}]], "state_before": "case h.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nx : H\nhxK : x \u2208 K\nhinner : inner (b - z) (x + z - z) \u2264 0\n\u22a2 0 \u2264 inner x (z - b)", "state_after": "no goals"}, {"tactic": "have hinner\u2080 := hinner 0 (K.pointed_of_nonempty_of_isClosed ne hc)", "annotated_tactic": ["have hinner\u2080 := hinner 0 (K.pointed_of_nonempty_of_isClosed ne hc)", []], "state_before": "case h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\n\u22a2 inner (z - b) b < 0", "state_after": "case h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : inner (b - z) (0 - z) \u2264 0\n\u22a2 inner (z - b) b < 0"}, {"tactic": "rw [zero_sub, inner_neg_right, Right.neg_nonpos_iff] at hinner\u2080", "annotated_tactic": ["rw [<a>zero_sub</a>, <a>inner_neg_right</a>, <a>Right.neg_nonpos_iff</a>] at hinner\u2080", [{"full_name": "zero_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [317, 3], "def_end_pos": [317, 14]}, {"full_name": "inner_neg_right", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [635, 9], "def_end_pos": [635, 24]}, {"full_name": "Right.neg_nonpos_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [214, 3], "def_end_pos": [214, 14]}]], "state_before": "case h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : inner (b - z) (0 - z) \u2264 0\n\u22a2 inner (z - b) b < 0", "state_after": "case h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\n\u22a2 inner (z - b) b < 0"}, {"tactic": "have hbz : b - z \u2260 0 := by\n  rw [sub_ne_zero]\n  contrapose! hzK\n  rwa [\u2190 hzK]", "annotated_tactic": ["have hbz : b - z \u2260 0 := by\n      rw [<a>sub_ne_zero</a>]\n      contrapose! hzK\n      rwa [\u2190 hzK]", [{"full_name": "sub_ne_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [813, 3], "def_end_pos": [813, 14]}]], "state_before": "case h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\n\u22a2 inner (z - b) b < 0", "state_after": "case h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\nhbz : b - z \u2260 0\n\u22a2 inner (z - b) b < 0"}, {"tactic": "rw [\u2190 neg_zero, lt_neg, \u2190 neg_one_mul, \u2190 real_inner_smul_left, smul_sub, neg_smul, one_smul,\n  neg_smul, neg_sub_neg, one_smul]", "annotated_tactic": ["rw [\u2190 <a>neg_zero</a>, <a>lt_neg</a>, \u2190 <a>neg_one_mul</a>, \u2190 <a>real_inner_smul_left</a>, <a>smul_sub</a>, <a>neg_smul</a>, <a>one_smul</a>,\n      <a>neg_smul</a>, <a>neg_sub_neg</a>, <a>one_smul</a>]", [{"full_name": "neg_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1014, 3], "def_end_pos": [1014, 14]}, {"full_name": "lt_neg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [393, 15], "def_end_pos": [393, 21]}, {"full_name": "neg_one_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [328, 9], "def_end_pos": [328, 20]}, {"full_name": "real_inner_smul_left", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [474, 9], "def_end_pos": [474, 29]}, {"full_name": "smul_sub", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [988, 9], "def_end_pos": [988, 17]}, {"full_name": "neg_smul", "def_path": "Mathlib/Algebra/Module/Basic.lean", "def_pos": [278, 9], "def_end_pos": [278, 17]}, {"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}, {"full_name": "neg_smul", "def_path": "Mathlib/Algebra/Module/Basic.lean", "def_pos": [278, 9], "def_end_pos": [278, 17]}, {"full_name": "neg_sub_neg", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [506, 3], "def_end_pos": [506, 14]}, {"full_name": "one_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [492, 9], "def_end_pos": [492, 17]}]], "state_before": "case h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\nhbz : b - z \u2260 0\n\u22a2 inner (z - b) b < 0", "state_after": "case h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\nhbz : b - z \u2260 0\n\u22a2 0 < inner (b - z) b"}, {"tactic": "calc\n  0 < \u27eab - z, b - z\u27eb_\u211d := lt_of_not_le ((Iff.not real_inner_self_nonpos).2 hbz)\n  _ = \u27eab - z, b - z\u27eb_\u211d + 0 := (add_zero _).symm\n  _ \u2264 \u27eab - z, b - z\u27eb_\u211d + \u27eab - z, z\u27eb_\u211d := (add_le_add rfl.ge hinner\u2080)\n  _ = \u27eab - z, b - z + z\u27eb_\u211d := (inner_add_right _ _ _).symm\n  _ = \u27eab - z, b\u27eb_\u211d := by rw [sub_add_cancel]", "annotated_tactic": ["calc\n      0 < \u27eab - z, b - z\u27eb_\u211d := <a>lt_of_not_le</a> ((<a>Iff.not</a> <a>real_inner_self_nonpos</a>).2 hbz)\n      _ = \u27eab - z, b - z\u27eb_\u211d + 0 := (<a>add_zero</a> _).<a>symm</a>\n      _ \u2264 \u27eab - z, b - z\u27eb_\u211d + \u27eab - z, z\u27eb_\u211d := (<a>add_le_add</a> rfl.ge hinner\u2080)\n      _ = \u27eab - z, b - z + z\u27eb_\u211d := (<a>inner_add_right</a> _ _ _).<a>symm</a>\n      _ = \u27eab - z, b\u27eb_\u211d := by rw [<a>sub_add_cancel</a>]", [{"full_name": "lt_of_not_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [488, 9], "def_end_pos": [488, 21]}, {"full_name": "Iff.not", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [274, 7], "def_end_pos": [274, 14]}, {"full_name": "real_inner_self_nonpos", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [621, 9], "def_end_pos": [621, 31]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [205, 15], "def_end_pos": [205, 25]}, {"full_name": "inner_add_right", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [459, 9], "def_end_pos": [459, 24]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "sub_add_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [728, 30], "def_end_pos": [728, 44]}]], "state_before": "case h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\nhbz : b - z \u2260 0\n\u22a2 0 < inner (b - z) b", "state_after": "no goals"}, {"tactic": "rw [sub_ne_zero]", "annotated_tactic": ["rw [<a>sub_ne_zero</a>]", [{"full_name": "sub_ne_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [813, 3], "def_end_pos": [813, 14]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\n\u22a2 b - z \u2260 0", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\n\u22a2 b \u2260 z"}, {"tactic": "contrapose! hzK", "annotated_tactic": ["contrapose! hzK", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\n\u22a2 b \u2260 z", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\nhzK : b = z\n\u22a2 \u00acz \u2208 \u2191K"}, {"tactic": "rwa [\u2190 hzK]", "annotated_tactic": ["rwa [\u2190 hzK]", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\nhzK : b = z\n\u22a2 \u00acz \u2208 \u2191K", "state_after": "no goals"}, {"tactic": "rw [sub_add_cancel]", "annotated_tactic": ["rw [<a>sub_add_cancel</a>]", [{"full_name": "sub_add_cancel", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [728, 30], "def_end_pos": [728, 44]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : InnerProductSpace \u211d H\ns t : Set H\ninst\u271d : CompleteSpace H\nK : ConvexCone \u211d H\nne : Set.Nonempty \u2191K\nhc : IsClosed \u2191K\nb : H\ndisj : \u00acb \u2208 K\nz : H\nhzK : z \u2208 \u2191K\ninfi : \u2016b - z\u2016 = \u2a05 w, \u2016b - \u2191w\u2016\nhinner : \u2200 (w : H), w \u2208 \u2191K \u2192 inner (b - z) (w - z) \u2264 0\nhinner\u2080 : 0 \u2264 inner (b - z) z\nhbz : b - z \u2260 0\n\u22a2 inner (b - z) (b - z + z) = inner (b - z) b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Maps.lean", "full_name": "OpenEmbedding.comp", "start": [642, 11], "end": [644, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Iso.lean", "full_name": "CategoryTheory.Iso.cancel_iso_hom_right", "start": [547, 1], "end": [549, 26], "traced_tactics": [{"tactic": "simp only [cancel_mono]", "annotated_tactic": ["simp only [<a>cancel_mono</a>]", [{"full_name": "CategoryTheory.cancel_mono", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [292, 9], "def_end_pos": [292, 20]}]], "state_before": "C : Type 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[]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Logic/Basic.lean", "full_name": "forall_of_ball", "start": [1070, 1], "end": [1070, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/Matrix/SchurComplement.lean", "full_name": "Matrix.isUnit_fromBlocks_iff_of_invertible\u2081\u2081", "start": [378, 1], "end": [382, 68], "traced_tactics": [{"tactic": "simp only [\u2190 nonempty_invertible_iff_isUnit,\n  (invertibleEquivFromBlocks\u2081\u2081Invertible A B C D).nonempty_congr]", "annotated_tactic": ["simp only [\u2190 <a>nonempty_invertible_iff_isUnit</a>,\n    (<a>invertibleEquivFromBlocks\u2081\u2081Invertible</a> A B C D).<a>nonempty_congr</a>]", [{"full_name": "nonempty_invertible_iff_isUnit", "def_path": "Mathlib/Algebra/Invertible/Basic.lean", "def_pos": [65, 9], "def_end_pos": [65, 39]}, {"full_name": "Matrix.invertibleEquivFromBlocks\u2081\u2081Invertible", "def_path": "Mathlib/LinearAlgebra/Matrix/SchurComplement.lean", "def_pos": [358, 5], "def_end_pos": [358, 42]}, {"full_name": "Equiv.nonempty_congr", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [236, 9], "def_end_pos": [236, 23]}]], "state_before": "l : Type u_1\nm : Type u_2\nn : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u2077 : Fintype l\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : CommRing \u03b1\nA : Matrix m m \u03b1\nB : Matrix m n \u03b1\nC : Matrix n m \u03b1\nD : Matrix n n \u03b1\ninst\u271d : Invertible A\n\u22a2 IsUnit (fromBlocks A B C D) \u2194 IsUnit (D - C * \u215fA * B)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.sum_comm", "start": [2014, 1], "end": [2018, 25], "traced_tactics": [{"tactic": "ext1 s hs", "annotated_tactic": ["ext1 s hs", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\n\u03b9' : Type u_8\n\u03bc : \u03b9 \u2192 \u03b9' \u2192 Measure \u03b1\n\u22a2 (sum fun n => sum (\u03bc n)) = sum fun m => sum fun n => \u03bc n m", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b9' : Type u_8\n\u03bc : \u03b9 \u2192 \u03b9' \u2192 Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(sum fun n => sum (\u03bc n)) s = \u2191\u2191(sum fun m => sum fun n => \u03bc n m) s"}, {"tactic": "simp_rw [sum_apply _ hs]", "annotated_tactic": ["simp_rw [<a>sum_apply</a> _ hs]", [{"full_name": "MeasureTheory.Measure.sum_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1989, 9], "def_end_pos": [1989, 18]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b9' : Type u_8\n\u03bc : \u03b9 \u2192 \u03b9' \u2192 Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(sum fun n => sum (\u03bc n)) s = \u2191\u2191(sum fun m => sum fun n => \u03bc n m) s", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b9' : Type u_8\n\u03bc : \u03b9 \u2192 \u03b9' \u2192 Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2211' (i : \u03b9) (i_1 : \u03b9'), \u2191\u2191(\u03bc i i_1) s = \u2211' (i : \u03b9') (i_1 : \u03b9), \u2191\u2191(\u03bc i_1 i) s"}, {"tactic": "rw [ENNReal.tsum_comm]", "annotated_tactic": ["rw [<a>ENNReal.tsum_comm</a>]", [{"full_name": "ENNReal.tsum_comm", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [819, 19], "def_end_pos": [819, 28]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t : Set \u03b1\n\u03b9' : Type u_8\n\u03bc : \u03b9 \u2192 \u03b9' \u2192 Measure \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2211' (i : \u03b9) (i_1 : \u03b9'), \u2191\u2191(\u03bc i i_1) s = \u2211' (i : \u03b9') (i_1 : \u03b9), \u2191\u2191(\u03bc i_1 i) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": 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ha_lim hb_lim hua hva hub hvb", [{"full_name": "intervalIntegral.integral_const", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [638, 9], "def_end_pos": [638, 23]}, {"full_name": "intervalIntegral.measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae", "def_path": "Mathlib/MeasureTheory/Integral/FundThmCalculus.lean", "def_pos": [408, 9], "def_end_pos": [408, 73]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : CompleteSpace E\ninst\u271d\u00b2 : NormedSpace \u211d E\nf : \u211d \u2192 E\nc ca cb : E\nl l' la la' lb lb' : Filter \u211d\nlt : Filter \u03b9\na b z : \u211d\nu v ua ub va vb : \u03b9 \u2192 \u211d\ninst\u271d\u00b9 : FTCFilter a la la'\ninst\u271d : FTCFilter b lb lb'\nhab : IntervalIntegrable f volume a b\nhmeas_a : StronglyMeasurableAtFilter f la'\nhmeas_b : StronglyMeasurableAtFilter f lb'\nha_lim : Tendsto f (la' \u2293 Measure.ae volume) (\ud835\udcdd ca)\nhb_lim : Tendsto f (lb' \u2293 Measure.ae volume) (\ud835\udcdd cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\n\u22a2 (fun t =>\n      ((\u222b (x : \u211d) in va t..vb t, f x) - \u222b (x : \u211d) in ua t..ub t, f x) -\n        ((vb t - ub t) \u2022 cb - (va t - ua t) \u2022 ca)) =o[lt]\n    fun t => \u2016va t - ua t\u2016 + \u2016vb t - ub t\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Instances.lean", "full_name": "Set.Ioc.coe_pow", "start": [301, 1], "end": [302, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "full_name": 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F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\ninst\u271d\u00b2 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : finrank \ud835\udd5c E = n\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\na : Fin n\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\n\u22a2 \u2191(subordinateOrthonormalBasis hn hV hV') a \u2208 V (subordinateOrthonormalBasisIndex hn hV a hV')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/RingQuot.lean", "full_name": "RingQuot.ringQuot_ext'", "start": [640, 1], "end": [644, 29], "traced_tactics": [{"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "R : Type uR\ninst\u271d\u2076 : Semiring R\nS : Type uS\ninst\u271d\u2075 : CommSemiring S\nT : Type uT\nA : Type uA\ninst\u271d\u2074 : Semiring A\ninst\u271d\u00b3 : Algebra S A\nr : R \u2192 R \u2192 Prop\ninst\u271d\u00b2 : Semiring T\nB : Type u\u2084\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra S B\ns : A \u2192 A \u2192 Prop\nf g : RingQuot s \u2192\u2090[S] B\nw : AlgHom.comp f (mkAlgHom S s) = AlgHom.comp g (mkAlgHom S s)\n\u22a2 f = g", "state_after": "case H\nR : Type uR\ninst\u271d\u2076 : Semiring R\nS : Type uS\ninst\u271d\u2075 : CommSemiring S\nT : Type uT\nA : Type uA\ninst\u271d\u2074 : Semiring A\ninst\u271d\u00b3 : Algebra S A\nr : R \u2192 R \u2192 Prop\ninst\u271d\u00b2 : Semiring T\nB : Type u\u2084\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra S B\ns : A \u2192 A \u2192 Prop\nf g : RingQuot s \u2192\u2090[S] B\nw : AlgHom.comp f (mkAlgHom S s) = AlgHom.comp g (mkAlgHom S s)\nx : RingQuot s\n\u22a2 \u2191f x = \u2191g x"}, {"tactic": "rcases mkAlgHom_surjective S s x with \u27e8x, rfl\u27e9", "annotated_tactic": ["rcases <a>mkAlgHom_surjective</a> S s x with \u27e8x, rfl\u27e9", [{"full_name": "RingQuot.mkAlgHom_surjective", "def_path": "Mathlib/Algebra/RingQuot.lean", "def_pos": [630, 9], "def_end_pos": [630, 28]}]], "state_before": "case H\nR : Type uR\ninst\u271d\u2076 : Semiring R\nS : Type uS\ninst\u271d\u2075 : CommSemiring S\nT : Type uT\nA : Type uA\ninst\u271d\u2074 : Semiring A\ninst\u271d\u00b3 : Algebra S A\nr : R \u2192 R \u2192 Prop\ninst\u271d\u00b2 : Semiring T\nB : Type u\u2084\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra S B\ns : A \u2192 A \u2192 Prop\nf g : RingQuot s \u2192\u2090[S] B\nw : AlgHom.comp f (mkAlgHom S s) = AlgHom.comp g (mkAlgHom S s)\nx : RingQuot s\n\u22a2 \u2191f x = \u2191g x", "state_after": "case H.intro\nR : Type uR\ninst\u271d\u2076 : Semiring R\nS : Type uS\ninst\u271d\u2075 : CommSemiring S\nT : Type uT\nA : Type uA\ninst\u271d\u2074 : Semiring A\ninst\u271d\u00b3 : Algebra S A\nr : R \u2192 R \u2192 Prop\ninst\u271d\u00b2 : Semiring T\nB : Type u\u2084\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra S B\ns : A \u2192 A \u2192 Prop\nf g : RingQuot s \u2192\u2090[S] B\nw : AlgHom.comp f (mkAlgHom S s) = AlgHom.comp g (mkAlgHom S s)\nx : A\n\u22a2 \u2191f (\u2191(mkAlgHom S s) x) = \u2191g (\u2191(mkAlgHom S s) x)"}, {"tactic": "exact AlgHom.congr_fun w x", "annotated_tactic": ["exact <a>AlgHom.congr_fun</a> w x", [{"full_name": "AlgHom.congr_fun", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [213, 19], "def_end_pos": [213, 28]}]], "state_before": "case H.intro\nR : Type uR\ninst\u271d\u2076 : Semiring R\nS : Type uS\ninst\u271d\u2075 : CommSemiring S\nT : Type uT\nA : Type uA\ninst\u271d\u2074 : Semiring A\ninst\u271d\u00b3 : Algebra S A\nr : R \u2192 R \u2192 Prop\ninst\u271d\u00b2 : Semiring T\nB : Type u\u2084\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra S B\ns : A \u2192 A \u2192 Prop\nf g : RingQuot s \u2192\u2090[S] B\nw : AlgHom.comp f (mkAlgHom S s) = AlgHom.comp g (mkAlgHom S s)\nx : A\n\u22a2 \u2191f (\u2191(mkAlgHom S s) x) = \u2191g (\u2191(mkAlgHom S s) x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Monoidal/FunctorCategory.lean", "full_name": "CategoryTheory.Monoidal.rightUnitor_inv_app", "start": [142, 1], "end": [144, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Limits/Preserves/Shapes/Equalizers.lean", "full_name": "CategoryTheory.Limits.map_\u03c0_preserves_coequalizer_inv", "start": [202, 1], "end": [206, 13], "traced_tactics": [{"tactic": "rw [\u2190 \u03b9_comp_coequalizerComparison_assoc, \u2190 PreservesCoequalizer.iso_hom, Iso.hom_inv_id,\n  comp_id]", "annotated_tactic": ["rw [\u2190 <a>\u03b9_comp_coequalizerComparison_assoc</a>, \u2190 <a>PreservesCoequalizer.iso_hom</a>, <a>Iso.hom_inv_id</a>,\n    <a>comp_id</a>]", [{"full_name": "CategoryTheory.Limits.\u03b9_comp_coequalizerComparison_assoc", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [1144, 3], "def_end_pos": [1144, 25]}, {"full_name": "CategoryTheory.Limits.PreservesCoequalizer.iso_hom", "def_path": "Mathlib/CategoryTheory/Limits/Preserves/Shapes/Equalizers.lean", "def_pos": [184, 9], "def_end_pos": [184, 37]}, {"full_name": "CategoryTheory.Iso.hom_inv_id", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [57, 3], "def_end_pos": [57, 13]}, {"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [157, 3], "def_end_pos": [157, 10]}]], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y Z : C\nf g : X \u27f6 Y\nh : Y \u27f6 Z\nw : f \u226b h = g \u226b h\ninst\u271d\u00b2 : HasCoequalizer f g\ninst\u271d\u00b9 : HasCoequalizer (G.map f) (G.map g)\ninst\u271d : PreservesColimit (parallelPair f g) G\n\u22a2 G.map (coequalizer.\u03c0 f g) \u226b (PreservesCoequalizer.iso G f g).inv = coequalizer.\u03c0 (G.map f) (G.map g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Hom/GroupInstances.lean", "full_name": "AddMonoidHom.mul_apply", "start": [309, 1], "end": [310, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/FreeGroup/Basic.lean", "full_name": "FreeGroup.Red.cons_cons_iff", "start": [252, 1], "end": [272, 14], "traced_tactics": [{"tactic": "generalize eq\u2081 : (p :: L\u2081 : List _) = LL\u2081", "annotated_tactic": ["generalize eq\u2081 : (p :: L\u2081 : <a>List</a> _) = LL\u2081", [{"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}]], "state_before": "\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\n\u22a2 Red (p :: L\u2081) (p :: L\u2082) \u2192 Red L\u2081 L\u2082", "state_after": "\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nLL\u2081 : List (\u03b1 \u00d7 Bool)\neq\u2081 : p :: L\u2081 = LL\u2081\n\u22a2 Red LL\u2081 (p :: L\u2082) \u2192 Red L\u2081 L\u2082"}, {"tactic": "generalize eq\u2082 : (p :: L\u2082 : List _) = LL\u2082", "annotated_tactic": ["generalize eq\u2082 : (p :: L\u2082 : <a>List</a> _) = LL\u2082", [{"full_name": "List", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2182, 11], "def_end_pos": [2182, 15]}]], "state_before": "\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nLL\u2081 : List (\u03b1 \u00d7 Bool)\neq\u2081 : p :: L\u2081 = LL\u2081\n\u22a2 Red LL\u2081 (p :: L\u2082) \u2192 Red L\u2081 L\u2082", "state_after": "\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nLL\u2081 : List (\u03b1 \u00d7 Bool)\neq\u2081 : p :: L\u2081 = LL\u2081\nLL\u2082 : List (\u03b1 \u00d7 Bool)\neq\u2082 : p :: L\u2082 = LL\u2082\n\u22a2 Red LL\u2081 LL\u2082 \u2192 Red L\u2081 L\u2082"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nLL\u2081 : List (\u03b1 \u00d7 Bool)\neq\u2081 : p :: L\u2081 = LL\u2081\nLL\u2082 : List (\u03b1 \u00d7 Bool)\neq\u2082 : p :: L\u2082 = LL\u2082\n\u22a2 Red LL\u2081 LL\u2082 \u2192 Red L\u2081 L\u2082", "state_after": "\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nLL\u2081 : List (\u03b1 \u00d7 Bool)\neq\u2081 : p :: L\u2081 = LL\u2081\nLL\u2082 : List (\u03b1 \u00d7 Bool)\neq\u2082 : p :: L\u2082 = LL\u2082\nh : Red LL\u2081 LL\u2082\n\u22a2 Red L\u2081 L\u2082"}, {"tactic": "induction' h using Relation.ReflTransGen.head_induction_on\n  with L\u2081 L\u2082 h\u2081\u2082 h ih\n  generalizing L\u2081 L\u2082", "annotated_tactic": ["induction' h using <a>Relation.ReflTransGen.head_induction_on</a>\n        with L\u2081 L\u2082 h\u2081\u2082 h ih\n        generalizing L\u2081 L\u2082", [{"full_name": "Relation.ReflTransGen.head_induction_on", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [298, 9], "def_end_pos": [298, 26]}]], "state_before": "\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nLL\u2081 : List (\u03b1 \u00d7 Bool)\neq\u2081 : p :: L\u2081 = LL\u2081\nLL\u2082 : List (\u03b1 \u00d7 Bool)\neq\u2082 : p :: L\u2082 = LL\u2082\nh : Red LL\u2081 LL\u2082\n\u22a2 Red L\u2081 L\u2082", "state_after": "case refl\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nLL\u2081 : List (\u03b1 \u00d7 Bool)\neq\u2081\u271d : p :: L\u2081\u271d = LL\u2081\nLL\u2082 : List (\u03b1 \u00d7 Bool)\neq\u2082\u271d : p :: L\u2082\u271d = LL\u2082\nL\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)\neq\u2081 : p :: L\u2081 = LL\u2082\neq\u2082 : p :: L\u2082 = LL\u2082\n\u22a2 Red L\u2081 L\u2082\n\ncase head\n\u03b1 : Type u\nL L\u2081\u271d\u00b9 L\u2082\u271d\u00b9 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nLL\u2081 : List (\u03b1 \u00d7 Bool)\neq\u2081\u271d : p :: L\u2081\u271d\u00b9 = LL\u2081\nLL\u2082 : List (\u03b1 \u00d7 Bool)\neq\u2082\u271d : p :: L\u2082\u271d\u00b9 = LL\u2082\nL\u2081\u271d L\u2082\u271d : List (\u03b1 \u00d7 Bool)\nh\u2081\u2082 : Step L\u2081\u271d L\u2082\u271d\nh : ReflTransGen Step L\u2082\u271d LL\u2082\nih : \u2200 {L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)}, p :: L\u2081 = L\u2082\u271d \u2192 p :: L\u2082 = LL\u2082 \u2192 Red L\u2081 L\u2082\nL\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)\neq\u2081 : p :: L\u2081 = L\u2081\u271d\neq\u2082 : p :: L\u2082 = LL\u2082\n\u22a2 Red L\u2081 L\u2082"}, {"tactic": "subst_vars", "annotated_tactic": ["subst_vars", []], "state_before": "case refl\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nLL\u2081 : List (\u03b1 \u00d7 Bool)\neq\u2081\u271d : p :: L\u2081\u271d = LL\u2081\nLL\u2082 : List (\u03b1 \u00d7 Bool)\neq\u2082\u271d : p :: L\u2082\u271d = LL\u2082\nL\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)\neq\u2081 : p :: L\u2081 = LL\u2082\neq\u2082 : p :: L\u2082 = LL\u2082\n\u22a2 Red L\u2081 L\u2082", "state_after": "case refl\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nL\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)\neq\u2081 : p :: L\u2081 = p :: L\u2082\u271d\neq\u2082 : p :: L\u2082 = p :: L\u2082\u271d\n\u22a2 Red L\u2081 L\u2082"}, {"tactic": "cases eq\u2082", "annotated_tactic": ["cases eq\u2082", []], "state_before": "case refl\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nL\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)\neq\u2081 : p :: L\u2081 = p :: L\u2082\u271d\neq\u2082 : p :: L\u2082 = p :: L\u2082\u271d\n\u22a2 Red L\u2081 L\u2082", "state_after": "case refl.refl\n\u03b1 : Type u\nL L\u2081\u271d L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nL\u2081 : List (\u03b1 \u00d7 Bool)\neq\u2081 : p :: L\u2081 = p :: L\u2082\n\u22a2 Red L\u2081 L\u2082"}, {"tactic": "cases eq\u2081", "annotated_tactic": ["cases eq\u2081", []], "state_before": "case refl.refl\n\u03b1 : Type u\nL L\u2081\u271d L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nL\u2081 : List (\u03b1 \u00d7 Bool)\neq\u2081 : p :: L\u2081 = p :: L\u2082\n\u22a2 Red L\u2081 L\u2082", "state_after": "case refl.refl.refl\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\n\u22a2 Red L\u2082 L\u2082"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case refl.refl.refl\n\u03b1 : Type u\nL L\u2081 L\u2082 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\n\u22a2 Red L\u2082 L\u2082", "state_after": "no goals"}, {"tactic": "subst_vars", "annotated_tactic": ["subst_vars", []], "state_before": "case head\n\u03b1 : Type u\nL L\u2081\u271d\u00b9 L\u2082\u271d\u00b9 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nLL\u2081 : List (\u03b1 \u00d7 Bool)\neq\u2081\u271d : p :: L\u2081\u271d\u00b9 = LL\u2081\nLL\u2082 : List (\u03b1 \u00d7 Bool)\neq\u2082\u271d : p :: L\u2082\u271d\u00b9 = LL\u2082\nL\u2081\u271d L\u2082\u271d : List (\u03b1 \u00d7 Bool)\nh\u2081\u2082 : Step L\u2081\u271d L\u2082\u271d\nh : ReflTransGen Step L\u2082\u271d LL\u2082\nih : \u2200 {L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)}, p :: L\u2081 = L\u2082\u271d \u2192 p :: L\u2082 = LL\u2082 \u2192 Red L\u2081 L\u2082\nL\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)\neq\u2081 : p :: L\u2081 = L\u2081\u271d\neq\u2082 : p :: L\u2082 = LL\u2082\n\u22a2 Red L\u2081 L\u2082", "state_after": "case head\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d\u00b9 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nL\u2082\u271d L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)\nh : ReflTransGen Step L\u2082\u271d (p :: L\u2082\u271d\u00b9)\nih : \u2200 {L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)}, p :: L\u2081 = L\u2082\u271d \u2192 p :: L\u2082 = p :: L\u2082\u271d\u00b9 \u2192 Red L\u2081 L\u2082\neq\u2082 : p :: L\u2082 = p :: L\u2082\u271d\u00b9\nh\u2081\u2082 : Step (p :: L\u2081) L\u2082\u271d\n\u22a2 Red L\u2081 L\u2082"}, {"tactic": "cases eq\u2082", "annotated_tactic": ["cases eq\u2082", []], "state_before": "case head\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d\u00b9 L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nL\u2082\u271d L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)\nh : ReflTransGen Step L\u2082\u271d (p :: L\u2082\u271d\u00b9)\nih : \u2200 {L\u2081 L\u2082 : List (\u03b1 \u00d7 Bool)}, p :: L\u2081 = L\u2082\u271d \u2192 p :: L\u2082 = p :: L\u2082\u271d\u00b9 \u2192 Red L\u2081 L\u2082\neq\u2082 : p :: L\u2082 = p :: L\u2082\u271d\u00b9\nh\u2081\u2082 : Step (p :: L\u2081) L\u2082\u271d\n\u22a2 Red L\u2081 L\u2082", "state_after": "case head.refl\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nL\u2082 L\u2081 : List (\u03b1 \u00d7 Bool)\nh : ReflTransGen Step L\u2082 (p :: L\u2082\u271d)\nih : \u2200 {L\u2081 L\u2082_1 : List (\u03b1 \u00d7 Bool)}, p :: L\u2081 = L\u2082 \u2192 p :: L\u2082_1 = p :: L\u2082\u271d \u2192 Red L\u2081 L\u2082_1\nh\u2081\u2082 : Step (p :: L\u2081) L\u2082\n\u22a2 Red L\u2081 L\u2082\u271d"}, {"tactic": "cases' p with a b", "annotated_tactic": ["cases' p with a b", []], "state_before": "case head.refl\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 : List (\u03b1 \u00d7 Bool)\np : \u03b1 \u00d7 Bool\nL\u2082 L\u2081 : List (\u03b1 \u00d7 Bool)\nh : ReflTransGen Step L\u2082 (p :: L\u2082\u271d)\nih : \u2200 {L\u2081 L\u2082_1 : List (\u03b1 \u00d7 Bool)}, p :: L\u2081 = L\u2082 \u2192 p :: L\u2082_1 = p :: L\u2082\u271d \u2192 Red L\u2081 L\u2082_1\nh\u2081\u2082 : Step (p :: L\u2081) L\u2082\n\u22a2 Red L\u2081 L\u2082\u271d", "state_after": "case head.refl.mk\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 L\u2082 L\u2081 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\nh : ReflTransGen Step L\u2082 ((a, b) :: L\u2082\u271d)\nih : \u2200 {L\u2081 L\u2082_1 : List (\u03b1 \u00d7 Bool)}, (a, b) :: L\u2081 = L\u2082 \u2192 (a, b) :: L\u2082_1 = (a, b) :: L\u2082\u271d \u2192 Red L\u2081 L\u2082_1\nh\u2081\u2082 : Step ((a, b) :: L\u2081) L\u2082\n\u22a2 Red L\u2081 L\u2082\u271d"}, {"tactic": "rw [Step.cons_left_iff] at h\u2081\u2082", "annotated_tactic": ["rw [<a>Step.cons_left_iff</a>] at h\u2081\u2082", [{"full_name": "FreeGroup.Red.Step.cons_left_iff", "def_path": "Mathlib/GroupTheory/FreeGroup/Basic.lean", "def_pos": [160, 9], "def_end_pos": [160, 27]}]], "state_before": "case head.refl.mk\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 L\u2082 L\u2081 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\nh : ReflTransGen Step L\u2082 ((a, b) :: L\u2082\u271d)\nih : \u2200 {L\u2081 L\u2082_1 : List (\u03b1 \u00d7 Bool)}, (a, b) :: L\u2081 = L\u2082 \u2192 (a, b) :: L\u2082_1 = (a, b) :: L\u2082\u271d \u2192 Red L\u2081 L\u2082_1\nh\u2081\u2082 : Step ((a, b) :: L\u2081) L\u2082\n\u22a2 Red L\u2081 L\u2082\u271d", "state_after": "case head.refl.mk\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 L\u2082 L\u2081 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\nh : ReflTransGen Step L\u2082 ((a, b) :: L\u2082\u271d)\nih : \u2200 {L\u2081 L\u2082_1 : List (\u03b1 \u00d7 Bool)}, (a, b) :: L\u2081 = L\u2082 \u2192 (a, b) :: L\u2082_1 = (a, b) :: L\u2082\u271d \u2192 Red L\u2081 L\u2082_1\nh\u2081\u2082 : (\u2203 L, Step L\u2081 L \u2227 L\u2082 = (a, b) :: L) \u2228 L\u2081 = (a, !b) :: L\u2082\n\u22a2 Red L\u2081 L\u2082\u271d"}, {"tactic": "rcases h\u2081\u2082 with (\u27e8L, h\u2081\u2082, rfl\u27e9 | rfl)", "annotated_tactic": ["rcases h\u2081\u2082 with (\u27e8L, h\u2081\u2082, rfl\u27e9 | rfl)", []], "state_before": "case head.refl.mk\n\u03b1 : Type u\nL L\u2081\u271d L\u2082\u271d L\u2083 L\u2084 L\u2082 L\u2081 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\nh : ReflTransGen Step L\u2082 ((a, b) :: L\u2082\u271d)\nih : \u2200 {L\u2081 L\u2082_1 : List (\u03b1 \u00d7 Bool)}, (a, b) :: L\u2081 = L\u2082 \u2192 (a, b) :: L\u2082_1 = (a, b) :: L\u2082\u271d \u2192 Red L\u2081 L\u2082_1\nh\u2081\u2082 : (\u2203 L, Step L\u2081 L \u2227 L\u2082 = (a, b) :: L) \u2228 L\u2081 = (a, !b) :: L\u2082\n\u22a2 Red L\u2081 L\u2082\u271d", "state_after": "case head.refl.mk.inl.intro.intro\n\u03b1 : Type u\nL\u271d L\u2081\u271d L\u2082 L\u2083 L\u2084 L\u2081 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\nL : List (\u03b1 \u00d7 Bool)\nh\u2081\u2082 : Step L\u2081 L\nh : ReflTransGen Step ((a, b) :: L) ((a, b) :: L\u2082)\nih : \u2200 {L\u2081 L\u2082_1 : List (\u03b1 \u00d7 Bool)}, (a, b) :: L\u2081 = (a, b) :: L \u2192 (a, b) :: L\u2082_1 = (a, b) :: L\u2082 \u2192 Red L\u2081 L\u2082_1\n\u22a2 Red L\u2081 L\u2082\n\ncase head.refl.mk.inr\n\u03b1 : Type u\nL L\u2081 L\u2082\u271d L\u2083 L\u2084 L\u2082 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\nh : ReflTransGen Step L\u2082 ((a, b) :: L\u2082\u271d)\nih : \u2200 {L\u2081 L\u2082_1 : List (\u03b1 \u00d7 Bool)}, (a, b) :: L\u2081 = L\u2082 \u2192 (a, b) :: L\u2082_1 = (a, b) :: L\u2082\u271d \u2192 Red L\u2081 L\u2082_1\n\u22a2 Red ((a, !b) :: L\u2082) L\u2082\u271d"}, {"tactic": "exact (ih rfl rfl).head h\u2081\u2082", "annotated_tactic": ["exact (ih <a>rfl</a> <a>rfl</a>).<a>head</a> h\u2081\u2082", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Relation.ReflTransGen.head", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [280, 9], "def_end_pos": [280, 13]}]], "state_before": "case head.refl.mk.inl.intro.intro\n\u03b1 : Type u\nL\u271d L\u2081\u271d L\u2082 L\u2083 L\u2084 L\u2081 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\nL : List (\u03b1 \u00d7 Bool)\nh\u2081\u2082 : Step L\u2081 L\nh : ReflTransGen Step ((a, b) :: L) ((a, b) :: L\u2082)\nih : \u2200 {L\u2081 L\u2082_1 : List (\u03b1 \u00d7 Bool)}, (a, b) :: L\u2081 = (a, b) :: L \u2192 (a, b) :: L\u2082_1 = (a, b) :: L\u2082 \u2192 Red L\u2081 L\u2082_1\n\u22a2 Red L\u2081 L\u2082", "state_after": "no goals"}, {"tactic": "exact (cons_cons h).tail Step.cons_not_rev", "annotated_tactic": ["exact (<a>cons_cons</a> h).<a>tail</a> <a>Step.cons_not_rev</a>", [{"full_name": "FreeGroup.Red.cons_cons", "def_path": "Mathlib/GroupTheory/FreeGroup/Basic.lean", "def_pos": [246, 9], "def_end_pos": [246, 18]}, {"full_name": "Relation.ReflTransGen.tail", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [224, 5], "def_end_pos": [224, 9]}, {"full_name": "FreeGroup.Red.Step.cons_not_rev", "def_path": "Mathlib/GroupTheory/FreeGroup/Basic.lean", "def_pos": [127, 9], "def_end_pos": [127, 26]}]], "state_before": "case head.refl.mk.inr\n\u03b1 : Type u\nL L\u2081 L\u2082\u271d L\u2083 L\u2084 L\u2082 : List (\u03b1 \u00d7 Bool)\na : \u03b1\nb : Bool\nh : ReflTransGen Step L\u2082 ((a, b) :: L\u2082\u271d)\nih : \u2200 {L\u2081 L\u2082_1 : List (\u03b1 \u00d7 Bool)}, (a, b) :: L\u2081 = L\u2082 \u2192 (a, b) :: L\u2082_1 = (a, b) :: L\u2082\u271d \u2192 Red L\u2081 L\u2082_1\n\u22a2 Red ((a, !b) :: L\u2082) L\u2082\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Closure.lean", "full_name": "ClosureOperator.closure_le_closed_iff_le", "start": [203, 1], "end": [204, 62], "traced_tactics": [{"tactic": "rw [\u2190 c.closure_eq_self_of_mem_closed hy, \u2190 le_closure_iff]", "annotated_tactic": ["rw [\u2190 c.closure_eq_self_of_mem_closed hy, \u2190 <a>le_closure_iff</a>]", [{"full_name": "ClosureOperator.le_closure_iff", "def_path": "Mathlib/Order/Closure.lean", "def_pos": [164, 9], "def_end_pos": [164, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Sort u_2\n\u03ba : \u03b9 \u2192 Sort u_3\ninst\u271d\u00b9 inst\u271d : PartialOrder \u03b1\nc : ClosureOperator \u03b1\nx y : \u03b1\nhy : closed c y\n\u22a2 \u2191c x \u2264 y \u2194 x \u2264 y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean", "full_name": "finite_of_fin_dim_affineIndependent", "start": [81, 1], "end": [87, 100], "traced_tactics": [{"tactic": "nontriviality \u03b9", "annotated_tactic": ["nontriviality \u03b9", []], "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : FiniteDimensional k V\np : \u03b9 \u2192 P\nhi : AffineIndependent k p\n\u22a2 _root_.Finite \u03b9", "state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : FiniteDimensional k V\np : \u03b9 \u2192 P\nhi : AffineIndependent k p\n\u271d : Nontrivial \u03b9\n\u22a2 _root_.Finite \u03b9"}, {"tactic": "inhabit \u03b9", "annotated_tactic": ["inhabit \u03b9", []], "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : FiniteDimensional k V\np : \u03b9 \u2192 P\nhi : AffineIndependent k p\n\u271d : Nontrivial \u03b9\n\u22a2 _root_.Finite \u03b9", "state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : FiniteDimensional k V\np : \u03b9 \u2192 P\nhi : AffineIndependent k p\n\u271d : Nontrivial \u03b9\ninhabited_h : Inhabited \u03b9\n\u22a2 _root_.Finite \u03b9"}, {"tactic": "rw [affineIndependent_iff_linearIndependent_vsub k p default] at hi", "annotated_tactic": ["rw [<a>affineIndependent_iff_linearIndependent_vsub</a> k p <a>default</a>] at hi", [{"full_name": "affineIndependent_iff_linearIndependent_vsub", "def_path": "Mathlib/LinearAlgebra/AffineSpace/Independent.lean", "def_pos": [88, 9], "def_end_pos": [88, 53]}, {"full_name": "Inhabited.default", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [674, 3], "def_end_pos": [674, 10]}]], "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : FiniteDimensional k V\np : \u03b9 \u2192 P\nhi : AffineIndependent k p\n\u271d : Nontrivial \u03b9\ninhabited_h : Inhabited \u03b9\n\u22a2 _root_.Finite \u03b9", "state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : FiniteDimensional k V\np : \u03b9 \u2192 P\n\u271d : Nontrivial \u03b9\ninhabited_h : Inhabited \u03b9\nhi : LinearIndependent k fun i => p \u2191i -\u1d65 p default\n\u22a2 _root_.Finite \u03b9"}, {"tactic": "letI : IsNoetherian k V := IsNoetherian.iff_fg.2 inferInstance", "annotated_tactic": ["letI : <a>IsNoetherian</a> k V := <a>IsNoetherian.iff_fg</a>.2 <a>inferInstance</a>", [{"full_name": "IsNoetherian", "def_path": "Mathlib/RingTheory/Noetherian.lean", "def_pos": [67, 7], "def_end_pos": [67, 19]}, {"full_name": "IsNoetherian.iff_fg", "def_path": "Mathlib/FieldTheory/Finiteness.lean", "def_pos": [109, 9], "def_end_pos": [109, 15]}, {"full_name": "inferInstance", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [86, 8], "def_end_pos": [86, 21]}]], "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : FiniteDimensional k V\np : \u03b9 \u2192 P\n\u271d : Nontrivial \u03b9\ninhabited_h : Inhabited \u03b9\nhi : LinearIndependent k fun i => p \u2191i -\u1d65 p default\n\u22a2 _root_.Finite \u03b9", "state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : FiniteDimensional k V\np : \u03b9 \u2192 P\n\u271d : Nontrivial \u03b9\ninhabited_h : Inhabited \u03b9\nhi : LinearIndependent k fun i => p \u2191i -\u1d65 p default\nthis : IsNoetherian k V := IsNoetherian.iff_fg.mpr inferInstance\n\u22a2 _root_.Finite \u03b9"}, {"tactic": "exact\n  (Set.finite_singleton default).finite_of_compl (Set.finite_coe_iff.1 hi.finite_of_isNoetherian)", "annotated_tactic": ["exact\n    (<a>Set.finite_singleton</a> <a>default</a>).<a>finite_of_compl</a> (<a>Set.finite_coe_iff</a>.1 hi.finite_of_isNoetherian)", [{"full_name": "Set.finite_singleton", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [848, 9], "def_end_pos": [848, 25]}, {"full_name": "Inhabited.default", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [674, 3], "def_end_pos": [674, 10]}, {"full_name": "Set.Finite.finite_of_compl", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [739, 9], "def_end_pos": [739, 31]}, {"full_name": "Set.finite_coe_iff", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [77, 9], "def_end_pos": [77, 23]}]], "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u2074 : DivisionRing k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ninst\u271d : FiniteDimensional k V\np : \u03b9 \u2192 P\n\u271d : Nontrivial \u03b9\ninhabited_h : Inhabited \u03b9\nhi : LinearIndependent k fun i => p \u2191i -\u1d65 p default\nthis : IsNoetherian k V := IsNoetherian.iff_fg.mpr inferInstance\n\u22a2 _root_.Finite \u03b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/RingedSpace/Stalks.lean", "full_name": "AlgebraicGeometry.PresheafedSpace.stalkMap_germ'", "start": [64, 1], "end": [68, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/SimpleGraph/Density.lean", "full_name": "Rel.interedges_biUnion_left", "start": [106, 1], "end": [109, 77], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b2 : (a : \u03b1) \u2192 DecidablePred (r a)\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 : Finset \u03b2\na : \u03b1\nb : \u03b2\n\u03b4 : \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b9\nt : Finset \u03b2\nf : \u03b9 \u2192 Finset \u03b1\n\u22a2 interedges r (Finset.biUnion s f) t = Finset.biUnion s fun a => interedges r (f a) t", "state_after": "case a\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b2 : (a : \u03b1) \u2192 DecidablePred (r a)\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 : Finset \u03b2\na : \u03b1\nb : \u03b2\n\u03b4 : \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b9\nt : Finset \u03b2\nf : \u03b9 \u2192 Finset \u03b1\na\u271d : \u03b1 \u00d7 \u03b2\n\u22a2 a\u271d \u2208 interedges r (Finset.biUnion s f) t \u2194 a\u271d \u2208 Finset.biUnion s fun a => interedges r (f a) t"}, {"tactic": "simp only [mem_biUnion, mem_interedges_iff, exists_and_right, \u2190 and_assoc]", "annotated_tactic": ["simp only [<a>mem_biUnion</a>, <a>mem_interedges_iff</a>, <a>exists_and_right</a>, \u2190 <a>and_assoc</a>]", [{"full_name": "Finset.mem_biUnion", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3613, 9], "def_end_pos": [3613, 20]}, {"full_name": "Rel.mem_interedges_iff", "def_path": "Mathlib/Combinatorics/SimpleGraph/Density.lean", "def_pos": [58, 9], "def_end_pos": [58, 27]}, {"full_name": "exists_and_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [468, 17], "def_end_pos": [468, 33]}, {"full_name": "and_assoc", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [177, 9], "def_end_pos": [177, 18]}]], "state_before": "case a\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03ba : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b3 : LinearOrderedField \ud835\udd5c\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ninst\u271d\u00b2 : (a : \u03b1) \u2192 DecidablePred (r a)\ns\u271d s\u2081 s\u2082 : Finset \u03b1\nt\u271d t\u2081 t\u2082 : Finset \u03b2\na : \u03b1\nb : \u03b2\n\u03b4 : \ud835\udd5c\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\ns : Finset \u03b9\nt : Finset \u03b2\nf : \u03b9 \u2192 Finset \u03b1\na\u271d : \u03b1 \u00d7 \u03b2\n\u22a2 a\u271d \u2208 interedges r (Finset.biUnion s f) t \u2194 a\u271d \u2208 Finset.biUnion s fun a => interedges r (f a) t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Category/ModuleCat/Projective.lean", "full_name": "IsProjective.iff_projective", "start": [33, 1], "end": [43, 82], "traced_tactics": [{"tactic": "refine' \u27e8fun h => _, fun h => _\u27e9", "annotated_tactic": ["refine' \u27e8fun h => _, fun h => _\u27e9", []], "state_before": "R : Type u\ninst\u271d\u00b2 : Ring R\nP : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup P\ninst\u271d : Module R P\n\u22a2 Module.Projective R P \u2194 Projective (of R P)", "state_after": "case refine'_1\nR : Type u\ninst\u271d\u00b2 : Ring R\nP : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup P\ninst\u271d : Module R P\nh : Module.Projective R P\n\u22a2 Projective (of R P)\n\ncase refine'_2\nR : Type u\ninst\u271d\u00b2 : Ring R\nP : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup P\ninst\u271d : Module R P\nh : Projective (of R P)\n\u22a2 Module.Projective R P"}, {"tactic": "letI : Module.Projective R (ModuleCat.of R P) := h", "annotated_tactic": ["letI : <a>Module.Projective</a> R (<a>ModuleCat.of</a> R P) := h", [{"full_name": "Module.Projective", "def_path": "Mathlib/Algebra/Module/Projective.lean", "def_pos": [74, 7], "def_end_pos": [74, 24]}, {"full_name": "ModuleCat.of", "def_path": "Mathlib/Algebra/Category/ModuleCat/Basic.lean", "def_pos": [138, 5], "def_end_pos": [138, 7]}]], "state_before": "case refine'_1\nR : Type u\ninst\u271d\u00b2 : Ring R\nP : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup P\ninst\u271d : Module R P\nh : Module.Projective R P\n\u22a2 Projective (of R P)", "state_after": "case refine'_1\nR : Type u\ninst\u271d\u00b2 : Ring R\nP : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup P\ninst\u271d : Module R P\nh : Module.Projective R P\nthis : Module.Projective R \u2191(of R P) := h\n\u22a2 Projective (of R P)"}, {"tactic": "exact \u27e8fun E X epi => Module.projective_lifting_property _ _\n  ((ModuleCat.epi_iff_surjective _).mp epi)\u27e9", "annotated_tactic": ["exact \u27e8fun E X epi => <a>Module.projective_lifting_property</a> _ _\n      ((<a>ModuleCat.epi_iff_surjective</a> _).<a>mp</a> epi)\u27e9", [{"full_name": "Module.projective_lifting_property", "def_path": "Mathlib/Algebra/Module/Projective.lean", "def_pos": [97, 9], "def_end_pos": [97, 36]}, {"full_name": "ModuleCat.epi_iff_surjective", "def_path": "Mathlib/Algebra/Category/ModuleCat/EpiMono.lean", "def_pos": [56, 9], "def_end_pos": [56, 27]}, {"full_name": "Iff.mp", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [90, 3], "def_end_pos": [90, 5]}]], "state_before": "case refine'_1\nR : Type u\ninst\u271d\u00b2 : Ring R\nP : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup P\ninst\u271d : Module R P\nh : Module.Projective R P\nthis : Module.Projective R \u2191(of R P) := h\n\u22a2 Projective (of R P)", "state_after": "no goals"}, {"tactic": "refine' Module.Projective.of_lifting_property.{u,v} _", "annotated_tactic": ["refine' <a>Module.Projective.of_lifting_property</a>.{u,v} _", [{"full_name": "Module.Projective.of_lifting_property", "def_path": "Mathlib/Algebra/Module/Projective.lean", "def_pos": [191, 9], "def_end_pos": [191, 39]}]], "state_before": "case refine'_2\nR : Type u\ninst\u271d\u00b2 : Ring R\nP : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup P\ninst\u271d : Module R P\nh : Projective (of R P)\n\u22a2 Module.Projective R P", "state_after": "case refine'_2\nR : Type u\ninst\u271d\u00b2 : Ring R\nP : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup P\ninst\u271d : Module R P\nh : Projective (of R P)\n\u22a2 \u2200 {M : Type (max v u)} {N : Type (max u v)} [inst : AddCommGroup M] [inst_1 : AddCommGroup N] [inst_2 : Module R M]\n    [inst_3 : Module R N] (f : M \u2192\u2097[R] N) (g : P \u2192\u2097[R] N), Function.Surjective \u2191f \u2192 \u2203 h, comp f h = g"}, {"tactic": "intro E X mE mX sE sX f g s", "annotated_tactic": ["intro E X mE mX sE sX f g s", []], "state_before": "case refine'_2\nR : Type u\ninst\u271d\u00b2 : Ring R\nP : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup P\ninst\u271d : Module R P\nh : Projective (of R P)\n\u22a2 \u2200 {M : Type (max v u)} {N : Type (max u v)} [inst : AddCommGroup M] [inst_1 : AddCommGroup N] [inst_2 : Module R M]\n    [inst_3 : Module R N] (f : M \u2192\u2097[R] N) (g : P \u2192\u2097[R] N), Function.Surjective \u2191f \u2192 \u2203 h, comp f h = g", "state_after": "case refine'_2\nR : Type u\ninst\u271d\u00b2 : Ring R\nP : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup P\ninst\u271d : Module R P\nh : Projective (of R P)\nE : Type (max v u)\nX : Type (max u v)\nmE : AddCommGroup E\nmX : AddCommGroup X\nsE : Module R E\nsX : Module R X\nf : E \u2192\u2097[R] X\ng : P \u2192\u2097[R] X\ns : Function.Surjective \u2191f\n\u22a2 \u2203 h, comp f h = g"}, {"tactic": "haveI : Epi (\u219ff) := (ModuleCat.epi_iff_surjective (\u219ff)).mpr s", "annotated_tactic": ["haveI : <a>Epi</a> (\u219ff) := (<a>ModuleCat.epi_iff_surjective</a> (\u219ff)).<a>mpr</a> s", [{"full_name": "CategoryTheory.Epi", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [267, 7], "def_end_pos": [267, 10]}, {"full_name": "ModuleCat.epi_iff_surjective", "def_path": "Mathlib/Algebra/Category/ModuleCat/EpiMono.lean", "def_pos": [56, 9], "def_end_pos": [56, 27]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "case refine'_2\nR : Type u\ninst\u271d\u00b2 : Ring R\nP : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup P\ninst\u271d : Module R P\nh : Projective (of R P)\nE : Type (max v u)\nX : Type (max u v)\nmE : AddCommGroup E\nmX : AddCommGroup X\nsE : Module R E\nsX : Module R X\nf : E \u2192\u2097[R] X\ng : P \u2192\u2097[R] X\ns : Function.Surjective \u2191f\n\u22a2 \u2203 h, comp f h = g", "state_after": "case refine'_2\nR : Type u\ninst\u271d\u00b2 : Ring R\nP : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup P\ninst\u271d : Module R P\nh : Projective (of R P)\nE : Type (max v u)\nX : Type (max u v)\nmE : AddCommGroup E\nmX : AddCommGroup X\nsE : Module R E\nsX : Module R X\nf : E \u2192\u2097[R] X\ng : P \u2192\u2097[R] X\ns : Function.Surjective \u2191f\nthis : Epi (\u219ff)\n\u22a2 \u2203 h, comp f h = g"}, {"tactic": "letI : Projective (ModuleCat.of R P) := h", "annotated_tactic": ["letI : <a>Projective</a> (<a>ModuleCat.of</a> R P) := h", [{"full_name": "CategoryTheory.Projective", "def_path": "Mathlib/CategoryTheory/Preadditive/Projective.lean", "def_pos": [45, 7], "def_end_pos": [45, 17]}, {"full_name": "ModuleCat.of", "def_path": "Mathlib/Algebra/Category/ModuleCat/Basic.lean", "def_pos": [138, 5], "def_end_pos": [138, 7]}]], "state_before": "case refine'_2\nR : Type u\ninst\u271d\u00b2 : Ring R\nP : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup P\ninst\u271d : Module R P\nh : Projective (of R P)\nE : Type (max v u)\nX : Type (max u v)\nmE : AddCommGroup E\nmX : AddCommGroup X\nsE : Module R E\nsX : Module R X\nf : E \u2192\u2097[R] X\ng : P \u2192\u2097[R] X\ns : Function.Surjective \u2191f\nthis : Epi (\u219ff)\n\u22a2 \u2203 h, comp f h = g", "state_after": "case refine'_2\nR : Type u\ninst\u271d\u00b2 : Ring R\nP : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup P\ninst\u271d : Module R P\nh : Projective (of R P)\nE : Type (max v u)\nX : Type (max u v)\nmE : AddCommGroup E\nmX : AddCommGroup X\nsE : Module R E\nsX : Module R X\nf : E \u2192\u2097[R] X\ng : P \u2192\u2097[R] X\ns : Function.Surjective \u2191f\nthis\u271d : Epi (\u219ff)\nthis : Projective (of R P) := h\n\u22a2 \u2203 h, comp f h = g"}, {"tactic": "exact \u27e8Projective.factorThru (\u219fg) (\u219ff), Projective.factorThru_comp (\u219fg) (\u219ff)\u27e9", "annotated_tactic": ["exact \u27e8<a>Projective.factorThru</a> (\u219fg) (\u219ff), <a>Projective.factorThru_comp</a> (\u219fg) (\u219ff)\u27e9", [{"full_name": "CategoryTheory.Projective.factorThru", "def_path": "Mathlib/CategoryTheory/Preadditive/Projective.lean", "def_pos": [79, 5], "def_end_pos": [79, 15]}, {"full_name": "CategoryTheory.Projective.factorThru_comp", "def_path": "Mathlib/CategoryTheory/Preadditive/Projective.lean", "def_pos": [84, 9], "def_end_pos": [84, 24]}]], "state_before": "case refine'_2\nR : Type u\ninst\u271d\u00b2 : Ring R\nP : Type (max u v)\ninst\u271d\u00b9 : AddCommGroup P\ninst\u271d : Module R P\nh : Projective (of R P)\nE : Type (max v u)\nX : Type (max u v)\nmE : AddCommGroup E\nmX : AddCommGroup X\nsE : Module R E\nsX : Module R X\nf : E \u2192\u2097[R] X\ng : P \u2192\u2097[R] X\ns : Function.Surjective \u2191f\nthis\u271d : Epi (\u219ff)\nthis : Projective (of R P) := h\n\u22a2 \u2203 h, comp f h = g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean", "full_name": "CategoryTheory.Limits.biproduct.lift_map", "start": [640, 1], "end": [643, 12], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "J : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf g : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nP : C\nk : (j : J) \u2192 P \u27f6 f j\np : (j : J) \u2192 f j \u27f6 g j\n\u22a2 lift k \u226b map p = lift fun j => k j \u226b p j", "state_after": "case w\nJ : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf g : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nP : C\nk : (j : J) \u2192 P \u27f6 f j\np : (j : J) \u2192 f j \u27f6 g j\nj\u271d : J\n\u22a2 (lift k \u226b map p) \u226b \u03c0 (fun b => g b) j\u271d = (lift fun j => k j \u226b p j) \u226b \u03c0 (fun b => g b) j\u271d"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case w\nJ : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf g : J \u2192 C\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nP : C\nk : (j : J) \u2192 P \u27f6 f j\np : (j : J) \u2192 f j \u27f6 g j\nj\u271d : J\n\u22a2 (lift k \u226b map p) \u226b \u03c0 (fun b => g b) j\u271d = (lift fun j => k j \u226b p j) \u226b \u03c0 (fun b => g b) j\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "full_name": "ContinuousLinearEquiv.equivOfInverse_apply", "start": [2253, 1], "end": [2255, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Bases.lean", "full_name": "TopologicalSpace.IsTopologicalBasis.sigma", "start": [841, 1], "end": [855, 31], "traced_tactics": [{"tactic": "apply isTopologicalBasis_of_open_of_nhds", "annotated_tactic": ["apply <a>isTopologicalBasis_of_open_of_nhds</a>", [{"full_name": "TopologicalSpace.isTopologicalBasis_of_open_of_nhds", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [127, 9], "def_end_pos": [127, 43]}]], "state_before": "\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\n\u22a2 IsTopologicalBasis (\u22c3 i, (fun u => Sigma.mk i '' u) '' s i)", "state_after": "case h_open\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\n\u22a2 \u2200 (u : Set ((i : \u03b9) \u00d7 E i)), u \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i \u2192 IsOpen u\n\ncase h_nhds\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\n\u22a2 \u2200 (a : (i : \u03b9) \u00d7 E i) (u : Set ((i : \u03b9) \u00d7 E i)),\n    a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, v \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i \u2227 a \u2208 v \u2227 v \u2286 u"}, {"tactic": "intro u hu", "annotated_tactic": ["intro u hu", []], "state_before": "case h_open\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\n\u22a2 \u2200 (u : Set ((i : \u03b9) \u00d7 E i)), u \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i \u2192 IsOpen u", "state_after": "case h_open\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\nu : Set ((i : \u03b9) \u00d7 E i)\nhu : u \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i\n\u22a2 IsOpen u"}, {"tactic": "obtain \u27e8i, t, ts, rfl\u27e9 : \u2203 (i : \u03b9) (t : Set (E i)), t \u2208 s i \u2227 Sigma.mk i '' t = u := by\n  simpa only [mem_iUnion, mem_image] using hu", "annotated_tactic": ["obtain \u27e8i, t, ts, rfl\u27e9 : \u2203 (i : \u03b9) (t : <a>Set</a> (E i)), t \u2208 s i \u2227 <a>Sigma.mk</a> i '' t = u := by\n      simpa only [<a>mem_iUnion</a>, <a>mem_image</a>] using hu", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Sigma.mk", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [140, 3], "def_end_pos": [140, 5]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}]], "state_before": "case h_open\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\nu : Set ((i : \u03b9) \u00d7 E i)\nhu : u \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i\n\u22a2 IsOpen u", "state_after": "case h_open.intro.intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\ni : \u03b9\nt : Set (E i)\nts : t \u2208 s i\nhu : Sigma.mk i '' t \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i\n\u22a2 IsOpen (Sigma.mk i '' t)"}, {"tactic": "exact isOpenMap_sigmaMk _ ((hs i).isOpen ts)", "annotated_tactic": ["exact <a>isOpenMap_sigmaMk</a> _ ((hs i).<a>isOpen</a> ts)", [{"full_name": "isOpenMap_sigmaMk", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1505, 9], "def_end_pos": [1505, 26]}, {"full_name": "TopologicalSpace.IsTopologicalBasis.isOpen", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [166, 19], "def_end_pos": [166, 44]}]], "state_before": "case h_open.intro.intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\ni : \u03b9\nt : Set (E i)\nts : t \u2208 s i\nhu : Sigma.mk i '' t \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i\n\u22a2 IsOpen (Sigma.mk i '' t)", "state_after": "no goals"}, {"tactic": "simpa only [mem_iUnion, mem_image] using hu", "annotated_tactic": ["simpa only [<a>mem_iUnion</a>, <a>mem_image</a>] using hu", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [231, 9], "def_end_pos": [231, 18]}]], "state_before": "\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\nu : Set ((i : \u03b9) \u00d7 E i)\nhu : u \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i\n\u22a2 \u2203 i t, t \u2208 s i \u2227 Sigma.mk i '' t = u", "state_after": "no goals"}, {"tactic": "rintro \u27e8i, x\u27e9 u hxu u_open", "annotated_tactic": ["rintro \u27e8i, x\u27e9 u hxu u_open", []], "state_before": "case h_nhds\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\n\u22a2 \u2200 (a : (i : \u03b9) \u00d7 E i) (u : Set ((i : \u03b9) \u00d7 E i)),\n    a \u2208 u \u2192 IsOpen u \u2192 \u2203 v, v \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i \u2227 a \u2208 v \u2227 v \u2286 u", "state_after": "case h_nhds.mk\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\ni : \u03b9\nx : E i\nu : Set ((i : \u03b9) \u00d7 E i)\nhxu : { fst := i, snd := x } \u2208 u\nu_open : IsOpen u\n\u22a2 \u2203 v, v \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i \u2227 { fst := i, snd := x } \u2208 v \u2227 v \u2286 u"}, {"tactic": "have hx : x \u2208 Sigma.mk i \u207b\u00b9' u := hxu", "annotated_tactic": ["have hx : x \u2208 <a>Sigma.mk</a> i \u207b\u00b9' u := hxu", [{"full_name": "Sigma.mk", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [140, 3], "def_end_pos": [140, 5]}]], "state_before": "case h_nhds.mk\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\ni : \u03b9\nx : E i\nu : Set ((i : \u03b9) \u00d7 E i)\nhxu : { fst := i, snd := x } \u2208 u\nu_open : IsOpen u\n\u22a2 \u2203 v, v \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i \u2227 { fst := i, snd := x } \u2208 v \u2227 v \u2286 u", "state_after": "case h_nhds.mk\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\ni : \u03b9\nx : E i\nu : Set ((i : \u03b9) \u00d7 E i)\nhxu : { fst := i, snd := x } \u2208 u\nu_open : IsOpen u\nhx : x \u2208 Sigma.mk i \u207b\u00b9' u\n\u22a2 \u2203 v, v \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i \u2227 { fst := i, snd := x } \u2208 v \u2227 v \u2286 u"}, {"tactic": "obtain \u27e8v, vs, xv, hv\u27e9 : \u2203 (v : Set (E i)), v \u2208 s i \u2227 x \u2208 v \u2227 v \u2286 Sigma.mk i \u207b\u00b9' u :=\n  (hs i).exists_subset_of_mem_open hx (isOpen_sigma_iff.1 u_open i)", "annotated_tactic": ["obtain \u27e8v, vs, xv, hv\u27e9 : \u2203 (v : <a>Set</a> (E i)), v \u2208 s i \u2227 x \u2208 v \u2227 v \u2286 <a>Sigma.mk</a> i \u207b\u00b9' u :=\n      (hs i).<a>exists_subset_of_mem_open</a> hx (<a>isOpen_sigma_iff</a>.1 u_open i)", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}, {"full_name": "Sigma.mk", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [140, 3], "def_end_pos": [140, 5]}, {"full_name": "TopologicalSpace.IsTopologicalBasis.exists_subset_of_mem_open", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [177, 9], "def_end_pos": [177, 53]}, {"full_name": "isOpen_sigma_iff", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1495, 9], "def_end_pos": [1495, 25]}]], "state_before": "case h_nhds.mk\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\ni : \u03b9\nx : E i\nu : Set ((i : \u03b9) \u00d7 E i)\nhxu : { fst := i, snd := x } \u2208 u\nu_open : IsOpen u\nhx : x \u2208 Sigma.mk i \u207b\u00b9' u\n\u22a2 \u2203 v, v \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i \u2227 { fst := i, snd := x } \u2208 v \u2227 v \u2286 u", "state_after": "case h_nhds.mk.intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\ni : \u03b9\nx : E i\nu : Set ((i : \u03b9) \u00d7 E i)\nhxu : { fst := i, snd := x } \u2208 u\nu_open : IsOpen u\nhx : x \u2208 Sigma.mk i \u207b\u00b9' u\nv : Set (E i)\nvs : v \u2208 s i\nxv : x \u2208 v\nhv : v \u2286 Sigma.mk i \u207b\u00b9' u\n\u22a2 \u2203 v, v \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i \u2227 { fst := i, snd := x } \u2208 v \u2227 v \u2286 u"}, {"tactic": "exact\n  \u27e8Sigma.mk i '' v, mem_iUnion.2 \u27e8i, mem_image_of_mem _ vs\u27e9, mem_image_of_mem _ xv,\n    image_subset_iff.2 hv\u27e9", "annotated_tactic": ["exact\n      \u27e8<a>Sigma.mk</a> i '' v, <a>mem_iUnion</a>.2 \u27e8i, <a>mem_image_of_mem</a> _ vs\u27e9, <a>mem_image_of_mem</a> _ xv,\n        <a>image_subset_iff</a>.2 hv\u27e9", [{"full_name": "Sigma.mk", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [140, 3], "def_end_pos": [140, 5]}, {"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_image_of_mem", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [240, 9], "def_end_pos": [240, 25]}, {"full_name": "Set.mem_image_of_mem", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [240, 9], "def_end_pos": [240, 25]}, {"full_name": "Set.image_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [497, 9], "def_end_pos": [497, 25]}]], "state_before": "case h_nhds.mk.intro.intro.intro\n\u03b1 : Type u\nt : TopologicalSpace \u03b1\n\u03b9 : Type u_1\nE : \u03b9 \u2192 Type u_2\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (E i)\ns : (i : \u03b9) \u2192 Set (Set (E i))\nhs : \u2200 (i : \u03b9), IsTopologicalBasis (s i)\ni : \u03b9\nx : E i\nu : Set ((i : \u03b9) \u00d7 E i)\nhxu : { fst := i, snd := x } \u2208 u\nu_open : IsOpen u\nhx : x \u2208 Sigma.mk i \u207b\u00b9' u\nv : Set (E i)\nvs : v \u2208 s i\nxv : x \u2208 v\nhv : v \u2286 Sigma.mk i \u207b\u00b9' u\n\u22a2 \u2203 v, v \u2208 \u22c3 i, (fun u => Sigma.mk i '' u) '' s i \u2227 { fst := i, snd := x } \u2208 v \u2227 v \u2286 u", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Yoneda.lean", "full_name": "CategoryTheory.yonedaEquiv_apply", "start": [401, 1], "end": [403, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Logic.lean", "full_name": "Or.symm", "start": [247, 1], "end": [247, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/Calculus/ContDiff.lean", "full_name": "ContDiff.restrict_scalars", "start": [2277, 1], "end": [2278, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "full_name": "integral_withDensity_eq_integral_smul", "start": [1269, 1], "end": [1314, 57], "traced_tactics": [{"tactic": "by_cases hE : CompleteSpace E", "annotated_tactic": ["by_cases hE : <a>CompleteSpace</a> E", [{"full_name": "CompleteSpace", "def_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "def_pos": [397, 7], "def_end_pos": [397, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : \u00acCompleteSpace E\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : \u00acCompleteSpace E\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : \u00acCompleteSpace E\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc"}, {"tactic": "by_cases hg : Integrable g (\u03bc.withDensity fun x => f x)", "annotated_tactic": ["by_cases hg : <a>Integrable</a> g (\u03bc.withDensity fun x => f x)", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 5], "def_end_pos": [442, 15]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : \u00acIntegrable g\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : \u00acIntegrable g\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : \u00acIntegrable g\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc\n\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc"}, {"tactic": "refine' Integrable.induction\n  (P := fun g => \u222b a, g a \u2202\u03bc.withDensity (fun x => f x) = \u222b a, f a \u2022 g a \u2202\u03bc) _ _ _ _ hg", "annotated_tactic": ["refine' <a>Integrable.induction</a>\n    (P := fun g => \u222b a, g a \u2202\u03bc.withDensity (fun x => f x) = \u222b a, f a \u2022 g a \u2202\u03bc) _ _ _ _ hg", [{"full_name": "MeasureTheory.Integrable.induction", "def_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "def_pos": [1058, 9], "def_end_pos": [1058, 29]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc", "state_after": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192\n      \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4 \u2192\n        (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc)\n          (indicator s fun x => c)\n\ncase pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 \u2200 \u2983f_1 g : \u03b1 \u2192 E\u2984,\n    Disjoint (support f_1) (support g) \u2192\n      Integrable f_1 \u2192\n        Integrable g \u2192\n          (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) f_1 \u2192\n            (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) g \u2192\n              (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) (f_1 + g)\n\ncase pos.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 IsClosed {f_1 | (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) \u2191\u2191f_1}\n\ncase pos.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 \u2200 \u2983f_1 g : \u03b1 \u2192 E\u2984,\n    f_1 =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] g \u2192\n      Integrable f_1 \u2192\n        (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) f_1 \u2192\n          (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) g"}, {"tactic": "simp [integral, hE]", "annotated_tactic": ["simp [<a>integral</a>, hE]", [{"full_name": "MeasureTheory.integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [791, 17], "def_end_pos": [791, 25]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : \u00acCompleteSpace E\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [integral_undef hg, integral_undef]", "annotated_tactic": ["rw [<a>integral_undef</a> hg, <a>integral_undef</a>]", [{"full_name": "MeasureTheory.integral_undef", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [836, 9], "def_end_pos": [836, 23]}, {"full_name": "MeasureTheory.integral_undef", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [836, 9], "def_end_pos": [836, 23]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : \u00acIntegrable g\n\u22a2 (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : \u00acIntegrable g\n\u22a2 \u00acIntegrable fun a => f a \u2022 g a"}, {"tactic": "rwa [\u2190 integrable_withDensity_iff_integrable_smul f_meas]", "annotated_tactic": ["rwa [\u2190 <a>integrable_withDensity_iff_integrable_smul</a> f_meas]", [{"full_name": "MeasureTheory.integrable_withDensity_iff_integrable_smul", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [918, 9], "def_end_pos": [918, 51]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : \u00acIntegrable g\n\u22a2 \u00acIntegrable fun a => f a \u2022 g a", "state_after": "no goals"}, {"tactic": "intro c s s_meas hs", "annotated_tactic": ["intro c s s_meas hs", []], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192\n      \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4 \u2192\n        (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc)\n          (indicator s fun x => c)", "state_after": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 (\u222b (a : \u03b1), indicator s (fun x => c) a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) =\n    \u222b (a : \u03b1), f a \u2022 indicator s (fun x => c) a \u2202\u03bc"}, {"tactic": "rw [integral_indicator s_meas]", "annotated_tactic": ["rw [<a>integral_indicator</a> s_meas]", [{"full_name": "MeasureTheory.integral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [169, 9], "def_end_pos": [169, 27]}]], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 (\u222b (a : \u03b1), indicator s (fun x => c) a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) =\n    \u222b (a : \u03b1), f a \u2022 indicator s (fun x => c) a \u2202\u03bc", "state_after": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 (\u222b (x : \u03b1) in s, c \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 indicator s (fun x => c) a \u2202\u03bc"}, {"tactic": "simp_rw [\u2190 indicator_smul_apply, integral_indicator s_meas]", "annotated_tactic": ["simp_rw [\u2190 <a>indicator_smul_apply</a>, <a>integral_indicator</a> s_meas]", [{"full_name": "Set.indicator_smul_apply", "def_path": "Mathlib/Algebra/IndicatorFunction.lean", "def_pos": [481, 9], "def_end_pos": [481, 29]}, {"full_name": "MeasureTheory.integral_indicator", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [169, 9], "def_end_pos": [169, 27]}]], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 (\u222b (x : \u03b1) in s, c \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 indicator s (fun x => c) a \u2202\u03bc", "state_after": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 (\u222b (x : \u03b1) in s, c \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (x : \u03b1) in s, f x \u2022 c \u2202\u03bc"}, {"tactic": "simp only [s_meas, integral_const, Measure.restrict_apply', univ_inter, withDensity_apply]", "annotated_tactic": ["simp only [s_meas, <a>integral_const</a>, <a>Measure.restrict_apply'</a>, <a>univ_inter</a>, <a>withDensity_apply</a>]", [{"full_name": "MeasureTheory.integral_const", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1409, 9], "def_end_pos": [1409, 23]}, {"full_name": "MeasureTheory.Measure.restrict_apply'", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1567, 9], "def_end_pos": [1567, 24]}, {"full_name": "Set.univ_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1017, 9], "def_end_pos": [1017, 19]}, {"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}]], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 (\u222b (x : \u03b1) in s, c \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (x : \u03b1) in s, f x \u2022 c \u2202\u03bc", "state_after": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 ENNReal.toReal (\u222b\u207b (x : \u03b1) in s, \u2191(f x) \u2202\u03bc) \u2022 c = \u222b (x : \u03b1) in s, f x \u2022 c \u2202\u03bc"}, {"tactic": "rw [lintegral_coe_eq_integral, ENNReal.toReal_ofReal, \u2190 integral_smul_const]", "annotated_tactic": ["rw [<a>lintegral_coe_eq_integral</a>, <a>ENNReal.toReal_ofReal</a>, \u2190 <a>integral_smul_const</a>]", [{"full_name": "MeasureTheory.lintegral_coe_eq_integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1208, 9], "def_end_pos": [1208, 34]}, {"full_name": "ENNReal.toReal_ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [191, 9], "def_end_pos": [191, 22]}, {"full_name": "integral_smul_const", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [1257, 9], "def_end_pos": [1257, 28]}]], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 ENNReal.toReal (\u222b\u207b (x : \u03b1) in s, \u2191(f x) \u2202\u03bc) \u2022 c = \u222b (x : \u03b1) in s, f x \u2022 c \u2202\u03bc", "state_after": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191(f x) \u2022 c \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2022 c \u2202\u03bc\n\ncase pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 0 \u2264 \u222b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc\n\ncase pos.refine'_1.hfi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 Integrable fun x => \u2191(f x)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191(f x) \u2022 c \u2202\u03bc = \u222b (x : \u03b1) in s, f x \u2022 c \u2202\u03bc", "state_after": "no goals"}, {"tactic": "exact integral_nonneg fun x => NNReal.coe_nonneg _", "annotated_tactic": ["exact <a>integral_nonneg</a> fun x => <a>NNReal.coe_nonneg</a> _", [{"full_name": "MeasureTheory.integral_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 24]}, {"full_name": "NNReal.coe_nonneg", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [134, 9], "def_end_pos": [134, 19]}]], "state_before": "case pos.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 0 \u2264 \u222b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc", "state_after": "no goals"}, {"tactic": "refine' \u27e8f_meas.coe_nnreal_real.aemeasurable.aestronglyMeasurable, _\u27e9", "annotated_tactic": ["refine' \u27e8f_meas.coe_nnreal_real.aemeasurable.aestronglyMeasurable, _\u27e9", []], "state_before": "case pos.refine'_1.hfi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 Integrable fun x => \u2191(f x)", "state_after": "case pos.refine'_1.hfi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 HasFiniteIntegral fun x => \u2191(f x)"}, {"tactic": "rw [withDensity_apply _ s_meas] at hs", "annotated_tactic": ["rw [<a>withDensity_apply</a> _ s_meas] at hs", [{"full_name": "MeasureTheory.withDensity_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}]], "state_before": "case pos.refine'_1.hfi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u2191\u2191(Measure.withDensity \u03bc fun x => \u2191(f x)) s < \u22a4\n\u22a2 HasFiniteIntegral fun x => \u2191(f x)", "state_after": "case pos.refine'_1.hfi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u222b\u207b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc < \u22a4\n\u22a2 HasFiniteIntegral fun x => \u2191(f x)"}, {"tactic": "rw [HasFiniteIntegral]", "annotated_tactic": ["rw [<a>HasFiniteIntegral</a>]", [{"full_name": "MeasureTheory.HasFiniteIntegral", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [106, 5], "def_end_pos": [106, 22]}]], "state_before": "case pos.refine'_1.hfi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u222b\u207b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc < \u22a4\n\u22a2 HasFiniteIntegral fun x => \u2191(f x)", "state_after": "case pos.refine'_1.hfi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u222b\u207b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2016\u2191(f a)\u2016\u208a \u2202\u03bc < \u22a4"}, {"tactic": "convert hs with x", "annotated_tactic": ["convert hs with x", []], "state_before": "case pos.refine'_1.hfi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u222b\u207b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc < \u22a4\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2016\u2191(f a)\u2016\u208a \u2202\u03bc < \u22a4", "state_after": "case h.e'_3.h.e'_4.h.h.e'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u222b\u207b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc < \u22a4\nx : \u03b1\n\u22a2 \u2016\u2191(f x)\u2016\u208a = f x"}, {"tactic": "simp only [NNReal.nnnorm_eq]", "annotated_tactic": ["simp only [<a>NNReal.nnnorm_eq</a>]", [{"full_name": "NNReal.nnnorm_eq", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [830, 9], "def_end_pos": [830, 18]}]], "state_before": "case h.e'_3.h.e'_4.h.h.e'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nc : E\ns : Set \u03b1\ns_meas : MeasurableSet s\nhs : \u222b\u207b (a : \u03b1) in s, \u2191(f a) \u2202\u03bc < \u22a4\nx : \u03b1\n\u22a2 \u2016\u2191(f x)\u2016\u208a = f x", "state_after": "no goals"}, {"tactic": "intro u u' _ u_int u'_int h h'", "annotated_tactic": ["intro u u' _ u_int u'_int h h'", []], "state_before": "case pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 \u2200 \u2983f_1 g : \u03b1 \u2192 E\u2984,\n    Disjoint (support f_1) (support g) \u2192\n      Integrable f_1 \u2192\n        Integrable g \u2192\n          (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) f_1 \u2192\n            (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) g \u2192\n              (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) (f_1 + g)", "state_after": "case pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu u' : \u03b1 \u2192 E\na\u271d : Disjoint (support u) (support u')\nu_int : Integrable u\nu'_int : Integrable u'\nh : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nh' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc\n\u22a2 (\u222b (a : \u03b1), (u + u') a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 (u + u') a \u2202\u03bc"}, {"tactic": "change\n  (\u222b a : \u03b1, u a + u' a \u2202\u03bc.withDensity fun x : \u03b1 => \u2191(f x)) = \u222b a : \u03b1, f a \u2022 (u a + u' a) \u2202\u03bc", "annotated_tactic": ["change\n      (\u222b a : \u03b1, u a + u' a \u2202\u03bc.withDensity fun x : \u03b1 => \u2191(f x)) = \u222b a : \u03b1, f a \u2022 (u a + u' a) \u2202\u03bc", []], "state_before": "case pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu u' : \u03b1 \u2192 E\na\u271d : Disjoint (support u) (support u')\nu_int : Integrable u\nu'_int : Integrable u'\nh : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nh' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc\n\u22a2 (\u222b (a : \u03b1), (u + u') a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 (u + u') a \u2202\u03bc", "state_after": "case pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu u' : \u03b1 \u2192 E\na\u271d : Disjoint (support u) (support u')\nu_int : Integrable u\nu'_int : Integrable u'\nh : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nh' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc\n\u22a2 (\u222b (a : \u03b1), u a + u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 (u a + u' a) \u2202\u03bc"}, {"tactic": "simp_rw [smul_add]", "annotated_tactic": ["simp_rw [<a>smul_add</a>]", [{"full_name": "smul_add", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [807, 9], "def_end_pos": [807, 17]}]], "state_before": "case pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu u' : \u03b1 \u2192 E\na\u271d : Disjoint (support u) (support u')\nu_int : Integrable u\nu'_int : Integrable u'\nh : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nh' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc\n\u22a2 (\u222b (a : \u03b1), u a + u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 (u a + u' a) \u2202\u03bc", "state_after": "case pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu u' : \u03b1 \u2192 E\na\u271d : Disjoint (support u) (support u')\nu_int : Integrable u\nu'_int : Integrable u'\nh : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nh' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc\n\u22a2 (\u222b (a : \u03b1), u a + u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a + f a \u2022 u' a \u2202\u03bc"}, {"tactic": "rw [integral_add u_int u'_int, h, h', integral_add]", "annotated_tactic": ["rw [<a>integral_add</a> u_int u'_int, h, h', <a>integral_add</a>]", [{"full_name": "MeasureTheory.integral_add", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [868, 9], "def_end_pos": [868, 21]}, {"full_name": "MeasureTheory.integral_add", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [868, 9], "def_end_pos": [868, 21]}]], "state_before": "case pos.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu u' : \u03b1 \u2192 E\na\u271d : Disjoint (support u) (support u')\nu_int : Integrable u\nu'_int : Integrable u'\nh : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nh' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc\n\u22a2 (\u222b (a : \u03b1), u a + u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a + f a \u2022 u' a \u2202\u03bc", "state_after": "case pos.refine'_2.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu u' : \u03b1 \u2192 E\na\u271d : Disjoint (support u) (support u')\nu_int : Integrable u\nu'_int : Integrable u'\nh : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nh' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc\n\u22a2 Integrable fun a => f a \u2022 u a\n\ncase pos.refine'_2.hg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu u' : \u03b1 \u2192 E\na\u271d : Disjoint (support u) (support u')\nu_int : Integrable u\nu'_int : Integrable u'\nh : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nh' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc\n\u22a2 Integrable fun a => f a \u2022 u' a"}, {"tactic": "exact (integrable_withDensity_iff_integrable_smul f_meas).1 u_int", "annotated_tactic": ["exact (<a>integrable_withDensity_iff_integrable_smul</a> f_meas).1 u_int", [{"full_name": "MeasureTheory.integrable_withDensity_iff_integrable_smul", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [918, 9], "def_end_pos": [918, 51]}]], "state_before": "case pos.refine'_2.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu u' : \u03b1 \u2192 E\na\u271d : Disjoint (support u) (support u')\nu_int : Integrable u\nu'_int : Integrable u'\nh : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nh' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc\n\u22a2 Integrable fun a => f a \u2022 u a", "state_after": "no goals"}, {"tactic": "exact (integrable_withDensity_iff_integrable_smul f_meas).1 u'_int", "annotated_tactic": ["exact (<a>integrable_withDensity_iff_integrable_smul</a> f_meas).1 u'_int", [{"full_name": "MeasureTheory.integrable_withDensity_iff_integrable_smul", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [918, 9], "def_end_pos": [918, 51]}]], "state_before": "case pos.refine'_2.hg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu u' : \u03b1 \u2192 E\na\u271d : Disjoint (support u) (support u')\nu_int : Integrable u\nu'_int : Integrable u'\nh : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nh' : (\u222b (a : \u03b1), u' a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u' a \u2202\u03bc\n\u22a2 Integrable fun a => f a \u2022 u' a", "state_after": "no goals"}, {"tactic": "have C1 :\n  Continuous fun u : Lp E 1 (\u03bc.withDensity fun x => f x) =>\n    \u222b x, u x \u2202\u03bc.withDensity fun x => f x :=\n  continuous_integral", "annotated_tactic": ["have C1 :\n      <a>Continuous</a> fun u : <a>Lp</a> E 1 (\u03bc.withDensity fun x => f x) =>\n        \u222b x, u x \u2202\u03bc.withDensity fun x => f x :=\n      <a>continuous_integral</a>", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1591, 11], "def_end_pos": [1591, 21]}, {"full_name": "MeasureTheory.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}, {"full_name": "MeasureTheory.continuous_integral", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [957, 9], "def_end_pos": [957, 28]}]], "state_before": "case pos.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 IsClosed {f_1 | (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) \u2191\u2191f_1}", "state_after": "case pos.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nC1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x)\n\u22a2 IsClosed {f_1 | (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) \u2191\u2191f_1}"}, {"tactic": "have C2 : Continuous fun u : Lp E 1 (\u03bc.withDensity fun x => f x) => \u222b x, f x \u2022 u x \u2202\u03bc := by\n  have : Continuous ((fun u : Lp E 1 \u03bc => \u222b x, u x \u2202\u03bc) \u2218 withDensitySMulLI (E := E) \u03bc f_meas) :=\n    continuous_integral.comp (withDensitySMulLI (E := E) \u03bc f_meas).continuous\n  convert this with u\n  simp only [Function.comp_apply, withDensitySMulLI_apply]\n  exact integral_congr_ae (mem\u21121_smul_of_L1_withDensity f_meas u).coeFn_toLp.symm", "annotated_tactic": ["have C2 : <a>Continuous</a> fun u : <a>Lp</a> E 1 (\u03bc.withDensity fun x => f x) => \u222b x, f x \u2022 u x \u2202\u03bc := by\n      have : <a>Continuous</a> ((fun u : <a>Lp</a> E 1 \u03bc => \u222b x, u x \u2202\u03bc) \u2218 <a>withDensitySMulLI</a> (E := E) \u03bc f_meas) :=\n        continuous_integral.comp (<a>withDensitySMulLI</a> (E := E) \u03bc f_meas).<a>continuous</a>\n      convert this with u\n      simp only [<a>Function.comp_apply</a>, <a>withDensitySMulLI_apply</a>]\n      exact <a>integral_congr_ae</a> (<a>mem\u21121_smul_of_L1_withDensity</a> f_meas u).coeFn_toLp.symm", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1591, 11], "def_end_pos": [1591, 21]}, {"full_name": "MeasureTheory.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}, {"full_name": "Continuous", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1591, 11], "def_end_pos": [1591, 21]}, {"full_name": "MeasureTheory.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}, {"full_name": "MeasureTheory.withDensitySMulLI", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [978, 19], "def_end_pos": [978, 36]}, {"full_name": "MeasureTheory.withDensitySMulLI", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [978, 19], "def_end_pos": [978, 36]}, {"full_name": "LinearIsometry.continuous", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [301, 19], "def_end_pos": [301, 29]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "MeasureTheory.withDensitySMulLI_apply", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1027, 9], "def_end_pos": [1027, 32]}, {"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}, {"full_name": "MeasureTheory.mem\u21121_smul_of_L1_withDensity", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [968, 9], "def_end_pos": [968, 37]}]], "state_before": "case pos.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nC1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x)\n\u22a2 IsClosed {f_1 | (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) \u2191\u2191f_1}", "state_after": "case pos.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nC1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x)\nC2 : Continuous fun u => \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc\n\u22a2 IsClosed {f_1 | (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) \u2191\u2191f_1}"}, {"tactic": "exact isClosed_eq C1 C2", "annotated_tactic": ["exact <a>isClosed_eq</a> C1 C2", [{"full_name": "isClosed_eq", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [1217, 9], "def_end_pos": [1217, 20]}]], "state_before": "case pos.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nC1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x)\nC2 : Continuous fun u => \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc\n\u22a2 IsClosed {f_1 | (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) \u2191\u2191f_1}", "state_after": "no goals"}, {"tactic": "have : Continuous ((fun u : Lp E 1 \u03bc => \u222b x, u x \u2202\u03bc) \u2218 withDensitySMulLI (E := E) \u03bc f_meas) :=\n  continuous_integral.comp (withDensitySMulLI (E := E) \u03bc f_meas).continuous", "annotated_tactic": ["have : <a>Continuous</a> ((fun u : <a>Lp</a> E 1 \u03bc => \u222b x, u x \u2202\u03bc) \u2218 <a>withDensitySMulLI</a> (E := E) \u03bc f_meas) :=\n        continuous_integral.comp (<a>withDensitySMulLI</a> (E := E) \u03bc f_meas).<a>continuous</a>", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1591, 11], "def_end_pos": [1591, 21]}, {"full_name": "MeasureTheory.Lp", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [98, 5], "def_end_pos": [98, 7]}, {"full_name": "MeasureTheory.withDensitySMulLI", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [978, 19], "def_end_pos": [978, 36]}, {"full_name": "MeasureTheory.withDensitySMulLI", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [978, 19], "def_end_pos": [978, 36]}, {"full_name": "LinearIsometry.continuous", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [301, 19], "def_end_pos": [301, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nC1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x)\n\u22a2 Continuous fun u => \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nC1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x)\nthis : Continuous ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas))\n\u22a2 Continuous fun u => \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc"}, {"tactic": "convert this with u", "annotated_tactic": ["convert this with u", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nC1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x)\nthis : Continuous ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas))\n\u22a2 Continuous fun u => \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc", "state_after": "case h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nC1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x)\nthis : Continuous ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas))\nu : { x // x \u2208 Lp E 1 }\n\u22a2 \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc = ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas)) u"}, {"tactic": "simp only [Function.comp_apply, withDensitySMulLI_apply]", "annotated_tactic": ["simp only [<a>Function.comp_apply</a>, <a>withDensitySMulLI_apply</a>]", [{"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "MeasureTheory.withDensitySMulLI_apply", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1027, 9], "def_end_pos": [1027, 32]}]], "state_before": "case h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nC1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x)\nthis : Continuous ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas))\nu : { x // x \u2208 Lp E 1 }\n\u22a2 \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc = ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas)) u", "state_after": "case h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nC1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x)\nthis : Continuous ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas))\nu : { x // x \u2208 Lp E 1 }\n\u22a2 \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc = \u222b (x : \u03b1), \u2191\u2191(Mem\u2112p.toLp (fun x => f x \u2022 \u2191\u2191u x) (_ : Mem\u2112p (fun x => f x \u2022 \u2191\u2191u x) 1)) x \u2202\u03bc"}, {"tactic": "exact integral_congr_ae (mem\u21121_smul_of_L1_withDensity f_meas u).coeFn_toLp.symm", "annotated_tactic": ["exact <a>integral_congr_ae</a> (<a>mem\u21121_smul_of_L1_withDensity</a> f_meas u).coeFn_toLp.symm", [{"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}, {"full_name": "MeasureTheory.mem\u21121_smul_of_L1_withDensity", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [968, 9], "def_end_pos": [968, 37]}]], "state_before": "case h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nC1 : Continuous fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202Measure.withDensity \u03bc fun x => \u2191(f x)\nthis : Continuous ((fun u => \u222b (x : \u03b1), \u2191\u2191u x \u2202\u03bc) \u2218 \u2191(withDensitySMulLI \u03bc f_meas))\nu : { x // x \u2208 Lp E 1 }\n\u22a2 \u222b (x : \u03b1), f x \u2022 \u2191\u2191u x \u2202\u03bc = \u222b (x : \u03b1), \u2191\u2191(Mem\u2112p.toLp (fun x => f x \u2022 \u2191\u2191u x) (_ : Mem\u2112p (fun x => f x \u2022 \u2191\u2191u x) 1)) x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "intro u v huv _ hu", "annotated_tactic": ["intro u v huv _ hu", []], "state_before": "case pos.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\n\u22a2 \u2200 \u2983f_1 g : \u03b1 \u2192 E\u2984,\n    f_1 =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] g \u2192\n      Integrable f_1 \u2192\n        (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) f_1 \u2192\n          (fun g => (\u222b (a : \u03b1), g a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 g a \u2202\u03bc) g", "state_after": "case pos.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\n\u22a2 (\u222b (a : \u03b1), v a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 v a \u2202\u03bc"}, {"tactic": "rw [\u2190 integral_congr_ae huv, hu]", "annotated_tactic": ["rw [\u2190 <a>integral_congr_ae</a> huv, hu]", [{"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}]], "state_before": "case pos.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\n\u22a2 (\u222b (a : \u03b1), v a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 v a \u2202\u03bc", "state_after": "case pos.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\n\u22a2 \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc = \u222b (a : \u03b1), f a \u2022 v a \u2202\u03bc"}, {"tactic": "apply integral_congr_ae", "annotated_tactic": ["apply <a>integral_congr_ae</a>", [{"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}]], "state_before": "case pos.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\n\u22a2 \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc = \u222b (a : \u03b1), f a \u2022 v a \u2202\u03bc", "state_after": "case pos.refine'_4.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\n\u22a2 (fun a => f a \u2022 u a) =\u1d50[\u03bc] fun a => f a \u2022 v a"}, {"tactic": "filter_upwards [(ae_withDensity_iff f_meas.coe_nnreal_ennreal).1 huv] with x hx", "annotated_tactic": ["filter_upwards [(<a>ae_withDensity_iff</a> f_meas.coe_nnreal_ennreal).1 huv] with x hx", [{"full_name": "MeasureTheory.ae_withDensity_iff", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [223, 9], "def_end_pos": [223, 27]}]], "state_before": "case pos.refine'_4.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\n\u22a2 (fun a => f a \u2022 u a) =\u1d50[\u03bc] fun a => f a \u2022 v a", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nx : \u03b1\nhx : \u2191(f x) \u2260 0 \u2192 u x = v x\n\u22a2 f x \u2022 u x = f x \u2022 v x"}, {"tactic": "rcases eq_or_ne (f x) 0 with (h'x | h'x)", "annotated_tactic": ["rcases <a>eq_or_ne</a> (f x) 0 with (h'x | h'x)", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nx : \u03b1\nhx : \u2191(f x) \u2260 0 \u2192 u x = v x\n\u22a2 f x \u2022 u x = f x \u2022 v x", "state_after": "case h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nx : \u03b1\nhx : \u2191(f x) \u2260 0 \u2192 u x = v x\nh'x : f x = 0\n\u22a2 f x \u2022 u x = f x \u2022 v x\n\ncase h.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nx : \u03b1\nhx : \u2191(f x) \u2260 0 \u2192 u x = v x\nh'x : f x \u2260 0\n\u22a2 f x \u2022 u x = f x \u2022 v x"}, {"tactic": "simp only [h'x, zero_smul]", "annotated_tactic": ["simp only [h'x, <a>zero_smul</a>]", [{"full_name": "zero_smul", "def_path": "Mathlib/Algebra/SMulWithZero.lean", "def_pos": [70, 9], "def_end_pos": [70, 18]}]], "state_before": "case h.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nx : \u03b1\nhx : \u2191(f x) \u2260 0 \u2192 u x = v x\nh'x : f x = 0\n\u22a2 f x \u2022 u x = f x \u2022 v x", "state_after": "no goals"}, {"tactic": "rw [hx _]", "annotated_tactic": ["rw [hx _]", []], "state_before": "case h.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nx : \u03b1\nhx : \u2191(f x) \u2260 0 \u2192 u x = v x\nh'x : f x \u2260 0\n\u22a2 f x \u2022 u x = f x \u2022 v x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nx : \u03b1\nhx : \u2191(f x) \u2260 0 \u2192 u x = v x\nh'x : f x \u2260 0\n\u22a2 \u2191(f x) \u2260 0"}, {"tactic": "simpa only [Ne.def, ENNReal.coe_eq_zero] using h'x", "annotated_tactic": ["simpa only [<a>Ne.def</a>, <a>ENNReal.coe_eq_zero</a>] using h'x", [{"full_name": "Ne.def", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [59, 9], "def_end_pos": [59, 15]}, {"full_name": "ENNReal.coe_eq_zero", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [368, 28], "def_end_pos": [368, 39]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03b9 : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : NormedSpace \u211d F\nf : \u03b1 \u2192 \u211d\u22650\nf_meas : Measurable f\ng : \u03b1 \u2192 E\nhE : CompleteSpace E\nhg : Integrable g\nu v : \u03b1 \u2192 E\nhuv : u =\u1d50[Measure.withDensity \u03bc fun x => \u2191(f x)] v\na\u271d : Integrable u\nhu : (\u222b (a : \u03b1), u a \u2202Measure.withDensity \u03bc fun x => \u2191(f x)) = \u222b (a : \u03b1), f a \u2022 u a \u2202\u03bc\nx : \u03b1\nhx : \u2191(f x) \u2260 0 \u2192 u x = v x\nh'x : f x \u2260 0\n\u22a2 \u2191(f x) \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Order/Filter/Ultrafilter.lean", "full_name": "Ultrafilter.compl_mem_iff_not_mem", "start": [134, 1], "end": [134, 91], "traced_tactics": [{"tactic": "rw [\u2190 compl_not_mem_iff, compl_compl]", "annotated_tactic": ["rw [\u2190 <a>compl_not_mem_iff</a>, <a>compl_compl</a>]", [{"full_name": "Ultrafilter.compl_not_mem_iff", "def_path": "Mathlib/Order/Filter/Ultrafilter.lean", "def_pos": [119, 9], "def_end_pos": [119, 26]}, {"full_name": "compl_compl", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [634, 9], "def_end_pos": [634, 20]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf g : Ultrafilter \u03b1\ns t : Set \u03b1\np q : \u03b1 \u2192 Prop\n\u22a2 s\u1d9c \u2208 f \u2194 \u00acs \u2208 f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "acc_iff_cluster", "start": [1189, 1], "end": [1190, 47], "traced_tactics": [{"tactic": "rw [AccPt, nhdsWithin, ClusterPt, inf_assoc]", "annotated_tactic": ["rw [<a>AccPt</a>, <a>nhdsWithin</a>, <a>ClusterPt</a>, <a>inf_assoc</a>]", [{"full_name": "AccPt", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1185, 5], "def_end_pos": [1185, 10]}, {"full_name": "nhdsWithin", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [838, 5], "def_end_pos": [838, 15]}, {"full_name": "ClusterPt", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1096, 5], "def_end_pos": [1096, 14]}, {"full_name": "inf_assoc", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [507, 9], "def_end_pos": [507, 18]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Sort w\na : \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\np p\u2081 p\u2082 : \u03b1 \u2192 Prop\ninst\u271d : TopologicalSpace \u03b1\nx : \u03b1\nF : Filter \u03b1\n\u22a2 AccPt x F \u2194 ClusterPt x (\ud835\udcdf {x}\u1d9c \u2293 F)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/ModelTheory/Definability.lean", "full_name": "Set.Definable.preimage_comp", "start": [166, 1], "end": [171, 75], "traced_tactics": [{"tactic": "obtain \u27e8\u03c6, rfl\u27e9 := h", "annotated_tactic": ["obtain \u27e8\u03c6, rfl\u27e9 := h", []], "state_before": "M : Type w\nA : Set M\nL : Language\ninst\u271d : Structure L M\n\u03b1 : Type u\u2081\n\u03b2 : Type u_1\nB : Set M\ns\u271d : Set (\u03b1 \u2192 M)\nf : \u03b1 \u2192 \u03b2\ns : Set (\u03b1 \u2192 M)\nh : Definable A L s\n\u22a2 Definable A L ((fun g => g \u2218 f) \u207b\u00b9' s)", "state_after": "case intro\nM : Type w\nA : Set M\nL : Language\ninst\u271d : Structure L M\n\u03b1 : Type u\u2081\n\u03b2 : Type u_1\nB : Set M\ns : Set (\u03b1 \u2192 M)\nf : \u03b1 \u2192 \u03b2\n\u03c6 : Formula (L[[\u2191A]]) \u03b1\n\u22a2 Definable A L ((fun g => g \u2218 f) \u207b\u00b9' setOf (Formula.Realize \u03c6))"}, {"tactic": "refine' \u27e8\u03c6.relabel f, _\u27e9", "annotated_tactic": ["refine' \u27e8\u03c6.relabel f, _\u27e9", []], "state_before": "case intro\nM : Type w\nA : Set M\nL : Language\ninst\u271d : Structure L M\n\u03b1 : Type u\u2081\n\u03b2 : Type u_1\nB : Set M\ns : Set (\u03b1 \u2192 M)\nf : \u03b1 \u2192 \u03b2\n\u03c6 : Formula (L[[\u2191A]]) \u03b1\n\u22a2 Definable A L ((fun g => g \u2218 f) \u207b\u00b9' setOf (Formula.Realize \u03c6))", "state_after": "case intro\nM : Type w\nA : Set M\nL : Language\ninst\u271d : Structure L M\n\u03b1 : Type u\u2081\n\u03b2 : Type u_1\nB : Set M\ns : Set (\u03b1 \u2192 M)\nf : \u03b1 \u2192 \u03b2\n\u03c6 : Formula (L[[\u2191A]]) \u03b1\n\u22a2 (fun g => g \u2218 f) \u207b\u00b9' setOf (Formula.Realize \u03c6) = setOf (Formula.Realize (Formula.relabel f \u03c6))"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "case intro\nM : Type w\nA : Set M\nL : Language\ninst\u271d : Structure L M\n\u03b1 : Type u\u2081\n\u03b2 : Type u_1\nB : Set M\ns : Set (\u03b1 \u2192 M)\nf : \u03b1 \u2192 \u03b2\n\u03c6 : Formula (L[[\u2191A]]) \u03b1\n\u22a2 (fun g => g \u2218 f) \u207b\u00b9' setOf (Formula.Realize \u03c6) = setOf (Formula.Realize (Formula.relabel f \u03c6))", "state_after": "case intro.h\nM : Type w\nA : Set M\nL : Language\ninst\u271d : Structure L M\n\u03b1 : Type u\u2081\n\u03b2 : Type u_1\nB : Set M\ns : Set (\u03b1 \u2192 M)\nf : \u03b1 \u2192 \u03b2\n\u03c6 : Formula (L[[\u2191A]]) \u03b1\nx\u271d : \u03b2 \u2192 M\n\u22a2 x\u271d \u2208 (fun g => g \u2218 f) \u207b\u00b9' setOf (Formula.Realize \u03c6) \u2194 x\u271d \u2208 setOf (Formula.Realize (Formula.relabel f \u03c6))"}, {"tactic": "simp only [Set.preimage_setOf_eq, mem_setOf_eq, Formula.realize_relabel]", "annotated_tactic": ["simp only [<a>Set.preimage_setOf_eq</a>, <a>mem_setOf_eq</a>, <a>Formula.realize_relabel</a>]", [{"full_name": "Set.preimage_setOf_eq", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [118, 9], "def_end_pos": [118, 26]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}, {"full_name": "FirstOrder.Language.Formula.realize_relabel", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [674, 9], "def_end_pos": [674, 24]}]], "state_before": "case intro.h\nM : Type w\nA : Set M\nL : Language\ninst\u271d : Structure L M\n\u03b1 : Type u\u2081\n\u03b2 : Type u_1\nB : Set M\ns : Set (\u03b1 \u2192 M)\nf : \u03b1 \u2192 \u03b2\n\u03c6 : Formula (L[[\u2191A]]) \u03b1\nx\u271d : \u03b2 \u2192 M\n\u22a2 x\u271d \u2208 (fun g => g \u2218 f) \u207b\u00b9' setOf (Formula.Realize \u03c6) \u2194 x\u271d \u2208 setOf (Formula.Realize (Formula.relabel f \u03c6))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Tropical/BigOperators.lean", "full_name": "Finset.untrop_sum", "start": [143, 1], "end": [145, 52], "traced_tactics": [{"tactic": "simpa [\u2190 _root_.untrop_sum] using sum_attach.symm", "annotated_tactic": ["simpa [\u2190 <a>_root_.untrop_sum</a>] using sum_attach.symm", [{"full_name": "untrop_sum", "def_path": "Mathlib/Algebra/Tropical/BigOperators.lean", "def_pos": [135, 9], "def_end_pos": [135, 19]}]], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d : ConditionallyCompleteLinearOrder R\ns : Finset S\nf : S \u2192 Tropical (WithTop R)\n\u22a2 untrop (\u2211 i in s, f i) = \u2a05 i, untrop (f \u2191i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.nat_cast_pos", "start": [2340, 1], "end": [2341, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Quaternion.lean", "full_name": "QuaternionAlgebra.nat_cast_re", "start": [409, 1], "end": [410, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/PartitionOfUnity.lean", "full_name": "PartitionOfUnity.le_one", "start": [173, 1], "end": [174, 101], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/LocalHomeomorph.lean", "full_name": "LocalHomeomorph.image_open_of_open'", "start": [659, 1], "end": [660, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Nilpotent.lean", "full_name": "ascending_central_series_le_upper", "start": [193, 1], "end": [199, 76], "traced_tactics": [{"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "G : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d : Subgroup G\ninst\u271d : Normal H\u271d\nH : \u2115 \u2192 Subgroup G\nhH : IsAscendingCentralSeries H\nn : \u2115\n\u22a2 H (n + 1) \u2264 upperCentralSeries G (n + 1)", "state_after": "G : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d : Subgroup G\ninst\u271d : Normal H\u271d\nH : \u2115 \u2192 Subgroup G\nhH : IsAscendingCentralSeries H\nn : \u2115\nx : G\nhx : x \u2208 H (n + 1)\n\u22a2 x \u2208 upperCentralSeries G (n + 1)"}, {"tactic": "rw [mem_upperCentralSeries_succ_iff]", "annotated_tactic": ["rw [<a>mem_upperCentralSeries_succ_iff</a>]", [{"full_name": "mem_upperCentralSeries_succ_iff", "def_path": "Mathlib/GroupTheory/Nilpotent.lean", "def_pos": [160, 9], "def_end_pos": [160, 40]}]], "state_before": "G : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d : Subgroup G\ninst\u271d : Normal H\u271d\nH : \u2115 \u2192 Subgroup G\nhH : IsAscendingCentralSeries H\nn : \u2115\nx : G\nhx : x \u2208 H (n + 1)\n\u22a2 x \u2208 upperCentralSeries G (n + 1)", "state_after": "G : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d : Subgroup G\ninst\u271d : Normal H\u271d\nH : \u2115 \u2192 Subgroup G\nhH : IsAscendingCentralSeries H\nn : \u2115\nx : G\nhx : x \u2208 H (n + 1)\n\u22a2 \u2200 (y : G), x * y * x\u207b\u00b9 * y\u207b\u00b9 \u2208 upperCentralSeries G n"}, {"tactic": "exact fun y => ascending_central_series_le_upper H hH n (hH.2 x n hx y)", "annotated_tactic": ["exact fun y => ascending_central_series_le_upper H hH n (hH.2 x n hx y)", []], "state_before": "G : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d : Subgroup G\ninst\u271d : Normal H\u271d\nH : \u2115 \u2192 Subgroup G\nhH : IsAscendingCentralSeries H\nn : \u2115\nx : G\nhx : x \u2208 H (n + 1)\n\u22a2 \u2200 (y : G), x * y * x\u207b\u00b9 * y\u207b\u00b9 \u2208 upperCentralSeries G n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Sort.lean", "full_name": "Fin.sort_univ", "start": [265, 1], "end": [270, 34], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "n : \u2115\n\u22a2 List.toFinset (Finset.sort (fun x y => x \u2264 y) Finset.univ) = List.toFinset (List.finRange n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean", "full_name": "SimpleGraph.left_nonuniformWitnesses_subset", "start": [125, 1], "end": [128, 46], "traced_tactics": [{"tactic": "rw [nonuniformWitnesses, dif_pos h]", "annotated_tactic": ["rw [<a>nonuniformWitnesses</a>, <a>dif_pos</a> h]", [{"full_name": "SimpleGraph.nonuniformWitnesses", "def_path": "Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean", "def_pos": [119, 19], "def_end_pos": [119, 38]}, {"full_name": "dif_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [807, 9], "def_end_pos": [807, 16]}]], "state_before": "\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\nG : SimpleGraph \u03b1\ninst\u271d : DecidableRel G.Adj\n\u03b5 : \ud835\udd5c\ns t : Finset \u03b1\na b : \u03b1\nh : \u00acIsUniform G \u03b5 s t\n\u22a2 (nonuniformWitnesses G \u03b5 s t).1 \u2286 s", "state_after": "\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\nG : SimpleGraph \u03b1\ninst\u271d : DecidableRel G.Adj\n\u03b5 : \ud835\udd5c\ns t : Finset \u03b1\na b : \u03b1\nh : \u00acIsUniform G \u03b5 s t\n\u22a2 (Exists.choose\n          (_ :\n            \u2203 s',\n              s' \u2286 s \u2227\n                \u2203 t',\n                  t' \u2286 t \u2227\n                    \u2191(card s) * \u03b5 \u2264 \u2191(card s') \u2227\n                      \u2191(card t) * \u03b5 \u2264 \u2191(card t') \u2227 \u03b5 \u2264 \u2191|edgeDensity G s' t' - edgeDensity G s t|),\n        Exists.choose\n          (_ :\n            \u2203 t',\n              t' \u2286 t \u2227\n                \u2191(card s) * \u03b5 \u2264\n                    \u2191(card\n                        (Exists.choose\n                          (_ :\n                            \u2203 s',\n                              s' \u2286 s \u2227\n                                \u2203 t',\n                                  t' \u2286 t \u2227\n                                    \u2191(card s) * \u03b5 \u2264 \u2191(card s') \u2227\n                                      \u2191(card t) * \u03b5 \u2264 \u2191(card t') \u2227 \u03b5 \u2264 \u2191|edgeDensity G s' t' - edgeDensity G s t|))) \u2227\n                  \u2191(card t) * \u03b5 \u2264 \u2191(card t') \u2227\n                    \u03b5 \u2264\n                      \u2191|edgeDensity G\n                              (Exists.choose\n                                (_ :\n                                  \u2203 s',\n                                    s' \u2286 s \u2227\n                                      \u2203 t',\n                                        t' \u2286 t \u2227\n                                          \u2191(card s) * \u03b5 \u2264 \u2191(card s') \u2227\n                                            \u2191(card t) * \u03b5 \u2264 \u2191(card t') \u2227\n                                              \u03b5 \u2264 \u2191|edgeDensity G s' t' - edgeDensity G s t|))\n                              t' -\n                            edgeDensity G s t|)).1 \u2286\n    s"}, {"tactic": "exact (not_isUniform_iff.1 h).choose_spec.1", "annotated_tactic": ["exact (<a>not_isUniform_iff</a>.1 h).<a>choose_spec</a>.1", [{"full_name": "SimpleGraph.not_isUniform_iff", "def_path": "Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean", "def_pos": [105, 9], "def_end_pos": [105, 26]}, {"full_name": "Exists.choose_spec", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [445, 9], "def_end_pos": [445, 27]}]], "state_before": "\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b9 : LinearOrderedField \ud835\udd5c\nG : SimpleGraph \u03b1\ninst\u271d : DecidableRel G.Adj\n\u03b5 : \ud835\udd5c\ns t : Finset \u03b1\na b : \u03b1\nh : \u00acIsUniform G \u03b5 s t\n\u22a2 (Exists.choose\n          (_ :\n            \u2203 s',\n              s' \u2286 s \u2227\n                \u2203 t',\n                  t' \u2286 t \u2227\n                    \u2191(card s) * \u03b5 \u2264 \u2191(card s') \u2227\n                      \u2191(card t) * \u03b5 \u2264 \u2191(card t') \u2227 \u03b5 \u2264 \u2191|edgeDensity G s' t' - edgeDensity G s t|),\n        Exists.choose\n          (_ :\n            \u2203 t',\n              t' \u2286 t \u2227\n                \u2191(card s) * \u03b5 \u2264\n                    \u2191(card\n                        (Exists.choose\n                          (_ :\n                            \u2203 s',\n                              s' \u2286 s \u2227\n                                \u2203 t',\n                                  t' \u2286 t \u2227\n                                    \u2191(card s) * \u03b5 \u2264 \u2191(card s') \u2227\n                                      \u2191(card t) * \u03b5 \u2264 \u2191(card t') \u2227 \u03b5 \u2264 \u2191|edgeDensity G s' t' - edgeDensity G s t|))) \u2227\n                  \u2191(card t) * \u03b5 \u2264 \u2191(card t') \u2227\n                    \u03b5 \u2264\n                      \u2191|edgeDensity G\n                              (Exists.choose\n                                (_ :\n                                  \u2203 s',\n                                    s' \u2286 s \u2227\n                                      \u2203 t',\n                                        t' \u2286 t \u2227\n                                          \u2191(card s) * \u03b5 \u2264 \u2191(card s') \u2227\n                                            \u2191(card t) * \u03b5 \u2264 \u2191(card t') \u2227\n                                              \u03b5 \u2264 \u2191|edgeDensity G s' t' - edgeDensity G s t|))\n                              t' -\n                            edgeDensity G s t|)).1 \u2286\n    s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Complex/Basic.lean", "full_name": "Complex.abs_of_nat", "start": [993, 1], "end": [996, 55], "traced_tactics": [{"tactic": "rw [ofReal_nat_cast]", "annotated_tactic": ["rw [<a>ofReal_nat_cast</a>]", [{"full_name": "Complex.ofReal_nat_cast", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [787, 9], "def_end_pos": [787, 24]}]], "state_before": "n : \u2115\n\u22a2 \u2191abs \u2191n = \u2191abs \u2191\u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "full_name": "BoxIntegral.Box.withBotCoe_inj", "start": [304, 1], "end": [305, 76], "traced_tactics": [{"tactic": "simp only [Subset.antisymm_iff, \u2190 le_antisymm_iff, withBotCoe_subset_iff]", "annotated_tactic": ["simp only [<a>Subset.antisymm_iff</a>, \u2190 <a>le_antisymm_iff</a>, <a>withBotCoe_subset_iff</a>]", [{"full_name": "Set.Subset.antisymm_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [374, 9], "def_end_pos": [374, 28]}, {"full_name": "le_antisymm_iff", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [192, 9], "def_end_pos": [192, 24]}, {"full_name": "BoxIntegral.Box.withBotCoe_subset_iff", "def_path": "Mathlib/Analysis/BoxIntegral/Box/Basic.lean", "def_pos": [297, 9], "def_end_pos": [297, 30]}]], "state_before": "\u03b9 : Type u_1\nI\u271d J\u271d : Box \u03b9\nx y : \u03b9 \u2192 \u211d\nI J : WithBot (Box \u03b9)\n\u22a2 \u2191I = \u2191J \u2194 I = J", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Ordmap/Ordset.lean", "full_name": "Ordnode.dual_node3L", "start": [313, 1], "end": [315, 46], "traced_tactics": [{"tactic": "simp [node3L, node3R, dual_node', add_comm]", "annotated_tactic": ["simp [<a>node3L</a>, <a>node3R</a>, <a>dual_node'</a>, <a>add_comm</a>]", [{"full_name": "Ordnode.node3L", "def_path": "Mathlib/Data/Ordmap/Ordset.lean", "def_pos": [227, 5], "def_end_pos": [227, 11]}, {"full_name": "Ordnode.node3R", "def_path": "Mathlib/Data/Ordmap/Ordset.lean", "def_pos": [232, 5], "def_end_pos": [232, 11]}, {"full_name": "Ordnode.dual_node'", "def_path": 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"Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}]], "state_before": "X : Type u\nY : Type v\nZ : Type w\ninst\u271d\u00b9 : MetricSpace X\ninst\u271d : MetricSpace Y\n\u03a6\u271d : Z \u2192 X\n\u03a8\u271d : Z \u2192 Y\n\u03b5\u271d : \u211d\n\u03a6 : Z \u2192 X\n\u03a8 : Z \u2192 Y\n\u03b5 : \u211d\nval\u271d\u00b9 : Y\nval\u271d : X\n\u22a2 glueDist \u03a8 \u03a6 \u03b5 (Sum.swap (Sum.inr val\u271d\u00b9)) (Sum.swap (Sum.inl val\u271d)) = glueDist \u03a6 \u03a8 \u03b5 (Sum.inr val\u271d\u00b9) (Sum.inl val\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/FieldTheory/Fixed.lean", "full_name": "FixedPoints.minpoly_eq_minpoly", "start": [263, 1], "end": [265, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Ordmap/Ordset.lean", "full_name": "Ordnode.Valid.merge", "start": [1499, 1], "end": [1501, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Logic/Function/Basic.lean", "full_name": "Function.apply_update", "start": [648, 1], "end": [654, 13], "traced_tactics": [{"tactic": "by_cases h:j = i", "annotated_tactic": ["by_cases h:j = i", []], "state_before": "\u03b1\u271d : Sort u\n\u03b2\u271d : \u03b1\u271d \u2192 Sort v\n\u03b1' : Sort w\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\ninst\u271d\u00b9 : DecidableEq \u03b1'\nf\u271d g\u271d : (a : \u03b1\u271d) \u2192 \u03b2\u271d a\na : \u03b1\u271d\nb : \u03b2\u271d a\n\u03b9 : Sort u_1\ninst\u271d : DecidableEq \u03b9\n\u03b1 : \u03b9 \u2192 Sort u_2\n\u03b2 : \u03b9 \u2192 Sort u_3\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\ng : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\nv : \u03b1 i\nj : \u03b9\n\u22a2 f j (update g i v j) = update (fun k => f k (g k)) i (f i v) j", "state_after": "case pos\n\u03b1\u271d : Sort u\n\u03b2\u271d : \u03b1\u271d \u2192 Sort v\n\u03b1' : Sort w\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\ninst\u271d\u00b9 : DecidableEq \u03b1'\nf\u271d g\u271d : (a : \u03b1\u271d) \u2192 \u03b2\u271d a\na : \u03b1\u271d\nb : \u03b2\u271d a\n\u03b9 : Sort u_1\ninst\u271d : DecidableEq \u03b9\n\u03b1 : \u03b9 \u2192 Sort u_2\n\u03b2 : \u03b9 \u2192 Sort u_3\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\ng : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\nv : \u03b1 i\nj : \u03b9\nh : j = i\n\u22a2 f j (update g i v j) = update (fun k => f k (g k)) i (f i v) j\n\ncase neg\n\u03b1\u271d : Sort u\n\u03b2\u271d : \u03b1\u271d \u2192 Sort v\n\u03b1' : Sort w\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\ninst\u271d\u00b9 : DecidableEq \u03b1'\nf\u271d g\u271d : (a : \u03b1\u271d) \u2192 \u03b2\u271d a\na : \u03b1\u271d\nb : \u03b2\u271d a\n\u03b9 : Sort u_1\ninst\u271d : DecidableEq \u03b9\n\u03b1 : \u03b9 \u2192 Sort u_2\n\u03b2 : \u03b9 \u2192 Sort u_3\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\ng : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\nv : \u03b1 i\nj : \u03b9\nh : \u00acj = i\n\u22a2 f j (update g i v j) = update (fun k => f k (g k)) i (f i v) j"}, {"tactic": "subst j", "annotated_tactic": ["subst j", []], "state_before": "case pos\n\u03b1\u271d : Sort u\n\u03b2\u271d : \u03b1\u271d \u2192 Sort v\n\u03b1' : Sort w\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\ninst\u271d\u00b9 : DecidableEq \u03b1'\nf\u271d g\u271d : (a : \u03b1\u271d) \u2192 \u03b2\u271d a\na : \u03b1\u271d\nb : \u03b2\u271d a\n\u03b9 : Sort u_1\ninst\u271d : DecidableEq \u03b9\n\u03b1 : \u03b9 \u2192 Sort u_2\n\u03b2 : \u03b9 \u2192 Sort u_3\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\ng : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\nv : \u03b1 i\nj : \u03b9\nh : j = i\n\u22a2 f j (update g i v j) = update (fun k => f k (g k)) i (f i v) j", "state_after": "case pos\n\u03b1\u271d : Sort u\n\u03b2\u271d : \u03b1\u271d \u2192 Sort v\n\u03b1' : Sort w\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\ninst\u271d\u00b9 : DecidableEq \u03b1'\nf\u271d g\u271d : (a : \u03b1\u271d) \u2192 \u03b2\u271d a\na : \u03b1\u271d\nb : \u03b2\u271d a\n\u03b9 : Sort u_1\ninst\u271d : DecidableEq \u03b9\n\u03b1 : \u03b9 \u2192 Sort u_2\n\u03b2 : \u03b9 \u2192 Sort u_3\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\ng : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\nv : \u03b1 i\n\u22a2 f i (update g i v i) = update (fun k => f k (g k)) i (f i v) i"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case pos\n\u03b1\u271d : Sort u\n\u03b2\u271d : \u03b1\u271d \u2192 Sort v\n\u03b1' : Sort w\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\ninst\u271d\u00b9 : DecidableEq \u03b1'\nf\u271d g\u271d : (a : \u03b1\u271d) \u2192 \u03b2\u271d a\na : \u03b1\u271d\nb : \u03b2\u271d a\n\u03b9 : Sort u_1\ninst\u271d : DecidableEq \u03b9\n\u03b1 : \u03b9 \u2192 Sort u_2\n\u03b2 : \u03b9 \u2192 Sort u_3\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\ng : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\nv : \u03b1 i\n\u22a2 f i (update g i v i) = update (fun k => f k (g k)) i (f i v) i", "state_after": "no goals"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "case neg\n\u03b1\u271d : Sort u\n\u03b2\u271d : \u03b1\u271d \u2192 Sort v\n\u03b1' : Sort w\ninst\u271d\u00b2 : DecidableEq \u03b1\u271d\ninst\u271d\u00b9 : DecidableEq \u03b1'\nf\u271d g\u271d : (a : \u03b1\u271d) \u2192 \u03b2\u271d a\na : \u03b1\u271d\nb : \u03b2\u271d a\n\u03b9 : Sort u_1\ninst\u271d : DecidableEq \u03b9\n\u03b1 : \u03b9 \u2192 Sort u_2\n\u03b2 : \u03b9 \u2192 Sort u_3\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\ng : (i : \u03b9) \u2192 \u03b1 i\ni : \u03b9\nv : \u03b1 i\nj : \u03b9\nh : \u00acj = i\n\u22a2 f j (update g i v j) = update (fun k => f k (g k)) i (f i v) j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.mapsTo_empty", "start": [414, 1], "end": [415, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/DFinsupp/Basic.lean", "full_name": "DFinsupp.comapDomain_smul", "start": [1371, 1], "end": [1375, 68], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b2 : Monoid \u03b3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nhh : Function.Injective h\nr : \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\n\u22a2 comapDomain h hh (r \u2022 f) = r \u2022 comapDomain h hh f", "state_after": "case h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b2 : Monoid \u03b3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nhh : Function.Injective h\nr : \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni\u271d : \u03ba\n\u22a2 \u2191(comapDomain h hh (r \u2022 f)) i\u271d = \u2191(r \u2022 comapDomain h hh f) i\u271d"}, {"tactic": "rw [smul_apply, comapDomain_apply, smul_apply, comapDomain_apply]", "annotated_tactic": ["rw [<a>smul_apply</a>, <a>comapDomain_apply</a>, <a>smul_apply</a>, <a>comapDomain_apply</a>]", [{"full_name": "DFinsupp.smul_apply", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [360, 9], "def_end_pos": [360, 19]}, {"full_name": "DFinsupp.comapDomain_apply", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [1351, 9], "def_end_pos": [1351, 26]}, {"full_name": "DFinsupp.smul_apply", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [360, 9], "def_end_pos": [360, 19]}, {"full_name": "DFinsupp.comapDomain_apply", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [1351, 9], "def_end_pos": [1351, 26]}]], "state_before": "case h\n\u03b9 : Type u\n\u03b3 : Type w\n\u03b2 : \u03b9 \u2192 Type v\n\u03b2\u2081 : \u03b9 \u2192 Type v\u2081\n\u03b2\u2082 : \u03b9 \u2192 Type v\u2082\ndec : DecidableEq \u03b9\n\u03ba : Type u_1\ninst\u271d\u00b2 : Monoid \u03b3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddMonoid (\u03b2 i)\ninst\u271d : (i : \u03b9) \u2192 DistribMulAction \u03b3 (\u03b2 i)\nh : \u03ba \u2192 \u03b9\nhh : Function.Injective h\nr : \u03b3\nf : \u03a0\u2080 (i : \u03b9), \u03b2 i\ni\u271d : \u03ba\n\u22a2 \u2191(comapDomain h hh (r \u2022 f)) i\u271d = \u2191(r \u2022 comapDomain h hh f) i\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/PGroup.lean", "full_name": "IsPGroup.card_modEq_card_fixedPoints", "start": [182, 1], "end": [215, 73], "traced_tactics": [{"tactic": "classical\n  calc\n    card \u03b1 = card (\u03a3y : Quotient (orbitRel G \u03b1), { x // Quotient.mk'' x = y }) :=\n      card_congr (Equiv.sigmaFiberEquiv (@Quotient.mk'' _ (orbitRel G \u03b1))).symm\n    _ = \u2211 a : Quotient (orbitRel G \u03b1), card { x // Quotient.mk'' x = a } := (card_sigma _)\n    _ \u2261 \u2211 _a : fixedPoints G \u03b1, 1 [MOD p] := ?_\n    _ = _ := by simp; rfl\n  rw [\u2190 ZMod.eq_iff_modEq_nat p, Nat.cast_sum, Nat.cast_sum]\n  have key :\n    \u2200 x,\n      card { y // (Quotient.mk'' y : Quotient (orbitRel G \u03b1)) = Quotient.mk'' x } =\n        card (orbit G x) :=\n    fun x => by simp only [Quotient.eq'']; congr\n  refine'\n    Eq.symm\n      (Finset.sum_bij_ne_zero (fun a _ _ => Quotient.mk'' a.1) (fun _ _ _ => Finset.mem_univ _)\n        (fun a\u2081 a\u2082 _ _ _ _ h =>\n          Subtype.eq ((mem_fixedPoints' \u03b1).mp a\u2082.2 a\u2081.1 (Quotient.exact' h)))\n        (fun b => Quotient.inductionOn' b fun b _ hb => _) fun a ha _ => by\n        rw [key, mem_fixedPoints_iff_card_orbit_eq_one.mp a.2])\n  obtain \u27e8k, hk\u27e9 := hG.card_orbit b\n  have : k = 0 :=\n    le_zero_iff.1\n      (Nat.le_of_lt_succ\n        (lt_of_not_ge\n          (mt (pow_dvd_pow p)\n            (by\n              rwa [pow_one, \u2190 hk, \u2190 Nat.modEq_zero_iff_dvd, \u2190 ZMod.eq_iff_modEq_nat, \u2190 key,\n                Nat.cast_zero]))))\n  exact\n    \u27e8\u27e8b, mem_fixedPoints_iff_card_orbit_eq_one.2 <| by rw [hk, this, pow_zero]\u27e9,\n      Finset.mem_univ _, ne_of_eq_of_ne Nat.cast_one one_ne_zero, rfl\u27e9", "annotated_tactic": ["classical\n    calc\n      <a>card</a> \u03b1 = <a>card</a> (\u03a3y : <a>Quotient</a> (<a>orbitRel</a> G \u03b1), { x // <a>Quotient.mk''</a> x = y }) :=\n        <a>card_congr</a> (<a>Equiv.sigmaFiberEquiv</a> (@<a>Quotient.mk''</a> _ (<a>orbitRel</a> G \u03b1))).<a>symm</a>\n      _ = \u2211 a : <a>Quotient</a> (<a>orbitRel</a> G \u03b1), <a>card</a> { x // <a>Quotient.mk''</a> x = a } := (<a>card_sigma</a> _)\n      _ \u2261 \u2211 _a : <a>fixedPoints</a> G \u03b1, 1 [MOD p] := ?_\n      _ = _ := by simp; rfl\n    rw [\u2190 <a>ZMod.eq_iff_modEq_nat</a> p, <a>Nat.cast_sum</a>, <a>Nat.cast_sum</a>]\n    have key :\n      \u2200 x,\n        <a>card</a> { y // (<a>Quotient.mk''</a> y : <a>Quotient</a> (<a>orbitRel</a> G \u03b1)) = <a>Quotient.mk''</a> x } =\n          <a>card</a> (<a>orbit</a> G x) :=\n      fun x => by simp only [<a>Quotient.eq''</a>]; congr\n    refine'\n      <a>Eq.symm</a>\n        (<a>Finset.sum_bij_ne_zero</a> (fun a _ _ => <a>Quotient.mk''</a> a.1) (fun _ _ _ => <a>Finset.mem_univ</a> _)\n          (fun a\u2081 a\u2082 _ _ _ _ h =>\n            <a>Subtype.eq</a> ((<a>mem_fixedPoints'</a> \u03b1).<a>mp</a> a\u2082.2 a\u2081.1 (<a>Quotient.exact'</a> h)))\n          (fun b => <a>Quotient.inductionOn'</a> b fun b _ hb => _) fun a ha _ => by\n          rw [key, mem_fixedPoints_iff_card_orbit_eq_one.mp a.2])\n    obtain \u27e8k, hk\u27e9 := hG.card_orbit b\n    have : k = 0 :=\n      <a>le_zero_iff</a>.1\n        (<a>Nat.le_of_lt_succ</a>\n          (<a>lt_of_not_ge</a>\n            (<a>mt</a> (<a>pow_dvd_pow</a> p)\n              (by\n                rwa [<a>pow_one</a>, \u2190 hk, \u2190 <a>Nat.modEq_zero_iff_dvd</a>, \u2190 <a>ZMod.eq_iff_modEq_nat</a>, \u2190 key,\n                  <a>Nat.cast_zero</a>]))))\n    exact\n      \u27e8\u27e8b, <a>mem_fixedPoints_iff_card_orbit_eq_one</a>.2 <| by rw [hk, this, <a>pow_zero</a>]\u27e9,\n        <a>Finset.mem_univ</a> _, <a>ne_of_eq_of_ne</a> <a>Nat.cast_one</a> <a>one_ne_zero</a>, <a>rfl</a>\u27e9", [{"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "Quotient", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1323, 5], "def_end_pos": [1323, 13]}, {"full_name": "MulAction.orbitRel", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [275, 5], "def_end_pos": [275, 13]}, {"full_name": "Quotient.mk''", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [608, 15], "def_end_pos": [608, 19]}, {"full_name": "Fintype.card_congr", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [149, 9], "def_end_pos": [149, 19]}, {"full_name": "Equiv.sigmaFiberEquiv", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [543, 5], "def_end_pos": [543, 20]}, {"full_name": "Quotient.mk''", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [608, 15], "def_end_pos": [608, 19]}, {"full_name": "MulAction.orbitRel", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [275, 5], "def_end_pos": [275, 13]}, {"full_name": "Equiv.symm", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [157, 15], "def_end_pos": [157, 19]}, {"full_name": "Quotient", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1323, 5], "def_end_pos": [1323, 13]}, {"full_name": "MulAction.orbitRel", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [275, 5], "def_end_pos": [275, 13]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "Quotient.mk''", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [608, 15], "def_end_pos": [608, 19]}, {"full_name": "Fintype.card_sigma", "def_path": "Mathlib/Data/Fintype/BigOperators.lean", "def_pos": [123, 16], "def_end_pos": [123, 34]}, {"full_name": "MulAction.fixedPoints", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [109, 5], "def_end_pos": [109, 16]}, {"full_name": "ZMod.eq_iff_modEq_nat", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [719, 9], "def_end_pos": [719, 25]}, {"full_name": "Nat.cast_sum", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [2218, 9], "def_end_pos": [2218, 17]}, {"full_name": "Nat.cast_sum", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [2218, 9], "def_end_pos": [2218, 17]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "Quotient.mk''", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [608, 15], "def_end_pos": [608, 19]}, {"full_name": "Quotient", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1323, 5], "def_end_pos": [1323, 13]}, {"full_name": "MulAction.orbitRel", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [275, 5], "def_end_pos": [275, 13]}, {"full_name": "Quotient.mk''", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [608, 15], "def_end_pos": [608, 19]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "MulAction.orbit", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 10]}, {"full_name": "Quotient.eq''", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [783, 19], "def_end_pos": [783, 23]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "Finset.sum_bij_ne_zero", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1141, 3], "def_end_pos": [1141, 14]}, {"full_name": "Quotient.mk''", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [608, 15], "def_end_pos": [608, 19]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Subtype.eq", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [960, 19], "def_end_pos": [960, 21]}, {"full_name": "MulAction.mem_fixedPoints'", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [143, 9], "def_end_pos": [143, 25]}, {"full_name": "Iff.mp", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [90, 3], "def_end_pos": [90, 5]}, {"full_name": "Quotient.exact'", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [767, 9], "def_end_pos": [767, 15]}, {"full_name": "Quotient.inductionOn'", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [671, 19], "def_end_pos": [671, 31]}, {"full_name": "le_zero_iff", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [102, 9], "def_end_pos": [102, 20]}, {"full_name": "Nat.le_of_lt_succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1631, 9], "def_end_pos": [1631, 26]}, {"full_name": "lt_of_not_ge", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 21]}, {"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}, {"full_name": "pow_dvd_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [442, 9], "def_end_pos": [442, 20]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}, {"full_name": "Nat.modEq_zero_iff_dvd", "def_path": "Mathlib/Data/Nat/ModEq.lean", "def_pos": [77, 9], "def_end_pos": [77, 27]}, {"full_name": "ZMod.eq_iff_modEq_nat", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [719, 9], "def_end_pos": [719, 25]}, {"full_name": "Nat.cast_zero", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [114, 9], "def_end_pos": [114, 18]}, {"full_name": "MulAction.mem_fixedPoints_iff_card_orbit_eq_one", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [182, 9], "def_end_pos": [182, 46]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [639, 9], "def_end_pos": [639, 17]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "ne_of_eq_of_ne", "def_path": "Mathlib/Init/CCLemmas.lean", "def_pos": [118, 9], "def_end_pos": [118, 23]}, {"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [141, 9], "def_end_pos": [141, 17]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\n\u22a2 card \u03b1 \u2261 card \u2191(fixedPoints G \u03b1) [MOD p]", "state_after": "no goals"}, {"tactic": "calc\n  card \u03b1 = card (\u03a3y : Quotient (orbitRel G \u03b1), { x // Quotient.mk'' x = y }) :=\n    card_congr (Equiv.sigmaFiberEquiv (@Quotient.mk'' _ (orbitRel G \u03b1))).symm\n  _ = \u2211 a : Quotient (orbitRel G \u03b1), card { x // Quotient.mk'' x = a } := (card_sigma _)\n  _ \u2261 \u2211 _a : fixedPoints G \u03b1, 1 [MOD p] := ?_\n  _ = _ := by simp; rfl", "annotated_tactic": ["calc\n      <a>card</a> \u03b1 = <a>card</a> (\u03a3y : <a>Quotient</a> (<a>orbitRel</a> G \u03b1), { x // <a>Quotient.mk''</a> x = y }) :=\n        <a>card_congr</a> (<a>Equiv.sigmaFiberEquiv</a> (@<a>Quotient.mk''</a> _ (<a>orbitRel</a> G \u03b1))).<a>symm</a>\n      _ = \u2211 a : <a>Quotient</a> (<a>orbitRel</a> G \u03b1), <a>card</a> { x // <a>Quotient.mk''</a> x = a } := (<a>card_sigma</a> _)\n      _ \u2261 \u2211 _a : <a>fixedPoints</a> G \u03b1, 1 [MOD p] := ?_\n      _ = _ := by simp; rfl", [{"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "Quotient", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1323, 5], "def_end_pos": [1323, 13]}, {"full_name": "MulAction.orbitRel", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [275, 5], "def_end_pos": [275, 13]}, {"full_name": "Quotient.mk''", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [608, 15], "def_end_pos": [608, 19]}, {"full_name": "Fintype.card_congr", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [149, 9], "def_end_pos": [149, 19]}, {"full_name": "Equiv.sigmaFiberEquiv", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [543, 5], "def_end_pos": [543, 20]}, {"full_name": "Quotient.mk''", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [608, 15], "def_end_pos": [608, 19]}, {"full_name": "MulAction.orbitRel", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [275, 5], "def_end_pos": [275, 13]}, {"full_name": "Equiv.symm", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [157, 15], "def_end_pos": [157, 19]}, {"full_name": "Quotient", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1323, 5], "def_end_pos": [1323, 13]}, {"full_name": "MulAction.orbitRel", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [275, 5], "def_end_pos": [275, 13]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "Quotient.mk''", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [608, 15], "def_end_pos": [608, 19]}, {"full_name": "Fintype.card_sigma", "def_path": "Mathlib/Data/Fintype/BigOperators.lean", "def_pos": [123, 16], "def_end_pos": [123, 34]}, {"full_name": "MulAction.fixedPoints", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [109, 5], "def_end_pos": [109, 16]}]], "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\n\u22a2 card \u03b1 \u2261 card \u2191(fixedPoints G \u03b1) [MOD p]", "state_after": "p : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\n\u22a2 \u2211 a : Quotient (orbitRel G \u03b1), card { x // Quotient.mk'' x = a } \u2261 \u2211 _a : \u2191(fixedPoints G \u03b1), 1 [MOD p]"}, {"tactic": "rw [\u2190 ZMod.eq_iff_modEq_nat p, Nat.cast_sum, Nat.cast_sum]", "annotated_tactic": ["rw [\u2190 <a>ZMod.eq_iff_modEq_nat</a> p, <a>Nat.cast_sum</a>, <a>Nat.cast_sum</a>]", [{"full_name": "ZMod.eq_iff_modEq_nat", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [719, 9], "def_end_pos": [719, 25]}, {"full_name": "Nat.cast_sum", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [2218, 9], "def_end_pos": [2218, 17]}, {"full_name": "Nat.cast_sum", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [2218, 9], "def_end_pos": [2218, 17]}]], "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\n\u22a2 \u2211 a : Quotient (orbitRel G \u03b1), card { x // Quotient.mk'' x = a } \u2261 \u2211 _a : \u2191(fixedPoints G \u03b1), 1 [MOD p]", "state_after": "p : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\n\u22a2 \u2211 x : Quotient (orbitRel G \u03b1), \u2191(card { x_1 // Quotient.mk'' x_1 = x }) = \u2211 x : \u2191(fixedPoints G \u03b1), \u21911"}, {"tactic": "have key :\n  \u2200 x,\n    card { y // (Quotient.mk'' y : Quotient (orbitRel G \u03b1)) = Quotient.mk'' x } =\n      card (orbit G x) :=\n  fun x => by simp only [Quotient.eq'']; congr", "annotated_tactic": ["have key :\n      \u2200 x,\n        <a>card</a> { y // (<a>Quotient.mk''</a> y : <a>Quotient</a> (<a>orbitRel</a> G \u03b1)) = <a>Quotient.mk''</a> x } =\n          <a>card</a> (<a>orbit</a> G x) :=\n      fun x => by simp only [<a>Quotient.eq''</a>]; congr", [{"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "Quotient.mk''", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [608, 15], "def_end_pos": [608, 19]}, {"full_name": "Quotient", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [1323, 5], "def_end_pos": [1323, 13]}, {"full_name": "MulAction.orbitRel", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [275, 5], "def_end_pos": [275, 13]}, {"full_name": "Quotient.mk''", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [608, 15], "def_end_pos": [608, 19]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "MulAction.orbit", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 10]}, {"full_name": "Quotient.eq''", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [783, 19], "def_end_pos": [783, 23]}]], "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\n\u22a2 \u2211 x : Quotient (orbitRel G \u03b1), \u2191(card { x_1 // Quotient.mk'' x_1 = x }) = \u2211 x : \u2191(fixedPoints G \u03b1), \u21911", "state_after": "p : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\nkey : \u2200 (x : \u03b1), card { y // Quotient.mk'' y = Quotient.mk'' x } = card \u2191(orbit G x)\n\u22a2 \u2211 x : Quotient (orbitRel G \u03b1), \u2191(card { x_1 // Quotient.mk'' x_1 = x }) = \u2211 x : \u2191(fixedPoints G \u03b1), \u21911"}, {"tactic": "refine'\n  Eq.symm\n    (Finset.sum_bij_ne_zero (fun a _ _ => Quotient.mk'' a.1) (fun _ _ _ => Finset.mem_univ _)\n      (fun a\u2081 a\u2082 _ _ _ _ h =>\n        Subtype.eq ((mem_fixedPoints' \u03b1).mp a\u2082.2 a\u2081.1 (Quotient.exact' h)))\n      (fun b => Quotient.inductionOn' b fun b _ hb => _) fun a ha _ => by\n      rw [key, mem_fixedPoints_iff_card_orbit_eq_one.mp a.2])", "annotated_tactic": ["refine'\n      <a>Eq.symm</a>\n        (<a>Finset.sum_bij_ne_zero</a> (fun a _ _ => <a>Quotient.mk''</a> a.1) (fun _ _ _ => <a>Finset.mem_univ</a> _)\n          (fun a\u2081 a\u2082 _ _ _ _ h =>\n            <a>Subtype.eq</a> ((<a>mem_fixedPoints'</a> \u03b1).<a>mp</a> a\u2082.2 a\u2081.1 (<a>Quotient.exact'</a> h)))\n          (fun b => <a>Quotient.inductionOn'</a> b fun b _ hb => _) fun a ha _ => by\n          rw [key, mem_fixedPoints_iff_card_orbit_eq_one.mp a.2])", [{"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "Finset.sum_bij_ne_zero", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1141, 3], "def_end_pos": [1141, 14]}, {"full_name": "Quotient.mk''", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [608, 15], "def_end_pos": [608, 19]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Subtype.eq", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [960, 19], "def_end_pos": [960, 21]}, {"full_name": "MulAction.mem_fixedPoints'", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [143, 9], "def_end_pos": [143, 25]}, {"full_name": "Iff.mp", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [90, 3], "def_end_pos": [90, 5]}, {"full_name": "Quotient.exact'", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [767, 9], "def_end_pos": [767, 15]}, {"full_name": "Quotient.inductionOn'", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [671, 19], "def_end_pos": [671, 31]}]], "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\nkey : \u2200 (x : \u03b1), card { y // Quotient.mk'' y = Quotient.mk'' x } = card \u2191(orbit G x)\n\u22a2 \u2211 x : Quotient (orbitRel G \u03b1), \u2191(card { x_1 // Quotient.mk'' x_1 = x }) = \u2211 x : \u2191(fixedPoints G \u03b1), \u21911", "state_after": "p : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\nkey : \u2200 (x : \u03b1), card { y // Quotient.mk'' y = Quotient.mk'' x } = card \u2191(orbit G x)\nb\u271d : Quotient (orbitRel G \u03b1)\nb : \u03b1\nx\u271d : Quotient.mk'' b \u2208 Finset.univ\nhb : \u2191(card { x // Quotient.mk'' x = Quotient.mk'' b }) \u2260 0\n\u22a2 \u2203 a h\u2081 h\u2082, Quotient.mk'' b = (fun a x x => Quotient.mk'' \u2191a) a h\u2081 h\u2082"}, {"tactic": "obtain \u27e8k, hk\u27e9 := hG.card_orbit b", "annotated_tactic": ["obtain \u27e8k, hk\u27e9 := hG.card_orbit b", []], "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\nkey : \u2200 (x : \u03b1), card { y // Quotient.mk'' y = Quotient.mk'' x } = card \u2191(orbit G x)\nb\u271d : Quotient (orbitRel G \u03b1)\nb : \u03b1\nx\u271d : Quotient.mk'' b \u2208 Finset.univ\nhb : \u2191(card { x // Quotient.mk'' x = Quotient.mk'' b }) \u2260 0\n\u22a2 \u2203 a h\u2081 h\u2082, Quotient.mk'' b = (fun a x x => Quotient.mk'' \u2191a) a h\u2081 h\u2082", "state_after": "case intro\np : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\nkey : \u2200 (x : \u03b1), card { y // Quotient.mk'' y = Quotient.mk'' x } = card \u2191(orbit G x)\nb\u271d : Quotient (orbitRel G \u03b1)\nb : \u03b1\nx\u271d : Quotient.mk'' b \u2208 Finset.univ\nhb : \u2191(card { x // Quotient.mk'' x = Quotient.mk'' b }) \u2260 0\nk : \u2115\nhk : card \u2191(orbit G b) = p ^ k\n\u22a2 \u2203 a h\u2081 h\u2082, Quotient.mk'' b = (fun a x x => Quotient.mk'' \u2191a) a h\u2081 h\u2082"}, {"tactic": "have : k = 0 :=\n  le_zero_iff.1\n    (Nat.le_of_lt_succ\n      (lt_of_not_ge\n        (mt (pow_dvd_pow p)\n          (by\n            rwa [pow_one, \u2190 hk, \u2190 Nat.modEq_zero_iff_dvd, \u2190 ZMod.eq_iff_modEq_nat, \u2190 key,\n              Nat.cast_zero]))))", "annotated_tactic": ["have : k = 0 :=\n      <a>le_zero_iff</a>.1\n        (<a>Nat.le_of_lt_succ</a>\n          (<a>lt_of_not_ge</a>\n            (<a>mt</a> (<a>pow_dvd_pow</a> p)\n              (by\n                rwa [<a>pow_one</a>, \u2190 hk, \u2190 <a>Nat.modEq_zero_iff_dvd</a>, \u2190 <a>ZMod.eq_iff_modEq_nat</a>, \u2190 key,\n                  <a>Nat.cast_zero</a>]))))", [{"full_name": "le_zero_iff", "def_path": "Mathlib/Algebra/Order/WithZero.lean", "def_pos": [102, 9], "def_end_pos": [102, 20]}, {"full_name": "Nat.le_of_lt_succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1631, 9], "def_end_pos": [1631, 26]}, {"full_name": "lt_of_not_ge", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 21]}, {"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}, {"full_name": "pow_dvd_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [442, 9], "def_end_pos": [442, 20]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}, {"full_name": "Nat.modEq_zero_iff_dvd", "def_path": "Mathlib/Data/Nat/ModEq.lean", "def_pos": [77, 9], "def_end_pos": [77, 27]}, {"full_name": "ZMod.eq_iff_modEq_nat", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [719, 9], "def_end_pos": [719, 25]}, {"full_name": "Nat.cast_zero", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [114, 9], "def_end_pos": [114, 18]}]], "state_before": "case intro\np : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\nkey : \u2200 (x : \u03b1), card { y // Quotient.mk'' y = Quotient.mk'' x } = card \u2191(orbit G x)\nb\u271d : Quotient (orbitRel G \u03b1)\nb : \u03b1\nx\u271d : Quotient.mk'' b \u2208 Finset.univ\nhb : \u2191(card { x // Quotient.mk'' x = Quotient.mk'' b }) \u2260 0\nk : \u2115\nhk : card \u2191(orbit G b) = p ^ k\n\u22a2 \u2203 a h\u2081 h\u2082, Quotient.mk'' b = (fun a x x => Quotient.mk'' \u2191a) a h\u2081 h\u2082", "state_after": "case intro\np : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\nkey : \u2200 (x : \u03b1), card { y // Quotient.mk'' y = Quotient.mk'' x } = card \u2191(orbit G x)\nb\u271d : Quotient (orbitRel G \u03b1)\nb : \u03b1\nx\u271d : Quotient.mk'' b \u2208 Finset.univ\nhb : \u2191(card { x // Quotient.mk'' x = Quotient.mk'' b }) \u2260 0\nk : \u2115\nhk : card \u2191(orbit G b) = p ^ k\nthis : k = 0\n\u22a2 \u2203 a h\u2081 h\u2082, Quotient.mk'' b = (fun a x x => Quotient.mk'' \u2191a) a h\u2081 h\u2082"}, {"tactic": "exact\n  \u27e8\u27e8b, mem_fixedPoints_iff_card_orbit_eq_one.2 <| by rw [hk, this, pow_zero]\u27e9,\n    Finset.mem_univ _, ne_of_eq_of_ne Nat.cast_one one_ne_zero, rfl\u27e9", "annotated_tactic": ["exact\n      \u27e8\u27e8b, <a>mem_fixedPoints_iff_card_orbit_eq_one</a>.2 <| by rw [hk, this, <a>pow_zero</a>]\u27e9,\n        <a>Finset.mem_univ</a> _, <a>ne_of_eq_of_ne</a> <a>Nat.cast_one</a> <a>one_ne_zero</a>, <a>rfl</a>\u27e9", [{"full_name": "MulAction.mem_fixedPoints_iff_card_orbit_eq_one", "def_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "def_pos": [182, 9], "def_end_pos": [182, 46]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [639, 9], "def_end_pos": [639, 17]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "ne_of_eq_of_ne", "def_path": "Mathlib/Init/CCLemmas.lean", "def_pos": [118, 9], "def_end_pos": [118, 23]}, {"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [141, 9], "def_end_pos": [141, 17]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [55, 15], "def_end_pos": [55, 26]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}]], "state_before": "case intro\np : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\nkey : \u2200 (x : \u03b1), card { y // Quotient.mk'' y = Quotient.mk'' x } = card \u2191(orbit G x)\nb\u271d : Quotient (orbitRel G \u03b1)\nb : \u03b1\nx\u271d : Quotient.mk'' b \u2208 Finset.univ\nhb : \u2191(card { x // Quotient.mk'' x = Quotient.mk'' b }) \u2260 0\nk : \u2115\nhk : card \u2191(orbit G b) = p ^ k\nthis : k = 0\n\u22a2 \u2203 a h\u2081 h\u2082, Quotient.mk'' b = (fun a x x => Quotient.mk'' \u2191a) a h\u2081 h\u2082", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\n\u22a2 \u2211 _a : \u2191(fixedPoints G \u03b1), 1 = card \u2191(fixedPoints G \u03b1)", "state_after": "p : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\n\u22a2 Finset.card Finset.univ = card \u2191(fixedPoints G \u03b1)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\n\u22a2 Finset.card Finset.univ = card \u2191(fixedPoints G \u03b1)", "state_after": "no goals"}, {"tactic": "simp only [Quotient.eq'']", "annotated_tactic": ["simp only [<a>Quotient.eq''</a>]", [{"full_name": "Quotient.eq''", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [783, 19], "def_end_pos": [783, 23]}]], "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\nx : \u03b1\n\u22a2 card { y // Quotient.mk'' y = Quotient.mk'' x } = card \u2191(orbit G x)", "state_after": "p : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\nx : \u03b1\n\u22a2 card { y // Setoid.r y x } = card \u2191(orbit G x)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\nx : \u03b1\n\u22a2 card { y // Setoid.r y x } = card \u2191(orbit G x)", "state_after": "no goals"}, {"tactic": "rw [key, mem_fixedPoints_iff_card_orbit_eq_one.mp a.2]", "annotated_tactic": ["rw [key, mem_fixedPoints_iff_card_orbit_eq_one.mp a.2]", []], "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\nkey : \u2200 (x : \u03b1), card { y // Quotient.mk'' y = Quotient.mk'' x } = card \u2191(orbit G x)\na : \u2191(fixedPoints G \u03b1)\nha : a \u2208 Finset.univ\nx\u271d : \u21911 \u2260 0\n\u22a2 \u21911 = \u2191(card { x // Quotient.mk'' x = (fun a x x => Quotient.mk'' \u2191a) a ha x\u271d })", "state_after": "no goals"}, {"tactic": "rwa [pow_one, \u2190 hk, \u2190 Nat.modEq_zero_iff_dvd, \u2190 ZMod.eq_iff_modEq_nat, \u2190 key,\n  Nat.cast_zero]", "annotated_tactic": ["rwa [<a>pow_one</a>, \u2190 hk, \u2190 <a>Nat.modEq_zero_iff_dvd</a>, \u2190 <a>ZMod.eq_iff_modEq_nat</a>, \u2190 key,\n                  <a>Nat.cast_zero</a>]", [{"full_name": "pow_one", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 16]}, {"full_name": "Nat.modEq_zero_iff_dvd", "def_path": "Mathlib/Data/Nat/ModEq.lean", "def_pos": [77, 9], "def_end_pos": [77, 27]}, {"full_name": "ZMod.eq_iff_modEq_nat", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [719, 9], "def_end_pos": [719, 25]}, {"full_name": "Nat.cast_zero", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [114, 9], "def_end_pos": [114, 18]}]], "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\nkey : \u2200 (x : \u03b1), card { y // Quotient.mk'' y = Quotient.mk'' x } = card \u2191(orbit G x)\nb\u271d : Quotient (orbitRel G \u03b1)\nb : \u03b1\nx\u271d : Quotient.mk'' b \u2208 Finset.univ\nhb : \u2191(card { x // Quotient.mk'' x = Quotient.mk'' b }) \u2260 0\nk : \u2115\nhk : card \u2191(orbit G b) = p ^ k\n\u22a2 \u00acp ^ Nat.succ 0 \u2223 p ^ k", "state_after": "no goals"}, {"tactic": "rw [hk, this, pow_zero]", "annotated_tactic": ["rw [hk, this, <a>pow_zero</a>]", [{"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [639, 9], "def_end_pos": [639, 17]}]], "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b3 : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\n\u03b1 : Type u_2\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(fixedPoints G \u03b1)\nkey : \u2200 (x : \u03b1), card { y // Quotient.mk'' y = Quotient.mk'' x } = card \u2191(orbit G x)\nb\u271d : Quotient (orbitRel G \u03b1)\nb : \u03b1\nx\u271d : Quotient.mk'' b \u2208 Finset.univ\nhb : \u2191(card { x // Quotient.mk'' x = Quotient.mk'' b }) \u2260 0\nk : \u2115\nhk : card \u2191(orbit G b) = p ^ k\nthis : k = 0\n\u22a2 card \u2191(orbit G b) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Homology/Single.lean", "full_name": "CochainComplex.from_single\u2080_ext", "start": [453, 1], "end": [458, 15], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "V : Type u\ninst\u271d\u00b2 : Category.{v, u} V\ninst\u271d\u00b9 : HasZeroMorphisms V\ninst\u271d : HasZeroObject V\nC : CochainComplex V \u2115\nX : V\nf g : (single\u2080 V).obj X \u27f6 C\nh : Hom.f f 0 = Hom.f g 0\n\u22a2 \u2191(fromSingle\u2080Equiv C X) f = \u2191(fromSingle\u2080Equiv C X) g", "state_after": "case a\nV : Type u\ninst\u271d\u00b2 : Category.{v, u} V\ninst\u271d\u00b9 : HasZeroMorphisms V\ninst\u271d : HasZeroObject V\nC : CochainComplex V \u2115\nX : V\nf g : (single\u2080 V).obj X \u27f6 C\nh : Hom.f f 0 = Hom.f g 0\n\u22a2 \u2191(\u2191(fromSingle\u2080Equiv C X) f) = \u2191(\u2191(fromSingle\u2080Equiv C X) g)"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "case a\nV : Type u\ninst\u271d\u00b2 : Category.{v, u} V\ninst\u271d\u00b9 : HasZeroMorphisms V\ninst\u271d : HasZeroObject V\nC : CochainComplex V \u2115\nX : V\nf g : (single\u2080 V).obj X \u27f6 C\nh : Hom.f f 0 = Hom.f g 0\n\u22a2 \u2191(\u2191(fromSingle\u2080Equiv C X) f) = \u2191(\u2191(fromSingle\u2080Equiv C X) g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.EqOn.comp_right", "start": [426, 1], "end": [427, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Option/Lemmas.lean", "full_name": "Option.map_id''", "start": [143, 9], "end": [143, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "full_name": "MeasureTheory.measure_mono_null", "start": [197, 1], "end": [198, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/PGroup.lean", "full_name": "IsPGroup.of_injective", "start": [74, 1], "end": [77, 26], "traced_tactics": [{"tactic": "simp_rw [IsPGroup, \u2190 h\u03d5.eq_iff, \u03d5.map_pow, \u03d5.map_one]", "annotated_tactic": ["simp_rw [<a>IsPGroup</a>, \u2190 h\u03d5.eq_iff, \u03d5.map_pow, \u03d5.map_one]", [{"full_name": "IsPGroup", "def_path": "Mathlib/GroupTheory/PGroup.lean", "def_pos": [32, 5], "def_end_pos": [32, 13]}]], "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b9 : Group G\nhG : IsPGroup p G\nH : Type u_2\ninst\u271d : Group H\n\u03d5 : H \u2192* G\nh\u03d5 : Function.Injective \u2191\u03d5\n\u22a2 IsPGroup p H", "state_after": "p : \u2115\nG : Type u_1\ninst\u271d\u00b9 : Group G\nhG : IsPGroup p G\nH : Type u_2\ninst\u271d : Group H\n\u03d5 : H \u2192* G\nh\u03d5 : Function.Injective \u2191\u03d5\n\u22a2 \u2200 (g : H), \u2203 k, \u2191\u03d5 g ^ p ^ k = 1"}, {"tactic": "exact fun h => hG (\u03d5 h)", "annotated_tactic": ["exact fun h => hG (\u03d5 h)", []], "state_before": "p : \u2115\nG : Type u_1\ninst\u271d\u00b9 : Group G\nhG : IsPGroup p G\nH : Type u_2\ninst\u271d : Group H\n\u03d5 : H \u2192* G\nh\u03d5 : Function.Injective \u2191\u03d5\n\u22a2 \u2200 (g : H), \u2203 k, \u2191\u03d5 g ^ p ^ k = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/BigOperators/Finprod.lean", "full_name": "MonoidHom.map_finprod_of_injective", "start": [320, 1], "end": [322, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Finset/Powerset.lean", "full_name": "Finset.empty_mem_ssubsets", "start": [159, 1], "end": [161, 42], "traced_tactics": [{"tactic": "rw [mem_ssubsets, ssubset_iff_subset_ne]", "annotated_tactic": ["rw [<a>mem_ssubsets</a>, <a>ssubset_iff_subset_ne</a>]", [{"full_name": "Finset.mem_ssubsets", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [155, 9], "def_end_pos": [155, 21]}, {"full_name": "Finset.ssubset_iff_subset_ne", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [420, 9], "def_end_pos": [420, 30]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nh : Finset.Nonempty s\n\u22a2 \u2205 \u2208 ssubsets s", "state_after": "\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nh : Finset.Nonempty s\n\u22a2 \u2205 \u2286 s \u2227 \u2205 \u2260 s"}, {"tactic": "exact \u27e8empty_subset s, h.ne_empty.symm\u27e9", "annotated_tactic": ["exact \u27e8<a>empty_subset</a> s, h.ne_empty.symm\u27e9", [{"full_name": "Finset.empty_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [571, 9], "def_end_pos": [571, 21]}]], "state_before": "\u03b1 : Type u_1\ns\u271d t : Finset \u03b1\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nh : Finset.Nonempty s\n\u22a2 \u2205 \u2286 s \u2227 \u2205 \u2260 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Intervals/Instances.lean", "full_name": "Set.Ioc.coe_mul", "start": [296, 1], "end": [297, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Order/Floor.lean", "full_name": "tendsto_ceil_left_pure_ceil", "start": [77, 1], "end": [80, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.ae_tendsto_of_cauchy_snorm'", "start": [1547, 1], "end": [1587, 44], "traced_tactics": [{"tactic": "have h_summable : \u2200\u1d50 x \u2202\u03bc, Summable fun i : \u2115 => f (i + 1) x - f i x := by\n  have h1 :\n    \u2200 n, snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' i, B i :=\n    snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm' hf hp1 h_cau\n  have h2 :\n    \u2200 n,\n      (\u222b\u207b a, (\u2211 i in Finset.range (n + 1), \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) \u2264\n        (\u2211' i, B i) ^ p :=\n    fun n => lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum hp1 n (h1 n)\n  have h3 : (\u222b\u207b a, (\u2211' i, \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' i, B i :=\n    lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum hf hp1 h2\n  have h4 : \u2200\u1d50 x \u2202\u03bc, (\u2211' i, \u2016f (i + 1) x - f i x\u2016\u208a : \u211d\u22650\u221e) < \u221e :=\n    tsum_nnnorm_sub_ae_lt_top hf hp1 hB h3\n  exact\n    h4.mono fun x hx =>\n      summable_of_summable_nnnorm\n        (ENNReal.tsum_coe_ne_top_iff_summable.mp (lt_top_iff_ne_top.mp hx))", "annotated_tactic": ["have h_summable : \u2200\u1d50 x \u2202\u03bc, <a>Summable</a> fun i : \u2115 => f (i + 1) x - f i x := by\n    have h1 :\n      \u2200 n, <a>snorm'</a> (fun x => \u2211 i in <a>Finset.range</a> (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' i, B i :=\n      <a>snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm'</a> hf hp1 h_cau\n    have h2 :\n      \u2200 n,\n        (\u222b\u207b a, (\u2211 i in <a>Finset.range</a> (n + 1), \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) \u2264\n          (\u2211' i, B i) ^ p :=\n      fun n => <a>lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum</a> hp1 n (h1 n)\n    have h3 : (\u222b\u207b a, (\u2211' i, \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' i, B i :=\n      <a>lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum</a> hf hp1 h2\n    have h4 : \u2200\u1d50 x \u2202\u03bc, (\u2211' i, \u2016f (i + 1) x - f i x\u2016\u208a : \u211d\u22650\u221e) < \u221e :=\n      <a>tsum_nnnorm_sub_ae_lt_top</a> hf hp1 hB h3\n    exact\n      h4.mono fun x hx =>\n        <a>summable_of_summable_nnnorm</a>\n          (ENNReal.tsum_coe_ne_top_iff_summable.mp (lt_top_iff_ne_top.mp hx))", [{"full_name": "Summable", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [62, 5], "def_end_pos": [62, 13]}, {"full_name": "MeasureTheory.snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [73, 5], "def_end_pos": [73, 11]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1457, 17], "def_end_pos": [1457, 61]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1473, 17], "def_end_pos": [1473, 63]}, {"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1497, 17], "def_end_pos": [1497, 59]}, {"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.tsum_nnnorm_sub_ae_lt_top", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1531, 17], "def_end_pos": [1531, 42]}, {"full_name": "summable_of_summable_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/InfiniteSum.lean", "def_pos": [177, 9], "def_end_pos": [177, 36]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "have h :\n  \u2200\u1d50 x \u2202\u03bc, \u2203 l : E,\n    atTop.Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) (\ud835\udcdd l) := by\n  refine' h_summable.mono fun x hx => _\n  let hx_sum := hx.hasSum.tendsto_sum_nat\n  exact \u27e8\u2211' i, (f (i + 1) x - f i x), hx_sum\u27e9", "annotated_tactic": ["have h :\n    \u2200\u1d50 x \u2202\u03bc, \u2203 l : E,\n      atTop.Tendsto (fun n => \u2211 i in <a>Finset.range</a> n, (f (i + 1) x - f i x)) (\ud835\udcdd l) := by\n    refine' h_summable.mono fun x hx => _\n    let hx_sum := hx.hasSum.tendsto_sum_nat\n    exact \u27e8\u2211' i, (f (i + 1) x - f i x), hx_sum\u27e9", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "refine' h.mono fun x hx => _", "annotated_tactic": ["refine' h.mono fun x hx => _", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nhx : \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "cases' hx with l hx", "annotated_tactic": ["cases' hx with l hx", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nhx : \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "have h_rw_sum :\n    (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x := by\n  ext1 n\n  change\n    (\u2211 i : \u2115 in Finset.range n, ((fun m => f m x) (i + 1) - (fun m => f m x) i)) = f n x - f 0 x\n  rw [Finset.sum_range_sub (fun m => f m x)]", "annotated_tactic": ["have h_rw_sum :\n      (fun n => \u2211 i in <a>Finset.range</a> n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x := by\n    ext1 n\n    change\n      (\u2211 i : \u2115 in <a>Finset.range</a> n, ((fun m => f m x) (i + 1) - (fun m => f m x) i)) = f n x - f 0 x\n    rw [<a>Finset.sum_range_sub</a> (fun m => f m x)]", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "Finset.sum_range_sub", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1397, 3], "def_end_pos": [1397, 14]}]], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "rw [h_rw_sum] at hx", "annotated_tactic": ["rw [h_rw_sum] at hx", []], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "have hf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x := by\n  ext1 n\n  abel", "annotated_tactic": ["have hf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x := by\n    ext1 n\n    abel", []], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\nhf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)"}, {"tactic": "rw [hf_rw]", "annotated_tactic": ["rw [hf_rw]", []], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\nhf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)", "state_after": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\nhf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x - f 0 x + f 0 x) atTop (\ud835\udcdd l)"}, {"tactic": "exact \u27e8l + f 0 x, Tendsto.add_const _ hx\u27e9", "annotated_tactic": ["exact \u27e8l + f 0 x, <a>Tendsto.add_const</a> _ hx\u27e9", [{"full_name": "Filter.Tendsto.add_const", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [132, 3], "def_end_pos": [132, 14]}]], "state_before": "case intro\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\nhf_rw : (fun n => f n x) = fun n => f n x - f 0 x + f 0 x\n\u22a2 \u2203 l, Tendsto (fun n => f n x - f 0 x + f 0 x) atTop (\ud835\udcdd l)", "state_after": "no goals"}, {"tactic": "have h1 :\n  \u2200 n, snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' i, B i :=\n  snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm' hf hp1 h_cau", "annotated_tactic": ["have h1 :\n      \u2200 n, <a>snorm'</a> (fun x => \u2211 i in <a>Finset.range</a> (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' i, B i :=\n      <a>snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm'</a> hf hp1 h_cau", [{"full_name": "MeasureTheory.snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm.lean", "def_pos": [73, 5], "def_end_pos": [73, 11]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1457, 17], "def_end_pos": [1457, 61]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x"}, {"tactic": "have h2 :\n  \u2200 n,\n    (\u222b\u207b a, (\u2211 i in Finset.range (n + 1), \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) \u2264\n      (\u2211' i, B i) ^ p :=\n  fun n => lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum hp1 n (h1 n)", "annotated_tactic": ["have h2 :\n      \u2200 n,\n        (\u222b\u207b a, (\u2211 i in <a>Finset.range</a> (n + 1), \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) \u2264\n          (\u2211' i, B i) ^ p :=\n      fun n => <a>lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum</a> hp1 n (h1 n)", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}, {"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1473, 17], "def_end_pos": [1473, 63]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\nh2 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), (\u2211 i in Finset.range (n + 1), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc \u2264 (\u2211' (i : \u2115), B i) ^ p\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x"}, {"tactic": "have h3 : (\u222b\u207b a, (\u2211' i, \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' i, B i :=\n  lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum hf hp1 h2", "annotated_tactic": ["have h3 : (\u222b\u207b a, (\u2211' i, \u2016f (i + 1) a - f i a\u2016\u208a : \u211d\u22650\u221e) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' i, B i :=\n      <a>lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum</a> hf hp1 h2", [{"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1497, 17], "def_end_pos": [1497, 59]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\nh2 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), (\u2211 i in Finset.range (n + 1), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc \u2264 (\u2211' (i : \u2115), B i) ^ p\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\nh2 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), (\u2211 i in Finset.range (n + 1), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc \u2264 (\u2211' (i : \u2115), B i) ^ p\nh3 : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x"}, {"tactic": "have h4 : \u2200\u1d50 x \u2202\u03bc, (\u2211' i, \u2016f (i + 1) x - f i x\u2016\u208a : \u211d\u22650\u221e) < \u221e :=\n  tsum_nnnorm_sub_ae_lt_top hf hp1 hB h3", "annotated_tactic": ["have h4 : \u2200\u1d50 x \u2202\u03bc, (\u2211' i, \u2016f (i + 1) x - f i x\u2016\u208a : \u211d\u22650\u221e) < \u221e :=\n      <a>tsum_nnnorm_sub_ae_lt_top</a> hf hp1 hB h3", [{"full_name": "_private.Mathlib.MeasureTheory.Function.LpSpace.0.MeasureTheory.Lp.tsum_nnnorm_sub_ae_lt_top", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1531, 17], "def_end_pos": [1531, 42]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\nh2 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), (\u2211 i in Finset.range (n + 1), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc \u2264 (\u2211' (i : \u2115), B i) ^ p\nh3 : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\nh2 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), (\u2211 i in Finset.range (n + 1), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc \u2264 (\u2211' (i : \u2115), B i) ^ p\nh3 : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nh4 : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x"}, {"tactic": "exact\n  h4.mono fun x hx =>\n    summable_of_summable_nnnorm\n      (ENNReal.tsum_coe_ne_top_iff_summable.mp (lt_top_iff_ne_top.mp hx))", "annotated_tactic": ["exact\n      h4.mono fun x hx =>\n        <a>summable_of_summable_nnnorm</a>\n          (ENNReal.tsum_coe_ne_top_iff_summable.mp (lt_top_iff_ne_top.mp hx))", [{"full_name": "summable_of_summable_nnnorm", "def_path": "Mathlib/Analysis/Normed/Group/InfiniteSum.lean", "def_pos": [177, 9], "def_end_pos": [177, 36]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh1 : \u2200 (n : \u2115), snorm' (fun x => \u2211 i in Finset.range (n + 1), \u2016f (i + 1) x - f i x\u2016) p \u03bc \u2264 \u2211' (i : \u2115), B i\nh2 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), (\u2211 i in Finset.range (n + 1), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc \u2264 (\u2211' (i : \u2115), B i) ^ p\nh3 : (\u222b\u207b (a : \u03b1), (\u2211' (i : \u2115), \u2191\u2016f (i + 1) a - f i a\u2016\u208a) ^ p \u2202\u03bc) ^ (1 / p) \u2264 \u2211' (i : \u2115), B i\nh4 : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2211' (i : \u2115), \u2191\u2016f (i + 1) x - f i x\u2016\u208a < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x", "state_after": "no goals"}, {"tactic": "refine' h_summable.mono fun x hx => _", "annotated_tactic": ["refine' h_summable.mono fun x hx => _", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nx : \u03b1\nhx : Summable fun i => f (i + 1) x - f i x\n\u22a2 \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)"}, {"tactic": "let hx_sum := hx.hasSum.tendsto_sum_nat", "annotated_tactic": ["let hx_sum := hx.hasSum.tendsto_sum_nat", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nx : \u03b1\nhx : Summable fun i => f (i + 1) x - f i x\n\u22a2 \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nx : \u03b1\nhx : Summable fun i => f (i + 1) x - f i x\nhx_sum : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop\n  (\ud835\udcdd (\u2211' (b : \u2115), (f (b + 1) x - f b x))) :=\n  HasSum.tendsto_sum_nat (Summable.hasSum hx)\n\u22a2 \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)"}, {"tactic": "exact \u27e8\u2211' i, (f (i + 1) x - f i x), hx_sum\u27e9", "annotated_tactic": ["exact \u27e8\u2211' i, (f (i + 1) x - f i x), hx_sum\u27e9", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nx : \u03b1\nhx : Summable fun i => f (i + 1) x - f i x\nhx_sum : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop\n  (\ud835\udcdd (\u2211' (b : \u2115), (f (b + 1) x - f b x))) :=\n  HasSum.tendsto_sum_nat (Summable.hasSum hx)\n\u22a2 \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)", "state_after": "no goals"}, {"tactic": "ext1 n", "annotated_tactic": ["ext1 n", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\n\u22a2 (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nn : \u2115\n\u22a2 \u2211 i in Finset.range n, (f (i + 1) x - f i x) = f n x - f 0 x"}, {"tactic": "change\n  (\u2211 i : \u2115 in Finset.range n, ((fun m => f m x) (i + 1) - (fun m => f m x) i)) = f n x - f 0 x", "annotated_tactic": ["change\n      (\u2211 i : \u2115 in <a>Finset.range</a> n, ((fun m => f m x) (i + 1) - (fun m => f m x) i)) = f n x - f 0 x", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3027, 5], "def_end_pos": [3027, 10]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nn : \u2115\n\u22a2 \u2211 i in Finset.range n, (f (i + 1) x - f i x) = f n x - f 0 x", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nn : \u2115\n\u22a2 \u2211 i in Finset.range n, ((fun m => f m x) (i + 1) - (fun m => f m x) i) = f n x - f 0 x"}, {"tactic": "rw [Finset.sum_range_sub (fun m => f m x)]", "annotated_tactic": ["rw [<a>Finset.sum_range_sub</a> (fun m => f m x)]", [{"full_name": "Finset.sum_range_sub", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [1397, 3], "def_end_pos": [1397, 14]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nn : \u2115\n\u22a2 \u2211 i in Finset.range n, ((fun m => f m x) (i + 1) - (fun m => f m x) i) = f n x - f 0 x", "state_after": "no goals"}, {"tactic": "ext1 n", "annotated_tactic": ["ext1 n", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\n\u22a2 (fun n => f n x) = fun n => f n x - f 0 x + f 0 x", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\nn : \u2115\n\u22a2 f n x = f n x - f 0 x + f 0 x"}, {"tactic": "abel", "annotated_tactic": ["abel", []], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np\u271d : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03b1 \u2192 E\np : \u211d\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nhp1 : 1 \u2264 p\nB : \u2115 \u2192 \u211d\u22650\u221e\nhB : \u2211' (i : \u2115), B i \u2260 \u22a4\nh_cau : \u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 snorm' (f n - f m) p \u03bc < B N\nh_summable : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Summable fun i => f (i + 1) x - f i x\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) atTop (\ud835\udcdd l)\nx : \u03b1\nl : E\nhx : Tendsto (fun n => f n x - f 0 x) atTop (\ud835\udcdd l)\nh_rw_sum : (fun n => \u2211 i in Finset.range n, (f (i + 1) x - f i x)) = fun n => f n x - f 0 x\nn : \u2115\n\u22a2 f n x = f n x - f 0 x + f 0 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/SpecialFunctions/Log/Base.lean", "full_name": "Real.logb_le_iff_le_rpow", "start": [192, 1], "end": [193, 74], "traced_tactics": [{"tactic": "rw [\u2190 rpow_le_rpow_left_iff hb, rpow_logb (b_pos hb) (b_ne_one' hb) hx]", "annotated_tactic": ["rw [\u2190 <a>rpow_le_rpow_left_iff</a> hb, <a>rpow_logb</a> (<a>b_pos</a> hb) (<a>b_ne_one'</a> hb) hx]", [{"full_name": "Real.rpow_le_rpow_left_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [523, 9], "def_end_pos": [523, 30]}, {"full_name": "Real.rpow_logb", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Base.lean", "def_pos": [134, 9], "def_end_pos": [134, 18]}, {"full_name": "_private.Mathlib.Analysis.SpecialFunctions.Log.Base.0.Real.b_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Base.lean", "def_pos": [171, 17], "def_end_pos": [171, 22]}, {"full_name": "_private.Mathlib.Analysis.SpecialFunctions.Log.Base.0.Real.b_ne_one'", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Base.lean", "def_pos": [174, 17], "def_end_pos": [174, 26]}]], "state_before": "b x y : \u211d\nhb : 1 < b\nhx : 0 < x\n\u22a2 logb b x \u2264 y \u2194 x \u2264 b ^ y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/ContinuousFunction/Ideals.lean", "full_name": "ContinuousMap.idealOfSet_isMaximal_iff", "start": [360, 1], "end": [365, 50], "traced_tactics": [{"tactic": "rw [Ideal.isMaximal_def]", "annotated_tactic": ["rw [<a>Ideal.isMaximal_def</a>]", [{"full_name": "Ideal.isMaximal_def", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [294, 9], "def_end_pos": [294, 22]}]], "state_before": "X : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\ns : Opens X\n\u22a2 Ideal.IsMaximal (idealOfSet \ud835\udd5c \u2191s) \u2194 IsCoatom s", "state_after": "X : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\ns : Opens X\n\u22a2 IsCoatom (idealOfSet \ud835\udd5c \u2191s) \u2194 IsCoatom s"}, {"tactic": "refine' (idealOpensGI X \ud835\udd5c).isCoatom_iff (fun I hI => _) s", "annotated_tactic": ["refine' (<a>idealOpensGI</a> X \ud835\udd5c).<a>isCoatom_iff</a> (fun I hI => _) s", [{"full_name": "ContinuousMap.idealOpensGI", "def_path": "Mathlib/Topology/ContinuousFunction/Ideals.lean", "def_pos": [345, 5], "def_end_pos": [345, 17]}, {"full_name": "GaloisInsertion.isCoatom_iff", "def_path": "Mathlib/Order/Atoms.lean", "def_pos": [835, 9], "def_end_pos": [835, 21]}]], "state_before": "X : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\ns : Opens X\n\u22a2 IsCoatom (idealOfSet \ud835\udd5c \u2191s) \u2194 IsCoatom s", "state_after": "X : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\ns : Opens X\nI : Ideal C(X, \ud835\udd5c)\nhI : IsCoatom I\n\u22a2 idealOfSet \ud835\udd5c \u2191(opensOfIdeal I) = I"}, {"tactic": "rw [\u2190 Ideal.isMaximal_def] at hI", "annotated_tactic": ["rw [\u2190 <a>Ideal.isMaximal_def</a>] at hI", [{"full_name": "Ideal.isMaximal_def", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [294, 9], "def_end_pos": [294, 22]}]], "state_before": "X : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\ns : Opens X\nI : Ideal C(X, \ud835\udd5c)\nhI : IsCoatom I\n\u22a2 idealOfSet \ud835\udd5c \u2191(opensOfIdeal I) = I", "state_after": "X : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\ns : Opens X\nI : Ideal C(X, \ud835\udd5c)\nhI : Ideal.IsMaximal I\n\u22a2 idealOfSet \ud835\udd5c \u2191(opensOfIdeal I) = I"}, {"tactic": "exact idealOfSet_ofIdeal_isClosed inferInstance", "annotated_tactic": ["exact <a>idealOfSet_ofIdeal_isClosed</a> <a>inferInstance</a>", [{"full_name": "ContinuousMap.idealOfSet_ofIdeal_isClosed", "def_path": "Mathlib/Topology/ContinuousFunction/Ideals.lean", "def_pos": [306, 9], "def_end_pos": [306, 36]}, {"full_name": "inferInstance", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [86, 8], "def_end_pos": [86, 21]}]], "state_before": "X : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : CompactSpace X\ninst\u271d : T2Space X\ns : Opens X\nI : Ideal C(X, \ud835\udd5c)\nhI : Ideal.IsMaximal I\n\u22a2 idealOfSet \ud835\udd5c \u2191(opensOfIdeal I) = I", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/List/Nodup.lean", "full_name": "List.count_eq_of_nodup", "start": [200, 1], "end": [204, 37], "traced_tactics": [{"tactic": "split_ifs with h", "annotated_tactic": ["split_ifs with h", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nl\u271d l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d b : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\nd : Nodup l\n\u22a2 count a l = if a \u2208 l then 1 else 0", "state_after": "case pos\n\u03b1 : Type u\n\u03b2 : Type v\nl\u271d l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d b : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\nd : Nodup l\nh : a \u2208 l\n\u22a2 count a l = 1\n\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\nl\u271d l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d b : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\nd : Nodup l\nh : \u00aca \u2208 l\n\u22a2 count a l = 0"}, {"tactic": "exact count_eq_one_of_mem d h", "annotated_tactic": ["exact <a>count_eq_one_of_mem</a> d h", [{"full_name": "List.count_eq_one_of_mem", "def_path": "Mathlib/Data/List/Nodup.lean", "def_pos": [195, 9], "def_end_pos": [195, 28]}]], "state_before": "case pos\n\u03b1 : Type u\n\u03b2 : Type v\nl\u271d l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d b : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\nd : Nodup l\nh : a \u2208 l\n\u22a2 count a l = 1", "state_after": "no goals"}, {"tactic": "exact count_eq_zero_of_not_mem h", "annotated_tactic": ["exact <a>count_eq_zero_of_not_mem</a> h", [{"full_name": "List.count_eq_zero_of_not_mem", "def_path": "lake-packages/std/Std/Data/List/Count.lean", "def_pos": [170, 21], "def_end_pos": [170, 45]}]], "state_before": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\nl\u271d l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d b : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List \u03b1\nd : Nodup l\nh : \u00aca \u2208 l\n\u22a2 count a l = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean", "full_name": "CliffordAlgebra.\u03b9_comp_lift", "start": [164, 1], "end": [166, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/LinearAlgebra/AffineSpace/Ordered.lean", "full_name": "lineMap_le_left_iff_le", "start": [142, 1], "end": [143, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.iInter_comm", "start": [902, 1], "end": [903, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "full_name": "Associates.out_injective", "start": [260, 1], "end": [261, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/NormedSpace/OperatorNorm.lean", "full_name": "ContinuousLinearMap.op_norm_le_of_shell", "start": [232, 1], "end": [234, 100], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "Real.cosh_eq", "start": [1359, 1], "end": [1362, 57], "traced_tactics": [{"tactic": "rw [cosh, exp, exp, Complex.ofReal_neg, Complex.cosh, mul_two, \u2190 Complex.add_re, \u2190 mul_two,\n  div_mul_cancel _ (two_ne_zero' \u2102), Complex.add_re]", "annotated_tactic": ["rw [<a>cosh</a>, <a>exp</a>, <a>exp</a>, <a>Complex.ofReal_neg</a>, <a>Complex.cosh</a>, <a>mul_two</a>, \u2190 <a>Complex.add_re</a>, \u2190 <a>mul_two</a>,\n      <a>div_mul_cancel</a> _ (<a>two_ne_zero'</a> \u2102), <a>Complex.add_re</a>]", [{"full_name": "Real.cosh", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [464, 12], "def_end_pos": [464, 16]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [434, 12], "def_end_pos": [434, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [434, 12], "def_end_pos": [434, 15]}, {"full_name": "Complex.ofReal_neg", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [254, 9], "def_end_pos": [254, 19]}, {"full_name": "Complex.cosh", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [409, 5], "def_end_pos": [409, 9]}, {"full_name": "mul_two", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 16]}, {"full_name": "Complex.add_re", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [192, 9], "def_end_pos": [192, 15]}, {"full_name": "mul_two", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 16]}, {"full_name": "div_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [66, 9], "def_end_pos": [66, 23]}, {"full_name": "two_ne_zero'", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [81, 7], "def_end_pos": [81, 19]}, {"full_name": "Complex.add_re", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [192, 9], "def_end_pos": [192, 15]}]], "state_before": "x\u271d y x : \u211d\n\u22a2 cosh x * 2 = rexp x + rexp (-x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Polynomial/Basic.lean", "full_name": "Polynomial.forall_eq_iff_forall_eq", "start": [821, 1], "end": [822, 70], "traced_tactics": [{"tactic": "simpa only [\u2190 subsingleton_iff] using subsingleton_iff_subsingleton", "annotated_tactic": ["simpa only [\u2190 <a>subsingleton_iff</a>] using <a>subsingleton_iff_subsingleton</a>", [{"full_name": "subsingleton_iff", "def_path": "Mathlib/Logic/Nontrivial/Defs.lean", "def_pos": [75, 9], "def_end_pos": [75, 25]}, {"full_name": "Polynomial.subsingleton_iff_subsingleton", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [811, 9], "def_end_pos": [811, 38]}]], "state_before": "R : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 (\u2200 (f g : R[X]), f = g) \u2194 \u2200 (a b : R), a = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RepresentationTheory/Invariants.lean", "full_name": "GroupAlgebra.mul_average_right", "start": [57, 1], "end": [62, 66], "traced_tactics": [{"tactic": "simp only [mul_one, Finset.sum_mul, Algebra.smul_mul_assoc, average, MonoidAlgebra.of_apply,\n  Finset.sum_congr, MonoidAlgebra.single_mul_single]", "annotated_tactic": ["simp only [<a>mul_one</a>, <a>Finset.sum_mul</a>, <a>Algebra.smul_mul_assoc</a>, <a>average</a>, <a>MonoidAlgebra.of_apply</a>,\n    <a>Finset.sum_congr</a>, <a>MonoidAlgebra.single_mul_single</a>]", [{"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "Finset.sum_mul", "def_path": "Mathlib/Algebra/BigOperators/Ring.lean", "def_pos": [51, 9], "def_end_pos": [51, 16]}, {"full_name": "Algebra.smul_mul_assoc", "def_path": "Mathlib/Algebra/Algebra/Basic.lean", "def_pos": [394, 19], "def_end_pos": [394, 33]}, {"full_name": "GroupAlgebra.average", "def_path": "Mathlib/RepresentationTheory/Invariants.lean", "def_pos": [39, 19], "def_end_pos": [39, 26]}, {"full_name": "MonoidAlgebra.of_apply", "def_path": "Mathlib/Algebra/MonoidAlgebra/Basic.lean", "def_pos": [499, 3], "def_end_pos": [499, 8]}, {"full_name": "Finset.sum_congr", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [376, 3], "def_end_pos": [376, 14]}, {"full_name": "MonoidAlgebra.single_mul_single", "def_path": "Mathlib/Algebra/MonoidAlgebra/Basic.lean", "def_pos": [449, 9], "def_end_pos": [449, 26]}]], "state_before": "k : Type u_1\nG : Type u_2\ninst\u271d\u00b3 : CommSemiring k\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\ninst\u271d : Invertible \u2191(Fintype.card G)\ng : G\n\u22a2 (average k G * fun\u2080 | g => 1) = average k G", "state_after": "k : Type u_1\nG : Type u_2\ninst\u271d\u00b3 : CommSemiring k\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\ninst\u271d : Invertible \u2191(Fintype.card G)\ng : G\n\u22a2 \u215f\u2191(Fintype.card G) \u2022 \u2211 x : G, single (x * g) 1 = \u215f\u2191(Fintype.card G) \u2022 \u2211 x : G, single x 1"}, {"tactic": "set f : G \u2192 MonoidAlgebra k G := fun x => Finsupp.single x 1", "annotated_tactic": ["set f : G \u2192 <a>MonoidAlgebra</a> k G := fun x => <a>Finsupp.single</a> x 1", [{"full_name": "MonoidAlgebra", "def_path": "Mathlib/Algebra/MonoidAlgebra/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 18]}, {"full_name": "Finsupp.single", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [289, 5], "def_end_pos": [289, 11]}]], "state_before": "k : Type u_1\nG : Type u_2\ninst\u271d\u00b3 : CommSemiring k\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\ninst\u271d : Invertible \u2191(Fintype.card G)\ng : G\n\u22a2 \u215f\u2191(Fintype.card G) \u2022 \u2211 x : G, single (x * g) 1 = \u215f\u2191(Fintype.card G) \u2022 \u2211 x : G, single x 1", "state_after": "k : Type u_1\nG : Type u_2\ninst\u271d\u00b3 : CommSemiring k\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\ninst\u271d : Invertible \u2191(Fintype.card G)\ng : G\nf : G \u2192 MonoidAlgebra k G := fun x => fun\u2080 | x => 1\n\u22a2 \u215f\u2191(Fintype.card G) \u2022 \u2211 x : G, single (x * g) 1 = \u215f\u2191(Fintype.card G) \u2022 Finset.sum Finset.univ f"}, {"tactic": "show \u215f (Fintype.card G : k) \u2022 \u2211 x : G, f (x * g) = \u215f (Fintype.card G : k) \u2022 \u2211 x : G, f x", "annotated_tactic": ["show \u215f (<a>Fintype.card</a> G : k) \u2022 \u2211 x : G, f (x * g) = \u215f (<a>Fintype.card</a> G : k) \u2022 \u2211 x : G, f x", [{"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [61, 5], "def_end_pos": [61, 9]}]], "state_before": "k : Type u_1\nG : Type u_2\ninst\u271d\u00b3 : CommSemiring k\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\ninst\u271d : Invertible \u2191(Fintype.card G)\ng : G\nf : G \u2192 MonoidAlgebra k G := fun x => fun\u2080 | x => 1\n\u22a2 \u215f\u2191(Fintype.card G) \u2022 \u2211 x : G, single (x * g) 1 = \u215f\u2191(Fintype.card G) \u2022 Finset.sum Finset.univ f", "state_after": "k : Type u_1\nG : Type u_2\ninst\u271d\u00b3 : CommSemiring k\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\ninst\u271d : Invertible \u2191(Fintype.card G)\ng : G\nf : G \u2192 MonoidAlgebra k G := fun x => fun\u2080 | x => 1\n\u22a2 \u215f\u2191(Fintype.card G) \u2022 \u2211 x : G, f (x * g) = \u215f\u2191(Fintype.card G) \u2022 \u2211 x : G, f x"}, {"tactic": "rw [Function.Bijective.sum_comp (Group.mulRight_bijective g) _]", "annotated_tactic": ["rw [<a>Function.Bijective.sum_comp</a> (<a>Group.mulRight_bijective</a> g) _]", [{"full_name": "Function.Bijective.sum_comp", "def_path": "Mathlib/Data/Fintype/BigOperators.lean", "def_pos": [204, 3], "def_end_pos": [204, 14]}, {"full_name": "Group.mulRight_bijective", "def_path": "Mathlib/Algebra/Hom/Equiv/Units/Basic.lean", "def_pos": [178, 9], "def_end_pos": [178, 40]}]], "state_before": "k : Type u_1\nG : Type u_2\ninst\u271d\u00b3 : CommSemiring k\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Fintype G\ninst\u271d : Invertible \u2191(Fintype.card G)\ng : G\nf : G \u2192 MonoidAlgebra k G := fun x => fun\u2080 | x => 1\n\u22a2 \u215f\u2191(Fintype.card G) \u2022 \u2211 x : G, f (x * g) = \u215f\u2191(Fintype.card G) \u2022 \u2211 x : G, f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/ULift.lean", "full_name": "PLift.up_surjective", "start": [45, 1], "end": [46, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "full_name": "Int.mul_emod_right", "start": [471, 9], "end": [472, 35], "traced_tactics": [{"tactic": "rw [Int.mul_comm, mul_emod_left]", "annotated_tactic": ["rw [<a>Int.mul_comm</a>, <a>mul_emod_left</a>]", [{"full_name": "Int.mul_comm", "def_path": "lake-packages/std/Std/Data/Int/Lemmas.lean", "def_pos": [380, 19], "def_end_pos": [380, 27]}, {"full_name": "Int.mul_emod_left", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [462, 17], "def_end_pos": [462, 30]}]], "state_before": "a b : Int\n\u22a2 a * b % a = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/List/Count.lean", "full_name": "List.count_singleton'", "start": [160, 1], "end": [160, 98], "traced_tactics": [{"tactic": "simp [count_cons]", "annotated_tactic": ["simp [<a>count_cons</a>]", [{"full_name": "List.count_cons", "def_path": "lake-packages/std/Std/Data/List/Count.lean", "def_pos": [137, 9], "def_end_pos": [137, 19]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\n\u22a2 count a [b] = if a = b then 1 else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/SetTheory/Game/Basic.lean", "full_name": "SetTheory.Game.PGame.equiv_iff_game_eq", "start": [147, 1], "end": [148, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Dynamics/BirkhoffSum/Average.lean", "full_name": "birkhoffAverage_one", "start": [51, 1], "end": [52, 65], "traced_tactics": [{"tactic": "simp [birkhoffAverage]", "annotated_tactic": ["simp [<a>birkhoffAverage</a>]", [{"full_name": "birkhoffAverage", "def_path": "Mathlib/Dynamics/BirkhoffSum/Average.lean", "def_pos": [43, 5], "def_end_pos": [43, 20]}]], "state_before": "R : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\ninst\u271d\u00b2 : DivisionSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nf : \u03b1 \u2192 \u03b1\ng : \u03b1 \u2192 M\nx : \u03b1\n\u22a2 birkhoffAverage R f g 1 x = g x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Sigma.lean", "full_name": "Set.mk_preimage_sigma_eq_empty", "start": [184, 1], "end": [185, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Probability/Kernel/Composition.lean", "full_name": "ProbabilityTheory.kernel.set_lintegral_compProd_univ_left", "start": [469, 1], "end": [473, 89], "traced_tactics": [{"tactic": "simp_rw [set_lintegral_compProd \u03ba \u03b7 a hf MeasurableSet.univ ht, Measure.restrict_univ]", "annotated_tactic": ["simp_rw [<a>set_lintegral_compProd</a> \u03ba \u03b7 a hf <a>MeasurableSet.univ</a> ht, <a>Measure.restrict_univ</a>]", [{"full_name": "ProbabilityTheory.kernel.set_lintegral_compProd", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [454, 9], "def_end_pos": [454, 31]}, {"full_name": "MeasurableSet.univ", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [101, 19], "def_end_pos": [101, 37]}, {"full_name": "MeasureTheory.Measure.restrict_univ", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1701, 9], "def_end_pos": [1701, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03b3 : Type u_4\nm\u03b3 : MeasurableSpace \u03b3\ns : Set (\u03b2 \u00d7 \u03b3)\n\u03ba : { x // x \u2208 kernel \u03b1 \u03b2 }\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 }\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nt : Set \u03b3\nht : MeasurableSet t\n\u22a2 \u222b\u207b (z : \u03b2 \u00d7 \u03b3) in Set.univ \u00d7\u02e2 t, f z \u2202\u2191(\u03ba \u2297\u2096 \u03b7) a = \u222b\u207b (x : \u03b2), \u222b\u207b (y : \u03b3) in t, f (x, y) \u2202\u2191\u03b7 (a, x) \u2202\u2191\u03ba a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/GroupTheory/Perm/Fin.lean", "full_name": "Fin.cycleRange_of_eq", "start": [207, 1], "end": [208, 39], "traced_tactics": [{"tactic": "rw [cycleRange_of_le h.le, if_pos h]", "annotated_tactic": ["rw [<a>cycleRange_of_le</a> h.le, <a>if_pos</a> h]", [{"full_name": "Fin.cycleRange_of_le", "def_path": "Mathlib/GroupTheory/Perm/Fin.lean", "def_pos": [169, 9], "def_end_pos": [169, 25]}, {"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}]], "state_before": "n : \u2115\ni j : Fin (Nat.succ n)\nh : j = i\n\u22a2 \u2191(cycleRange i) j = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Functor/Category.lean", "full_name": "CategoryTheory.NatTrans.mono_of_mono_app", "start": [93, 1], "end": [96, 60], "traced_tactics": [{"tactic": "ext X", "annotated_tactic": ["ext X", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF G H I : C \u2964 D\n\u03b1 : F \u27f6 G\ninst\u271d : \u2200 (X : C), Mono (\u03b1.app X)\nZ\u271d : C \u2964 D\ng h : Z\u271d \u27f6 F\neq : g \u226b \u03b1 = h \u226b \u03b1\n\u22a2 g = h", "state_after": "case w.h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d\u00b9 : Category.{v\u2083, u\u2083} E\nF G H I : C \u2964 D\n\u03b1 : F \u27f6 G\ninst\u271d : \u2200 (X : C), Mono (\u03b1.app X)\nZ\u271d : C \u2964 D\ng h : Z\u271d \u27f6 F\neq : g \u226b \u03b1 = h \u226b \u03b1\nX : C\n\u22a2 g.app X = h.app X"}, {"tactic": "rw [\u2190 cancel_mono (\u03b1.app X), \u2190 comp_app, eq, comp_app]", "annotated_tactic": ["rw [\u2190 <a>cancel_mono</a> (\u03b1.app X), \u2190 <a>comp_app</a>, eq, <a>comp_app</a>]", [{"full_name": "CategoryTheory.cancel_mono", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [292, 9], "def_end_pos": [292, 20]}, {"full_name": "CategoryTheory.NatTrans.comp_app", "def_path": "Mathlib/CategoryTheory/Functor/Category.lean", "def_pos": [76, 9], "def_end_pos": 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1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "full_name": "EuclideanSpace.orthonormal_single", "start": [308, 1], "end": [313, 10], "traced_tactics": [{"tactic": "simp_rw [orthonormal_iff_ite, EuclideanSpace.inner_single_left, map_one, one_mul,\n  EuclideanSpace.single_apply]", "annotated_tactic": ["simp_rw [<a>orthonormal_iff_ite</a>, <a>EuclideanSpace.inner_single_left</a>, <a>map_one</a>, <a>one_mul</a>,\n    <a>EuclideanSpace.single_apply</a>]", [{"full_name": "orthonormal_iff_ite", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [752, 9], "def_end_pos": [752, 28]}, {"full_name": "EuclideanSpace.inner_single_left", "def_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "def_pos": [269, 9], "def_end_pos": [269, 41]}, {"full_name": "map_one", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [202, 9], "def_end_pos": [202, 16]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "EuclideanSpace.single_apply", "def_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "def_pos": [264, 9], "def_end_pos": [264, 36]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\n\u22a2 Orthonormal \ud835\udd5c fun i => single i 1", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\n\u22a2 \u03b9 \u2192 \u03b9 \u2192 True"}, {"tactic": "intros", "annotated_tactic": ["intros", []], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\n\u22a2 \u03b9 \u2192 \u03b9 \u2192 True", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\ni\u271d j\u271d : \u03b9\n\u22a2 True"}, {"tactic": "trivial", "annotated_tactic": ["trivial", []], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\ni\u271d j\u271d : \u03b9\n\u22a2 True", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Computability/Halting.lean", "full_name": "Nat.Partrec'.rfindOpt", "start": [367, 1], "end": [387, 12], "traced_tactics": [{"tactic": "simp only [Nat.rfindOpt, exists_prop, tsub_eq_zero_iff_le, PFun.coe_val, Part.mem_bind_iff,\n  Part.mem_some_iff, Option.mem_def, Part.mem_coe]", "annotated_tactic": ["simp only [<a>Nat.rfindOpt</a>, <a>exists_prop</a>, <a>tsub_eq_zero_iff_le</a>, <a>PFun.coe_val</a>, <a>Part.mem_bind_iff</a>,\n        <a>Part.mem_some_iff</a>, <a>Option.mem_def</a>, <a>Part.mem_coe</a>]", [{"full_name": "Nat.rfindOpt", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [124, 5], "def_end_pos": [124, 13]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "tsub_eq_zero_iff_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [324, 9], "def_end_pos": [324, 28]}, {"full_name": "PFun.coe_val", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [144, 9], "def_end_pos": [144, 16]}, {"full_name": "Part.mem_bind_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [494, 9], "def_end_pos": [494, 21]}, {"full_name": "Part.mem_some_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [170, 9], "def_end_pos": [170, 21]}, {"full_name": "Option.mem_def", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [19, 17], "def_end_pos": [19, 24]}, {"full_name": "Part.mem_coe", "def_path": "Mathlib/Data/Part.lean", "def_pos": [351, 9], "def_end_pos": [351, 16]}]], "state_before": "n : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb : \u2115\n\u22a2 (b \u2208 Part.bind (Nat.rfind fun n_1 => Part.some (decide (1 - f (n_1 ::\u1d65 v) = 0))) fun a => \u2191pred (f (a ::\u1d65 v))) \u2194\n    b \u2208 Nat.rfindOpt fun a => ofNat (Option \u2115) (f (a ::\u1d65 v))", "state_after": "n : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb : \u2115\n\u22a2 (\u2203 a, (a \u2208 Nat.rfind fun n_1 => Part.some (decide (1 \u2264 f (n_1 ::\u1d65 v)))) \u2227 b = pred (f (a ::\u1d65 v))) \u2194\n    \u2203 a,\n      (a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))) \u2227\n        ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b"}, {"tactic": "refine'\n  exists_congr fun a => (and_congr (iff_of_eq _) Iff.rfl).trans (and_congr_right fun h => _)", "annotated_tactic": ["refine'\n        <a>exists_congr</a> fun a => (<a>and_congr</a> (<a>iff_of_eq</a> _) <a>Iff.rfl</a>).<a>trans</a> (<a>and_congr_right</a> fun h => _)", [{"full_name": "exists_congr", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [379, 9], "def_end_pos": [379, 21]}, {"full_name": "and_congr", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [156, 9], "def_end_pos": [156, 18]}, {"full_name": "iff_of_eq", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [47, 9], "def_end_pos": [47, 18]}, {"full_name": "Iff.rfl", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [663, 19], "def_end_pos": [663, 26]}, {"full_name": "Iff.trans", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [666, 9], "def_end_pos": [666, 18]}, {"full_name": "and_congr_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [161, 9], "def_end_pos": [161, 24]}]], "state_before": "n : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb : \u2115\n\u22a2 (\u2203 a, (a \u2208 Nat.rfind fun n_1 => Part.some (decide (1 \u2264 f (n_1 ::\u1d65 v)))) \u2227 b = pred (f (a ::\u1d65 v))) \u2194\n    \u2203 a,\n      (a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))) \u2227\n        ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b", "state_after": "case refine'_1\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\n\u22a2 (a \u2208 Nat.rfind fun n_1 => Part.some (decide (1 \u2264 f (n_1 ::\u1d65 v)))) =\n    (a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v))))))\n\ncase refine'_2\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\n\u22a2 b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case refine'_1\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\n\u22a2 (a \u2208 Nat.rfind fun n_1 => Part.some (decide (1 \u2264 f (n_1 ::\u1d65 v)))) =\n    (a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v))))))", "state_after": "case refine'_1.e_a.e_p\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\n\u22a2 (fun n_1 => Part.some (decide (1 \u2264 f (n_1 ::\u1d65 v)))) = fun n_1 =>\n    \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))"}, {"tactic": "funext n", "annotated_tactic": ["funext n", []], "state_before": "case refine'_1.e_a.e_p\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\n\u22a2 (fun n_1 => Part.some (decide (1 \u2264 f (n_1 ::\u1d65 v)))) = fun n_1 =>\n    \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))", "state_after": "case refine'_1.e_a.e_p.h\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\u271d\nb a n : \u2115\n\u22a2 Part.some (decide (1 \u2264 f (n ::\u1d65 v))) = \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n ::\u1d65 v)))))"}, {"tactic": "cases f (n ::\u1d65 v) <;> simp [Nat.succ_le_succ]", "annotated_tactic": ["cases f (n ::\u1d65 v) <;> simp [<a>Nat.succ_le_succ</a>]", [{"full_name": "Nat.succ_le_succ", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1582, 9], "def_end_pos": [1582, 25]}]], "state_before": "case refine'_1.e_a.e_p.h\nn\u271d : \u2115\nf : Vector \u2115 (n\u271d + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\u271d\nb a n : \u2115\n\u22a2 Part.some (decide (1 \u2264 f (n ::\u1d65 v))) = \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n ::\u1d65 v)))))", "state_after": "case refine'_1.e_a.e_p.h.succ\nn\u271d\u00b9 : \u2115\nf : Vector \u2115 (n\u271d\u00b9 + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\u271d\u00b9\nb a n n\u271d : \u2115\n\u22a2 true = Option.isSome (ofNat (Option \u2115) (succ n\u271d))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case refine'_1.e_a.e_p.h.succ\nn\u271d\u00b9 : \u2115\nf : Vector \u2115 (n\u271d\u00b9 + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\u271d\u00b9\nb a n n\u271d : \u2115\n\u22a2 true = Option.isSome (ofNat (Option \u2115) (succ n\u271d))", "state_after": "no goals"}, {"tactic": "have := Nat.rfind_spec h", "annotated_tactic": ["have := <a>Nat.rfind_spec</a> h", [{"full_name": "Nat.rfind_spec", "def_path": "Mathlib/Computability/Partrec.lean", "def_pos": [79, 9], "def_end_pos": [79, 19]}]], "state_before": "case refine'_2\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\n\u22a2 b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b", "state_after": "case refine'_2\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nthis : true \u2208 \u2191(some (Option.isSome (ofNat (Option \u2115) (f (a ::\u1d65 v)))))\n\u22a2 b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b"}, {"tactic": "simp only [Part.coe_some, Part.mem_some_iff] at this", "annotated_tactic": ["simp only [<a>Part.coe_some</a>, <a>Part.mem_some_iff</a>] at this", [{"full_name": "Part.coe_some", "def_path": "Mathlib/Data/Part.lean", "def_pos": [361, 9], "def_end_pos": [361, 17]}, {"full_name": "Part.mem_some_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [170, 9], "def_end_pos": [170, 21]}]], "state_before": "case refine'_2\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nthis : true \u2208 \u2191(some (Option.isSome (ofNat (Option \u2115) (f (a ::\u1d65 v)))))\n\u22a2 b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b", "state_after": "case refine'_2\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nthis : true = Option.isSome (ofNat (Option \u2115) (f (a ::\u1d65 v)))\n\u22a2 b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b"}, {"tactic": "revert this", "annotated_tactic": ["revert this", []], "state_before": "case refine'_2\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nthis : true = Option.isSome (ofNat (Option \u2115) (f (a ::\u1d65 v)))\n\u22a2 b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b", "state_after": "case refine'_2\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\n\u22a2 true = Option.isSome (ofNat (Option \u2115) (f (a ::\u1d65 v))) \u2192\n    (b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b)"}, {"tactic": "cases' f (a ::\u1d65 v) with c <;> intro this", "annotated_tactic": ["cases' f (a ::\u1d65 v) with c <;> intro this", []], "state_before": "case refine'_2\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\n\u22a2 true = Option.isSome (ofNat (Option \u2115) (f (a ::\u1d65 v))) \u2192\n    (b = pred (f (a ::\u1d65 v)) \u2194 ofNat (Option \u2115) (f (a ::\u1d65 v)) = some b)", "state_after": "case refine'_2.zero\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nthis : true = Option.isSome (ofNat (Option \u2115) zero)\n\u22a2 b = pred zero \u2194 ofNat (Option \u2115) zero = some b\n\ncase refine'_2.succ\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nc : \u2115\nthis : true = Option.isSome (ofNat (Option \u2115) (succ c))\n\u22a2 b = pred (succ c) \u2194 ofNat (Option \u2115) (succ c) = some b"}, {"tactic": "rw [\u2190 Option.some_inj, eq_comm]", "annotated_tactic": ["rw [\u2190 <a>Option.some_inj</a>, <a>eq_comm</a>]", [{"full_name": "Option.some_inj", "def_path": "lake-packages/std/Std/Data/Option/Basic.lean", "def_pos": [27, 9], "def_end_pos": [27, 17]}, {"full_name": "eq_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [104, 9], "def_end_pos": [104, 16]}]], "state_before": "case refine'_2.succ\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nc : \u2115\nthis : true = Option.isSome (ofNat (Option \u2115) (succ c))\n\u22a2 b = pred (succ c) \u2194 ofNat (Option \u2115) (succ c) = some b", "state_after": "case refine'_2.succ\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nc : \u2115\nthis : true = Option.isSome (ofNat (Option \u2115) (succ c))\n\u22a2 some (pred (succ c)) = some b \u2194 ofNat (Option \u2115) (succ c) = some b"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case refine'_2.succ\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nc : \u2115\nthis : true = Option.isSome (ofNat (Option \u2115) (succ c))\n\u22a2 some (pred (succ c)) = some b \u2194 ofNat (Option \u2115) (succ c) = some b", "state_after": "no goals"}, {"tactic": "cases this", "annotated_tactic": ["cases this", []], "state_before": "case refine'_2.zero\nn : \u2115\nf : Vector \u2115 (n + 1) \u2192 \u2115\nhf : Partrec' \u2191f\nv : Vector \u2115 n\nb a : \u2115\nh : a \u2208 Nat.rfind fun n_1 => \u2191(some (Option.isSome (ofNat (Option \u2115) (f (n_1 ::\u1d65 v)))))\nthis : true = Option.isSome (ofNat (Option \u2115) zero)\n\u22a2 b = pred zero \u2194 ofNat (Option \u2115) zero = some b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Euclidean/Inversion/Calculus.lean", "full_name": "ContDiffAt.inversion", "start": [49, 18], "end": [52, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/TypeVec.lean", "full_name": "TypeVec.dropFun_appendFun", "start": [213, 1], "end": [215, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "full_name": "Rat.add_def'", "start": [188, 1], "end": [189, 35], "traced_tactics": [{"tactic": "rw [add_def, normalize_eq_mkRat]", "annotated_tactic": ["rw [<a>add_def</a>, <a>normalize_eq_mkRat</a>]", [{"full_name": "Rat.add_def", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [173, 9], "def_end_pos": [173, 16]}, {"full_name": "Rat.normalize_eq_mkRat", "def_path": "lake-packages/std/Std/Data/Rat/Lemmas.lean", "def_pos": [87, 9], "def_end_pos": [87, 27]}]], "state_before": "a b : Rat\n\u22a2 a + b = mkRat (a.num * \u2191b.den + b.num * \u2191a.den) (a.den * b.den)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.image2_vsub", "start": [602, 1], "end": [603, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/GroupPower/Lemmas.lean", "full_name": "zpow_lt_zpow", "start": [353, 1], "end": [354, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Data/Nat/Parity.lean", "full_name": "Function.Involutive.iterate_bit0", "start": [298, 1], "end": [299, 82], "traced_tactics": [{"tactic": "rw [bit0, \u2190 two_mul, iterate_mul, involutive_iff_iter_2_eq_id.1 hf, iterate_id]", "annotated_tactic": ["rw [<a>bit0</a>, \u2190 <a>two_mul</a>, <a>iterate_mul</a>, <a>involutive_iff_iter_2_eq_id</a>.1 hf, <a>iterate_id</a>]", [{"full_name": "bit0", "def_path": "Mathlib/Init/ZeroOne.lean", "def_pos": [36, 34], "def_end_pos": [36, 38]}, {"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}, {"full_name": "Function.iterate_mul", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [90, 9], "def_end_pos": [90, 20]}, {"full_name": "Function.involutive_iff_iter_2_eq_id", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [251, 9], "def_end_pos": [251, 36]}, {"full_name": "Function.iterate_id", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [71, 9], "def_end_pos": [71, 19]}]], "state_before": "\u03b1 : Type u_1\nf : \u03b1 \u2192 \u03b1\nn\u271d : \u2115\nhf : Involutive f\nn : \u2115\n\u22a2 f^[bit0 n] = id", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "full_name": "MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonpos'", "start": [787, 1], "end": [790, 7], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean", "full_name": "nndist_lineMap_right", "start": [123, 1], "end": [125, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Topology/Algebra/Module/StrongTopology.lean", "full_name": "ContinuousLinearMap.strongTopology.t2Space", "start": [128, 1], "end": [134, 55], "traced_tactics": [{"tactic": "letI : UniformSpace F := TopologicalAddGroup.toUniformSpace F", "annotated_tactic": ["letI : <a>UniformSpace</a> F := <a>TopologicalAddGroup.toUniformSpace</a> F", [{"full_name": "UniformSpace", "def_path": "Mathlib/Topology/UniformSpace/Basic.lean", "def_pos": [305, 7], "def_end_pos": [305, 19]}, {"full_name": "TopologicalAddGroup.toUniformSpace", "def_path": "Mathlib/Topology/Algebra/UniformGroup.lean", "def_pos": [547, 3], "def_end_pos": [547, 14]}]], "state_before": "\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2074 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b2 : AddCommGroup E\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2070 : AddCommGroup E'\ninst\u271d\u2079 : Module \u211d E'\ninst\u271d\u2078 : AddCommGroup F\ninst\u271d\u2077 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \u211d F'\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : TopologicalSpace E'\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : TopologicalAddGroup F\ninst\u271d : T2Space F\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16 : \u22c3\u2080 \ud835\udd16 = Set.univ\n\u22a2 T2Space (E \u2192SL[\u03c3] F)", "state_after": "\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2074 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b2 : AddCommGroup E\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2070 : AddCommGroup E'\ninst\u271d\u2079 : Module \u211d E'\ninst\u271d\u2078 : AddCommGroup F\ninst\u271d\u2077 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \u211d F'\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : TopologicalSpace E'\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : TopologicalAddGroup F\ninst\u271d : T2Space F\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16 : \u22c3\u2080 \ud835\udd16 = Set.univ\nthis : UniformSpace F := TopologicalAddGroup.toUniformSpace F\n\u22a2 T2Space (E \u2192SL[\u03c3] F)"}, {"tactic": "haveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform", "annotated_tactic": ["haveI : <a>UniformAddGroup</a> F := <a>comm_topologicalAddGroup_is_uniform</a>", [{"full_name": "UniformAddGroup", "def_path": "Mathlib/Topology/Algebra/UniformGroup.lean", "def_pos": [57, 7], "def_end_pos": [57, 22]}, {"full_name": "comm_topologicalAddGroup_is_uniform", "def_path": "Mathlib/Topology/Algebra/UniformGroup.lean", "def_pos": [676, 3], "def_end_pos": [676, 14]}]], "state_before": "\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2074 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b2 : AddCommGroup E\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2070 : AddCommGroup E'\ninst\u271d\u2079 : Module \u211d E'\ninst\u271d\u2078 : AddCommGroup F\ninst\u271d\u2077 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \u211d F'\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : TopologicalSpace E'\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : TopologicalAddGroup F\ninst\u271d : T2Space F\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16 : \u22c3\u2080 \ud835\udd16 = Set.univ\nthis : UniformSpace F := TopologicalAddGroup.toUniformSpace F\n\u22a2 T2Space (E \u2192SL[\u03c3] F)", "state_after": "\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2074 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b2 : AddCommGroup E\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2070 : AddCommGroup E'\ninst\u271d\u2079 : Module \u211d E'\ninst\u271d\u2078 : AddCommGroup F\ninst\u271d\u2077 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \u211d F'\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : TopologicalSpace E'\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : TopologicalAddGroup F\ninst\u271d : T2Space F\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16 : \u22c3\u2080 \ud835\udd16 = Set.univ\nthis\u271d : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis : UniformAddGroup F\n\u22a2 T2Space (E \u2192SL[\u03c3] F)"}, {"tactic": "letI : TopologicalSpace (E \u2192SL[\u03c3] F) := strongTopology \u03c3 F \ud835\udd16", "annotated_tactic": ["letI : <a>TopologicalSpace</a> (E \u2192SL[\u03c3] F) := <a>strongTopology</a> \u03c3 F \ud835\udd16", [{"full_name": "TopologicalSpace", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [70, 7], "def_end_pos": [70, 23]}, {"full_name": "ContinuousLinearMap.strongTopology", "def_path": "Mathlib/Topology/Algebra/Module/StrongTopology.lean", "def_pos": [75, 5], "def_end_pos": [75, 19]}]], "state_before": "\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2074 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b2 : AddCommGroup E\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2070 : AddCommGroup E'\ninst\u271d\u2079 : Module \u211d E'\ninst\u271d\u2078 : AddCommGroup F\ninst\u271d\u2077 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \u211d F'\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : TopologicalSpace E'\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : TopologicalAddGroup F\ninst\u271d : T2Space F\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16 : \u22c3\u2080 \ud835\udd16 = Set.univ\nthis\u271d : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis : UniformAddGroup F\n\u22a2 T2Space (E \u2192SL[\u03c3] F)", "state_after": "\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2074 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b2 : AddCommGroup E\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2070 : AddCommGroup E'\ninst\u271d\u2079 : Module \u211d E'\ninst\u271d\u2078 : AddCommGroup F\ninst\u271d\u2077 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \u211d F'\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : TopologicalSpace E'\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : TopologicalAddGroup F\ninst\u271d : T2Space F\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16 : \u22c3\u2080 \ud835\udd16 = Set.univ\nthis\u271d\u00b9 : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis\u271d : UniformAddGroup F\nthis : TopologicalSpace (E \u2192SL[\u03c3] F) := strongTopology \u03c3 F \ud835\udd16\n\u22a2 T2Space (E \u2192SL[\u03c3] F)"}, {"tactic": "haveI : T2Space (E \u2192\u1d64[\ud835\udd16] F) := UniformOnFun.t2Space_of_covering h\ud835\udd16", "annotated_tactic": ["haveI : <a>T2Space</a> (E \u2192\u1d64[\ud835\udd16] F) := <a>UniformOnFun.t2Space_of_covering</a> h\ud835\udd16", [{"full_name": "T2Space", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [900, 7], "def_end_pos": [900, 14]}, {"full_name": "UniformOnFun.t2Space_of_covering", "def_path": "Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean", "def_pos": [843, 9], "def_end_pos": [843, 28]}]], "state_before": "\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2074 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b2 : AddCommGroup E\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2070 : AddCommGroup E'\ninst\u271d\u2079 : Module \u211d E'\ninst\u271d\u2078 : AddCommGroup F\ninst\u271d\u2077 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \u211d F'\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : TopologicalSpace E'\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : TopologicalAddGroup F\ninst\u271d : T2Space F\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16 : \u22c3\u2080 \ud835\udd16 = Set.univ\nthis\u271d\u00b9 : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis\u271d : UniformAddGroup F\nthis : TopologicalSpace (E \u2192SL[\u03c3] F) := strongTopology \u03c3 F \ud835\udd16\n\u22a2 T2Space (E \u2192SL[\u03c3] F)", "state_after": "\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2074 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b2 : AddCommGroup E\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2070 : AddCommGroup E'\ninst\u271d\u2079 : Module \u211d E'\ninst\u271d\u2078 : AddCommGroup F\ninst\u271d\u2077 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \u211d F'\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : TopologicalSpace E'\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : TopologicalAddGroup F\ninst\u271d : T2Space F\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16 : \u22c3\u2080 \ud835\udd16 = Set.univ\nthis\u271d\u00b2 : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis\u271d\u00b9 : UniformAddGroup F\nthis\u271d : TopologicalSpace (E \u2192SL[\u03c3] F) := strongTopology \u03c3 F \ud835\udd16\nthis : T2Space (E \u2192\u1d64[\ud835\udd16] F)\n\u22a2 T2Space (E \u2192SL[\u03c3] F)"}, {"tactic": "exact (strongTopology.embedding_coeFn \u03c3 F \ud835\udd16).t2Space", "annotated_tactic": ["exact (<a>strongTopology.embedding_coeFn</a> \u03c3 F \ud835\udd16).<a>t2Space</a>", [{"full_name": "ContinuousLinearMap.strongTopology.embedding_coeFn", "def_path": "Mathlib/Topology/Algebra/Module/StrongTopology.lean", "def_pos": [104, 9], "def_end_pos": [104, 39]}, {"full_name": "Embedding.t2Space", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [1183, 9], "def_end_pos": [1183, 26]}]], "state_before": "\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2074 : NormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b3 : NormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nE : Type u_3\nE' : Type u_4\nF : Type u_5\nF' : Type u_6\ninst\u271d\u00b9\u00b2 : AddCommGroup E\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c\u2081 E\ninst\u271d\u00b9\u2070 : AddCommGroup E'\ninst\u271d\u2079 : Module \u211d E'\ninst\u271d\u2078 : AddCommGroup F\ninst\u271d\u2077 : Module \ud835\udd5c\u2082 F\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \u211d F'\ninst\u271d\u2074 : TopologicalSpace E\ninst\u271d\u00b3 : TopologicalSpace E'\ninst\u271d\u00b2 : TopologicalSpace F\ninst\u271d\u00b9 : TopologicalAddGroup F\ninst\u271d : T2Space F\n\ud835\udd16 : Set (Set E)\nh\ud835\udd16 : \u22c3\u2080 \ud835\udd16 = Set.univ\nthis\u271d\u00b2 : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis\u271d\u00b9 : UniformAddGroup F\nthis\u271d : TopologicalSpace (E \u2192SL[\u03c3] F) := strongTopology \u03c3 F \ud835\udd16\nthis : T2Space (E \u2192\u1d64[\ud835\udd16] F)\n\u22a2 T2Space (E \u2192SL[\u03c3] F)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "full_name": "PowerSeries.order_eq_multiplicity_X", "start": [2547, 1], "end": [2567, 31], "traced_tactics": [{"tactic": "rcases eq_or_ne \u03c6 0 with (rfl | h\u03c6)", "annotated_tactic": ["rcases <a>eq_or_ne</a> \u03c6 0 with (rfl | h\u03c6)", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 17]}]], "state_before": "R\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\n\u22a2 order \u03c6 = multiplicity X \u03c6", "state_after": "case inl\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6 : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u22a2 order 0 = multiplicity X 0\n\ncase inr\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\n\u22a2 order \u03c6 = multiplicity X \u03c6"}, {"tactic": "induction' ho : order \u03c6 using PartENat.casesOn with n", "annotated_tactic": ["induction' ho : <a>order</a> \u03c6 using <a>PartENat.casesOn</a> with n", [{"full_name": "PowerSeries.order", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [2317, 5], "def_end_pos": [2317, 10]}, {"full_name": "PartENat.casesOn", "def_path": "Mathlib/Data/Nat/PartENat.lean", "def_pos": [148, 19], "def_end_pos": [148, 26]}]], "state_before": "case inr\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\n\u22a2 order \u03c6 = multiplicity X \u03c6", "state_after": "case inr.a\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nho : order \u03c6 = \u22a4\n\u22a2 \u22a4 = multiplicity X \u03c6\n\ncase inr.a\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\n\u22a2 \u2191n = multiplicity X \u03c6"}, {"tactic": "have hn : \u03c6.order.get (order_finite_iff_ne_zero.mpr h\u03c6) = n := by simp [ho]", "annotated_tactic": ["have hn : \u03c6.order.get (order_finite_iff_ne_zero.mpr h\u03c6) = n := by simp [ho]", []], "state_before": "case inr.a\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\n\u22a2 \u2191n = multiplicity X \u03c6", "state_after": "case inr.a\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\nhn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n\n\u22a2 \u2191n = multiplicity X \u03c6"}, {"tactic": "rw [\u2190 hn]", "annotated_tactic": ["rw [\u2190 hn]", []], "state_before": "case inr.a\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\nhn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n\n\u22a2 \u2191n = multiplicity X \u03c6", "state_after": "case inr.a\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\nhn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n\n\u22a2 \u2191(Part.get (order \u03c6) (_ : (order \u03c6).Dom)) = multiplicity X \u03c6"}, {"tactic": "refine'\n  le_antisymm (le_multiplicity_of_pow_dvd <| X_pow_order_dvd (order_finite_iff_ne_zero.mpr h\u03c6))\n    (PartENat.find_le _ _ _)", "annotated_tactic": ["refine'\n    <a>le_antisymm</a> (<a>le_multiplicity_of_pow_dvd</a> <| <a>X_pow_order_dvd</a> (order_finite_iff_ne_zero.mpr h\u03c6))\n      (<a>PartENat.find_le</a> _ _ _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "multiplicity.le_multiplicity_of_pow_dvd", "def_path": "Mathlib/RingTheory/Multiplicity.lean", "def_pos": [141, 9], "def_end_pos": [141, 35]}, {"full_name": "PowerSeries.X_pow_order_dvd", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [2534, 9], "def_end_pos": [2534, 24]}, {"full_name": "PartENat.find_le", "def_path": "Mathlib/Data/Nat/PartENat.lean", "def_pos": [808, 9], "def_end_pos": [808, 16]}]], "state_before": "case inr.a\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\nhn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n\n\u22a2 \u2191(Part.get (order \u03c6) (_ : (order \u03c6).Dom)) = multiplicity X \u03c6", "state_after": "case inr.a\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\nhn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n\n\u22a2 \u00acX ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) \u2223 \u03c6"}, {"tactic": "rintro \u27e8\u03c8, H\u27e9", "annotated_tactic": ["rintro \u27e8\u03c8, H\u27e9", []], "state_before": "case inr.a\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\nhn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n\n\u22a2 \u00acX ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) \u2223 \u03c6", "state_after": "case inr.a.intro\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\nhn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n\n\u03c8 : R\u27e6X\u27e7\nH : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8\n\u22a2 False"}, {"tactic": "have := congr_arg (coeff R n) H", "annotated_tactic": ["have := <a>congr_arg</a> (<a>coeff</a> R n) H", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "PowerSeries.coeff", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [1348, 5], "def_end_pos": [1348, 10]}]], "state_before": "case inr.a.intro\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\nhn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n\n\u03c8 : R\u27e6X\u27e7\nH : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8\n\u22a2 False", "state_after": "case inr.a.intro\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\nhn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n\n\u03c8 : R\u27e6X\u27e7\nH : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8\nthis : \u2191(coeff R n) \u03c6 = \u2191(coeff R n) (X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8)\n\u22a2 False"}, {"tactic": "rw [\u2190 (\u03c8.commute_X.pow_right _).eq, coeff_mul_of_lt_order, \u2190 hn] at this", "annotated_tactic": ["rw [\u2190 (\u03c8.commute_X.pow_right _).<a>eq</a>, <a>coeff_mul_of_lt_order</a>, \u2190 hn] at this", [{"full_name": "Commute.eq", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [47, 19], "def_end_pos": [47, 21]}, {"full_name": "PowerSeries.coeff_mul_of_lt_order", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [2503, 9], "def_end_pos": [2503, 30]}]], "state_before": "case inr.a.intro\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\nhn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n\n\u03c8 : R\u27e6X\u27e7\nH : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8\nthis : \u2191(coeff R n) \u03c6 = \u2191(coeff R n) (X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8)\n\u22a2 False", "state_after": "case inr.a.intro\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\nhn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n\n\u03c8 : R\u27e6X\u27e7\nH : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8\nthis : \u2191(coeff R (Part.get (order \u03c6) (_ : (order \u03c6).Dom))) \u03c6 = 0\n\u22a2 False\n\ncase inr.a.intro\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\nhn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n\n\u03c8 : R\u27e6X\u27e7\nH : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8\nthis : \u2191(coeff R n) \u03c6 = \u2191(coeff R n) (\u03c8 * X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1))\n\u22a2 \u2191n < order (X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1))"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case inl\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6 : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u22a2 order 0 = multiplicity X 0", "state_after": "no goals"}, {"tactic": "simp [h\u03c6] at ho", "annotated_tactic": ["simp [h\u03c6] at ho", []], "state_before": "case inr.a\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nho : order \u03c6 = \u22a4\n\u22a2 \u22a4 = multiplicity X \u03c6", "state_after": "no goals"}, {"tactic": "simp [ho]", "annotated_tactic": ["simp [ho]", []], "state_before": "R\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\n\u22a2 Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n", "state_after": "no goals"}, {"tactic": "exact coeff_order _ this", "annotated_tactic": ["exact <a>coeff_order</a> _ this", [{"full_name": "PowerSeries.coeff_order", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [2343, 9], "def_end_pos": [2343, 20]}]], "state_before": "case inr.a.intro\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\nhn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n\n\u03c8 : R\u27e6X\u27e7\nH : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8\nthis : \u2191(coeff R (Part.get (order \u03c6) (_ : (order \u03c6).Dom))) \u03c6 = 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "rw [X_pow_eq, order_monomial]", "annotated_tactic": ["rw [<a>X_pow_eq</a>, <a>order_monomial</a>]", [{"full_name": "PowerSeries.X_pow_eq", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [1493, 9], "def_end_pos": [1493, 17]}, {"full_name": "PowerSeries.order_monomial", "def_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "def_pos": [2482, 9], "def_end_pos": [2482, 23]}]], "state_before": "case inr.a.intro\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\nhn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n\n\u03c8 : R\u27e6X\u27e7\nH : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8\nthis : \u2191(coeff R n) \u03c6 = \u2191(coeff R n) (\u03c8 * X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1))\n\u22a2 \u2191n < order (X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1))", "state_after": "case inr.a.intro\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\nhn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n\n\u03c8 : R\u27e6X\u27e7\nH : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8\nthis : \u2191(coeff R n) \u03c6 = \u2191(coeff R n) (\u03c8 * X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1))\n\u22a2 \u2191n < if 1 = 0 then \u22a4 else \u2191(Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1)"}, {"tactic": "split_ifs", "annotated_tactic": ["split_ifs", []], "state_before": "case inr.a.intro\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\nhn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n\n\u03c8 : R\u27e6X\u27e7\nH : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8\nthis : \u2191(coeff R n) \u03c6 = \u2191(coeff R n) (\u03c8 * X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1))\n\u22a2 \u2191n < if 1 = 0 then \u22a4 else \u2191(Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1)", "state_after": "case pos\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\nhn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n\n\u03c8 : R\u27e6X\u27e7\nH : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8\nthis : \u2191(coeff R n) \u03c6 = \u2191(coeff R n) (\u03c8 * X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1))\nh\u271d : 1 = 0\n\u22a2 \u2191n < \u22a4\n\ncase neg\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\nhn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n\n\u03c8 : R\u27e6X\u27e7\nH : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8\nthis : \u2191(coeff R n) \u03c6 = \u2191(coeff R n) (\u03c8 * X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1))\nh\u271d : \u00ac1 = 0\n\u22a2 \u2191n < \u2191(Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1)"}, {"tactic": "exact PartENat.natCast_lt_top _", "annotated_tactic": ["exact <a>PartENat.natCast_lt_top</a> _", [{"full_name": "PartENat.natCast_lt_top", "def_path": "Mathlib/Data/Nat/PartENat.lean", "def_pos": [343, 9], "def_end_pos": [343, 23]}]], "state_before": "case pos\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\nhn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n\n\u03c8 : R\u27e6X\u27e7\nH : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8\nthis : \u2191(coeff R n) \u03c6 = \u2191(coeff R n) (\u03c8 * X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1))\nh\u271d : 1 = 0\n\u22a2 \u2191n < \u22a4", "state_after": "no goals"}, {"tactic": "rw [\u2190 hn, PartENat.coe_lt_coe]", "annotated_tactic": ["rw [\u2190 hn, <a>PartENat.coe_lt_coe</a>]", [{"full_name": "PartENat.coe_lt_coe", "def_path": "Mathlib/Data/Nat/PartENat.lean", "def_pos": [271, 9], "def_end_pos": [271, 19]}]], "state_before": "case neg\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\nhn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n\n\u03c8 : R\u27e6X\u27e7\nH : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8\nthis : \u2191(coeff R n) \u03c6 = \u2191(coeff R n) (\u03c8 * X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1))\nh\u271d : \u00ac1 = 0\n\u22a2 \u2191n < \u2191(Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1)", "state_after": "case neg\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\nhn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n\n\u03c8 : R\u27e6X\u27e7\nH : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8\nthis : \u2191(coeff R n) \u03c6 = \u2191(coeff R n) (\u03c8 * X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1))\nh\u271d : \u00ac1 = 0\n\u22a2 Part.get (order \u03c6) (_ : (order \u03c6).Dom) < Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1"}, {"tactic": "exact Nat.lt_succ_self _", "annotated_tactic": ["exact <a>Nat.lt_succ_self</a> _", [{"full_name": "Nat.lt_succ_self", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [294, 9], "def_end_pos": [294, 21]}]], "state_before": "case neg\nR\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\n\u03c6\u271d : R\u271d\u27e6X\u27e7\nR : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\n\u03c6 : R\u27e6X\u27e7\nh\u03c6 : \u03c6 \u2260 0\nx\u271d : PartENat\nho\u271d : order \u03c6 = x\u271d\nn : \u2115\nho : order \u03c6 = \u2191n\nhn : Part.get (order \u03c6) (_ : (order \u03c6).Dom) = n\n\u03c8 : R\u27e6X\u27e7\nH : \u03c6 = X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1) * \u03c8\nthis : \u2191(coeff R n) \u03c6 = \u2191(coeff R n) (\u03c8 * X ^ (Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1))\nh\u271d : \u00ac1 = 0\n\u22a2 Part.get (order \u03c6) (_ : (order \u03c6).Dom) < Part.get (order \u03c6) (_ : (order \u03c6).Dom) + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/CategoryTheory/Limits/FunctorCategory.lean", "full_name": "CategoryTheory.Limits.limit_map_limitObjIsoLimitCompEvaluation_hom", "start": [230, 1], "end": [235, 7], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "C : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasLimitsOfShape J C\ni j : K\nF : J \u2964 K \u2964 C\nf : i \u27f6 j\n\u22a2 (limit F).map f \u226b (limitObjIsoLimitCompEvaluation F j).hom =\n    (limitObjIsoLimitCompEvaluation F i).hom \u226b limMap (whiskerLeft F ((evaluation K C).map f))", "state_after": "case w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasLimitsOfShape J C\ni j : K\nF : J \u2964 K \u2964 C\nf : i \u27f6 j\nj\u271d : J\n\u22a2 ((limit F).map f \u226b (limitObjIsoLimitCompEvaluation F j).hom) \u226b limit.\u03c0 (F \u22d9 (evaluation K C).obj j) j\u271d =\n    ((limitObjIsoLimitCompEvaluation F i).hom \u226b limMap (whiskerLeft F ((evaluation K C).map f))) \u226b\n      limit.\u03c0 (F \u22d9 (evaluation K C).obj j) j\u271d"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasLimitsOfShape J C\ni j : K\nF : J \u2964 K \u2964 C\nf : i \u27f6 j\nj\u271d : J\n\u22a2 ((limit F).map f \u226b (limitObjIsoLimitCompEvaluation F j).hom) \u226b limit.\u03c0 (F \u22d9 (evaluation K C).obj j) j\u271d =\n    ((limitObjIsoLimitCompEvaluation F i).hom \u226b limMap (whiskerLeft F ((evaluation K C).map f))) \u226b\n      limit.\u03c0 (F \u22d9 (evaluation K C).obj j) j\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/RepresentationTheory/Action.lean", "full_name": "Action.sum_hom", "start": [415, 1], "end": [417, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Algebra/Quandle.lean", "full_name": "Rack.op_act_op_eq", "start": [274, 1], "end": [275, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover/std4", "commit": "869c615eb10130c0637a7bc038e2b80253559913", "file_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "full_name": "Nat.mod_add_mod", "start": [774, 9], "end": [776, 48], "traced_tactics": [{"tactic": "have := (add_mul_mod_self_left (m % n + k) n (m / n)).symm", "annotated_tactic": ["have := (<a>add_mul_mod_self_left</a> (m % n + k) n (m / n)).<a>symm</a>", [{"full_name": "Nat.add_mul_mod_self_left", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [716, 17], "def_end_pos": [716, 38]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "m n k : Nat\n\u22a2 (m % n + k) % n = (m + k) % n", "state_after": "m n k : Nat\nthis : (m % n + k) % n = (m % n + k + n * (m / n)) % n\n\u22a2 (m % n + k) % n = (m + k) % n"}, {"tactic": "rwa [Nat.add_right_comm, mod_add_div] at this", "annotated_tactic": ["rwa [<a>Nat.add_right_comm</a>, <a>mod_add_div</a>] at this", [{"full_name": "Nat.add_right_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [145, 19], "def_end_pos": [145, 33]}, {"full_name": "Nat.mod_add_div", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [550, 9], "def_end_pos": [550, 20]}]], "state_before": "m n k : Nat\nthis : (m % n + k) % n = (m % n + k + n * (m / n)) % n\n\u22a2 (m % n + k) % n = (m + k) % n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "3ce43c18f614b76e161f911b75a3e1ef641620ff", "file_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean", "full_name": "InnerProductGeometry.norm_sub_eq_abs_sub_norm_iff_angle_eq_zero", "start": [320, 1], "end": [330, 29], "traced_tactics": [{"tactic": "refine' \u27e8fun h => _, norm_sub_eq_abs_sub_norm_of_angle_eq_zero\u27e9", "annotated_tactic": ["refine' \u27e8fun h => _, <a>norm_sub_eq_abs_sub_norm_of_angle_eq_zero</a>\u27e9", [{"full_name": "InnerProductGeometry.norm_sub_eq_abs_sub_norm_of_angle_eq_zero", "def_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean", "def_pos": [285, 9], "def_end_pos": [285, 50]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016| \u2194 angle x y = 0", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016|\n\u22a2 angle x y = 0"}, {"tactic": "rw [\u2190 inner_eq_mul_norm_iff_angle_eq_zero hx hy]", "annotated_tactic": ["rw [\u2190 <a>inner_eq_mul_norm_iff_angle_eq_zero</a> hx hy]", [{"full_name": "InnerProductGeometry.inner_eq_mul_norm_iff_angle_eq_zero", "def_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean", "def_pos": [258, 9], "def_end_pos": [258, 44]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016|\n\u22a2 angle x y = 0", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016|\n\u22a2 inner x y = \u2016x\u2016 * \u2016y\u2016"}, {"tactic": "have h1 : \u2016x - y\u2016 ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2 := by\n  rw [h]\n  exact sq_abs (\u2016x\u2016 - \u2016y\u2016)", "annotated_tactic": ["have h1 : \u2016x - y\u2016 ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2 := by\n    rw [h]\n    exact <a>sq_abs</a> (\u2016x\u2016 - \u2016y\u2016)", [{"full_name": "sq_abs", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [680, 9], "def_end_pos": [680, 15]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016|\n\u22a2 inner x y = \u2016x\u2016 * \u2016y\u2016", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016|\nh1 : \u2016x - y\u2016 ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2\n\u22a2 inner x y = \u2016x\u2016 * \u2016y\u2016"}, {"tactic": "rw [norm_sub_pow_two_real] at h1", "annotated_tactic": ["rw [<a>norm_sub_pow_two_real</a>] at h1", [{"full_name": "norm_sub_pow_two_real", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [1072, 7], "def_end_pos": [1072, 28]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016|\nh1 : \u2016x - y\u2016 ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2\n\u22a2 inner x y = \u2016x\u2016 * \u2016y\u2016", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016|\nh1\u271d : \u2016x - y\u2016 ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2\nh1 : \u2016x\u2016 ^ 2 - 2 * inner x y + \u2016y\u2016 ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2\n\u22a2 inner x y = \u2016x\u2016 * \u2016y\u2016"}, {"tactic": "calc\n  \u27eax, y\u27eb = ((\u2016x\u2016 + \u2016y\u2016) ^ 2 - \u2016x\u2016 ^ 2 - \u2016y\u2016 ^ 2) / 2 := by linarith\n  _ = \u2016x\u2016 * \u2016y\u2016 := by ring", "annotated_tactic": ["calc\n    \u27eax, y\u27eb = ((\u2016x\u2016 + \u2016y\u2016) ^ 2 - \u2016x\u2016 ^ 2 - \u2016y\u2016 ^ 2) / 2 := by linarith\n    _ = \u2016x\u2016 * \u2016y\u2016 := by ring", []], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016|\nh1\u271d : \u2016x - y\u2016 ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2\nh1 : \u2016x\u2016 ^ 2 - 2 * inner x y + \u2016y\u2016 ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2\n\u22a2 inner x y = \u2016x\u2016 * \u2016y\u2016", "state_after": "no goals"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016|\n\u22a2 \u2016x - y\u2016 ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016|\n\u22a2 |\u2016x\u2016 - \u2016y\u2016| ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2"}, {"tactic": "exact sq_abs (\u2016x\u2016 - \u2016y\u2016)", "annotated_tactic": ["exact <a>sq_abs</a> (\u2016x\u2016 - \u2016y\u2016)", [{"full_name": "sq_abs", "def_path": "Mathlib/Algebra/GroupPower/Order.lean", "def_pos": [680, 9], "def_end_pos": [680, 15]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016|\n\u22a2 |\u2016x\u2016 - \u2016y\u2016| ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016|\nh1\u271d : \u2016x - y\u2016 ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2\nh1 : \u2016x\u2016 ^ 2 - 2 * inner x y + \u2016y\u2016 ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2\n\u22a2 inner x y = ((\u2016x\u2016 + \u2016y\u2016) ^ 2 - \u2016x\u2016 ^ 2 - \u2016y\u2016 ^ 2) / 2", "state_after": "no goals"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d x y : V\nhx : x \u2260 0\nhy : y \u2260 0\nh : \u2016x - y\u2016 = |\u2016x\u2016 - \u2016y\u2016|\nh1\u271d : \u2016x - y\u2016 ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2\nh1 : \u2016x\u2016 ^ 2 - 2 * inner x y + \u2016y\u2016 ^ 2 = (\u2016x\u2016 - \u2016y\u2016) ^ 2\n\u22a2 ((\u2016x\u2016 + \u2016y\u2016) ^ 2 - \u2016x\u2016 ^ 2 - \u2016y\u2016 ^ 2) / 2 = \u2016x\u2016 * \u2016y\u2016", "state_after": "no goals"}]}]
\ No newline at end of file